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s : Homs(A; E) ----- E given by ¢ !----> ¢(I). The inverse of cI> R is given by cI>RI (e) : a!----> ea. We thus find for homogeneous ¢ and a the relation
(L(¢). a = ((III¢}' a = (-I)(¢la)((all¢)) = ('r¢)(a) . We conclude that cI>RI 0
¢* is even and linear, as is the map HomL(E; F) ----. HomR(*F; *E), ¢ f-> *¢. (iv) \:j¢ E HomL(E; F) : 'r(('r¢)*) = 'ro*¢o'r-I. Proof Properties (i), (ii), and (iii) are elementary. For (iv) one has to realize which transposition operators are involved. In 'ro *¢ 0 'I-Ion the right hand side they represent the sequence F* ----. *F ----. *E ----. E*. In 'r( ('r¢ )*) on the left hand side the first one represents the switch HomL(F*; E*) ----. HomR(F*; E*) and the second one the switch HomdE; F) ----. HomR(E; F). Once one has this, the proof is elementary. IqEDI
2.22 Remark. In [2.21-iiJ we see again the advantage of the notation 0 for composition of left linear maps: we do not have to change the order of ¢ and 'ljJ in these formula;.
3.
DIRECT SUMS, FREE Qt-GRADED A-MODULES, AND QUOTIENTS
In the previous sections we have seen the construction ofthe Qi-graded A-modules submodule and morphisms; in this section we provide three new constructions of Qi-graded A-modules. In the first place the free Qi-graded A-module F( G, c) on a set G of homogeneous generators whose parity is given by c. The next construction is the direct sum of a family of Qi-graded A-modules. The third construction is that of the quotient ofan Qi-graded A-module by an Qi-waded sub module.
3.1 Construction (direct sums). If E i , i E I is a collection of Qi-graded A-modules, we define their direct sum E = EBiEI Ei as the subset of the direct product DiEI Ei
§3. Direct sums, free 2t-graded A-modules, aud quotients
15
consisting of those vectors (ei)iEI with ei = a except for finitely many indices i E I (recall that the direct sum of real vector spaces is defined exactly in this way). By defining a componentwise addition and (left) multiplication by elements of A, E becomes a left A-module. Finally we define the subsets En, a E Qt by En = En TIiEI(Ei)n. We leave it to the reader to verify that with these definitions E becomes an Qt-graded A-module. For each i E I we define maps 7ri : E ---- Ei and si : Ei ---- E by 7ri( (ej )jEr) = ei and (si(e))i = e, (si(e))j = a for j -I=- i. It follows immediately thatthe 7ri are sUljective even linear maps and that the Si are injective even linear maps, related by 7ri 0 Si = id(E i ). We will usually denote a general element (ei)iEI E E by EBiElei instead of by (ei)iEI, just to stress that it is not an arbitrary element of the direct product, but one with only finitely many non-zero entries. In case the index set I has a finite number n of elements, we will write El EB ... EB En for E = EB~=1 E i , and an arbitrary element will be denoted by el EB ... EB en' If the A-vector spaces Ei are all equal to a given one, Vi : Ei = P, the direct sum EB~=1 Ei is also denoted as pk. It is indeed the k-th power of F because for a finite index set I the direct sum equals the direct product. And if we define formally pO = {a}, then the equality p k EB pi = p k+i holds for all k, f!. E N.
3.2 Remark. One might ask why we do not define direct products of Qt-graded A-modules. There are several reasons. In the first place, ifboth the index set I and the abelian group Qt are infinite, one can easily find examples in which the direct product is not an Qt-graded A-module, the failure being that not every element can be written as a finite sum of homogeneous elements. In the second place, we never need infinite direct products. And in the third place, a direct product of finitely many Qt-graded A-modules is the same as the direct sum of these spaces.
3.3 Definition. If Pi, i E I is a family of Qt-graded submodules of a given Qt-graded A-module E, we can consider the map EBi Pi ---- E defined by (fi)iEI I--> I:i Ii- Note that this map is well defined because there are only finitely many f/s non-zero; its image is I:iEI Pi' One easily verifies that this map is even and linear by definition of Qt-graded submodules. Officially EBi Pi is never a submodule of E (but I:iEI Pi is); nevertheless, we will write E = EBi Pi whenever this map is an isomorphism onto E. As for real vector spaces, this is the case if and only if every element e E E can be written in a unique way as e = I:iEI fi with fi E Pi and only finitely many of them non-zero. If I contains two elements, we will write E = PI EB P2 . The Qt-graded submodules PI and P2 will be called supplements to each other.
3.4 Construction (free Qt-graded A-modules). Let c : G ____ Qt be a map from an abstract set G to Qt, and define G n C Gby G n = c 1 (a). We define the space P(G,c) as the set of all maps f : G ---- A with the property that f (g) = a for all 9 E G except finitely many. In P(G,c) we define an addition by (f + I')(g) = f(g) + I'(g), and a (left) multiplication by elements of A by (aJ)(g) = af(g). In this way P(G, c) becomes a left
Chapter 1. 2t-graded commutative linear algebra
16
A-module. One usually identifies each element 9 E G with the map ¢g : G -+ A defined by ¢g(g) = 1 and ¢g(h) = 0 for h -=I- g. It follows that each 1 E F(G, e) can be written in a unique way as 1 = I:9Ec Ig¢g == I:gEc f9 . 9 where Ig is defined as Ig = I(g) and where the sum is actually a finite sum by definition of F( G, e). To make F( G, e) into an Qt-graded A-module, we define F(G, e)a, (Y E Qt by
F(G,e)a = {I
E
F(G,e) I \:/g
E
G: I(g)
E
Aa-c(g)}.
1 = I:9Ec Ig . 9
has parity (Y if and only if the coefficient f9 has parity (Y e(g). In particular the element (map) ¢g has parity e(¢g) = e(g), justifying the use of the symbol e for the abstract map e : G -+ Qt. Decomposing the coefficients of an arbitrary element 1 E F(G, e) into homogeneous parts, it follows immediately that F(G,e) = EBaE21 F(G,e)a; since by construction AaF(G,e)(3 is contained in F(G, e)a+(3, we conclude that F(G, e) is an Qt-graded A-module. The Qt-graded A-module F(G, e) is usually called the free Qt-graded A-module on (homogeneous) generators G with parity e. Using the notion of Span, we can summarize the construction of F(G,e) by saying F(G,e) = Span(G). Using the notion of direct sums, we can write F(G, e) = EB9Ec F( {g}, eg), where F( {g}, eg) is the free Qt-graded A-module on the single generator 9 of parity eg = e(g). In words,
3.5 Nota Bene. We have seen that each element 1 E F(G,e) admits a unique decomposition 1 = I:9Ec Ig . 9 with Ig E A. Using the induced right action of A, it follows that there also exists a unique representation with the coefficients on the right of the ¢g, i.e., 1 = I:gEc 9 . I'g. In general the coefficients Ig and I'g are different; only if e(g) = 0 can we be sure that Ig = I'g. For any free Qt-graded A-module on a single homogeneous generator F({g},eg) we can define the map ¢ : A -+ F( {g}, eg) by a !----> ago This is a bijective linear map ofparity eg. It is an isomorphism if and only if eg = O. It follows that for eg -=I- 0 we cannot identify (in the naive and official sense of the word) the Qt-graded A-module F({g},eg) with A because left and right multiplication in the Qt-graded A-modules A and F( {g}, eg) are not related in the same way due to the difference in parity between 1 E A and 9 E F ( {g}, e g).
3.6 Corollary. Let F(G i , ei), i E I be afamily offree Qt-graded A-modules on generators G i (i -=I- j =? Gi n G j = ¢). Then EBiEI F( G i , ei) 9'! F(UiEIG i , e), where e is defined as elC i = ei.
3.7 Construction (quotients). Let E be an Qt-graded A-module and let F be an Qt-graded submodule. The quotient G = E / F with canonical projection 7r : E -+ G is defined in the sense of abelian groups, i.e., 7r(e) = 7r(e') {:} e - e' E F. As for abelian groups, the element 7r( e) E G will also be denoted as e mod F. We claim that G can be equipped with the structure of an Qt-graded A-module. Addition and (left) multiplication
§4. Tensor products
17
by elements in A are defined by 7r( e) + 7r( e') = 7r( e + e'), a7r( e) = 7r( ae). The subgroups en are defined by 7r(e) E en {:} 31 E F : e - 1 E En. It follows immediately that with this grading 7r is an even morphism. The only tricky point in proving that G is an S2t-graded A-module, is in the proof that the decomposition in homogeneous components is unique. Therefore, let us suppose I:nE217r(e n ) = owith 7r(e n ) E en (and of course only finitely many of them non-zero, which implies I:nE21 e a E F). By definition of en and the projection 7r we may assume that en E En. Since these en are homogeneous and F is an S2t-graded submodule, we have by [1.14] that en belongs to F, i.e., 7r( en) = O. This proves that the decomposition into homogeneous components is unique.
3.8 Lemma. Let E and H be S2t-graded A-modules, Fan S2t-graded submodule of E and c/J : E ---4 H a linear map that vanishes on F, i.e., c/J( F) = {O}. Then there exists a unique induced map
: H ---., H such that X = 0 X. Moreover, since X and X are even, it follows that cI> and ill also are even. Now denote W = 0 ~ : H ---., H to obtain the equality X = W0 X. Ifwe view the X on the left of this equation as arbitrary and apply the property with the pair (H, X), we see that W is the unique linear map given by [4.3]. Since the identity is also a solution, it follows that W = 0 ill = id(H). In exactly the same way one proves that ~ 0 cI> = id(H). We thus conclude that cI> and ~ are isomorphisms between Hand H. This finishes the proof IqEDI (modulo some small details that are left to the reader).
Chapter 1.
20
2t-graded
commutative linear algebra
4.5 Discussion. The tensor product symbol @ is not only used to denote the S2t-graded A-module E @ F, it is also used to replace the map X : Ex F -+ E @ F in the following way: e @ j == X( e, f) .
There is no symbol, not on the left nor on the right, to tell us that it is either right or left bilinear. But this is justified because the map X is even, and thus is left and right bilinear at the same time. Since F(G, c:) in [4.1] is generated by the elements ¢(e,J) with e and j homogeneous, it follows that the elements e @ j generate E @ F (but note that due to the quotient, a decomposition as linear combination of this kind of elements is not necessarily unique). It follows immediately that any linear map on E @ F is completely determined by its values on elements of the form e @ j with e and j homogeneous. Finally note that for homogeneous e E E and j E F we have c:(e @ f) = c:(e) + c:(f).
4.6 Lemma. Given three S2t-graded A-modules E, F and G, there exists a canonical
isomorphism (identification) between (E@F) @G and E@ (F@G) mapping (e@f) @g toe@(f@g). Proof Let us first distinguish the various maps X (or @) that intervene in the construction of these spaces: Xl ,2 : E x F -+ E @ F, Xl2 ,3 : (E @ F) x G -+ (E @ F) @ G, X2,3 : F x G -+ F @ G, and Xl ,23 : E x (F @ G) -+ E @ (F @ G). Next consider the trilinear map ¢ E Map(E, F, G; E @ (F @ G))o defined by
¢(e,j,g) = Xl ,23(e,X 2,3(f,g)). For a fixed element 9 E G we define the map ¢g E MapdE, F; E @ (F @ G)) by ¢g (e, f) = ¢( e, j, g) (Nota Bene. In general ¢g is not right bilinear). Thus there exists a unique induced linear map ¢g E MapdE@F;E @ (F@G)) such that ¢g = ¢g 0 Xl ,2' Defining ¢( h, g) = ¢g(h), one can easily show that ¢ E MaPL(E@F, G; E@(F@G)) is even. Thus there exists a unique induced even linear map cI> : (E@F) @G -+ E@ (F@G) with the property that ¢ = cI> 0 Xl2 ,3' or in other words,
cI>(X l2 ,3 (X l ,2 (e, f), g))
= Xl,23 (e, X2,3 (f, g)) == ¢( e, j, g) .
In exactly the same way, starting with the trilinear map 'ljJ E Map(E, F, G; (E@F) @G)o defined by 'ljJ(e,j,g) = Xl2 ,3(Xl,2(e,f),g), one shows the existence ofa unique even linear map W : E @ (F @ G) -+ (E @ F) @ G with the property
w(X l ,23(e,X 2,3(f,g))) = Xl2 ,3(X l ,2(e,f),g) == 'ljJ(e,j,g). We now claim that cI> and W are inverse to each other. Therefore we note that the map @ F) @ G)o is a map that satisfies (w 0 cI>)('ljJ(e, j, g)) = 'ljJ(e, j, g). Exactly as we showed the uniqueness of the maps cI> and W, one can show the uniqueness of a map that is the identity on elements of the form 'ljJ( e, j, g). But since the identity map satisfies this property, the uniqueness proves that W 0 cI> = id((E @ F) @ G). In exactly IQEDI the same way one proves cI> 0 W = id(E @ (F @ G)).
W 0 cI> E End((E
§4. Tensor products
21
4.7 Discussion. Using the identification given in [4.6] we are thus allowed to say that the operation of taking the tensor product is associative. It follows that we can write E l @ ... @ Ek without using parentheses, and that we can speak of elements el @ ... @ ek in this multiple tensor product. Note that (by an easy induction argument) the elements of the form el @ .. , @ ek with ei E Ei homogeneous (and then c:(el @ ... @ ek) = c:(el) + ... + c:(ek)) generate the tensor product E l @ .. , @ E k . It follows that a linear map defined on E l @ ... @ Ek is completely known once we know its values on the elements el @ ... @ ek.
4.8 Proposition. Given Qi-graded A-modules E l , ... , E k, there exists, up to isomorphism, a unique Qi-graded A-module H = E l @ ... @ Ek and an even k-linear map X E Hom(E l , ... , Ek; H)o, x(el, ... , ek) = el @ ... @ ek, with thefollowing property. Given any Qi-graded A-module F and any ¢ E MaPs(El, ... , E k ; F), there exists a unique E MaPs(H; F) such that ¢ = 0 x. If¢ has parity a, then so has .
Proof The proof of the uniqueness is a word by word copy of the same proof in the case k = 2, replacing bilinear by k-linear. To prove that this unique space is E l @ ... @ Eb it suffices to show that it has the announced property. The easiest way to do this is by induction on k. The principle of such a proof has been used in the proof of [4.6], where IqEDI essentially the case k = 3 has been shown.
4.9 Corollary. Given Qi-graded A-modules E l , .,. , Ek, and F, the map ¢ !----> ¢ 0 X from MaPs(E l @ ... @ E k ; F) to MaPs(El"'" E k ; F) is a bijection. Restricted to Homs(E l @ ... @ E k ; F) it is an isomorphism onto Homs(E l , ... , Ek; F).
4.10 Examples.• The map ~ : El x ... X Ek x HomL(E l , .. . , E k ; F) -+ F given by (el, ... ,ek,¢) !----> ~(el, ... ,ek)¢ is (k + I)-linear and even and thus induces an even linear map E l @ ... @ E k @ HomL( E l , ... , E k ; F) -+ F. • The evaluation map HomR(E; F) x E -+ F given by (¢, e) !----> ¢(e) is even and bilinear and thus induces an even linear map HomR(E; F) @ E -+ F which maps ¢ @ e to ¢(e). • The composition map : HomL(E; F) x HomL(F; G) -+ HomL(E; G) given by (¢, 'l/J) !----> ¢ 'l/J == 'l/J 0 ¢ is even and bilinear. We thus have an induced even liner map HomL(E; F) @ HomL(F; G) -+ HomL(E; G). • Left multiplication by elements of A in an Qi-graded A-module E, mL : A x E -+ E is bi -additive and, by definition of multiplication, left bilinear. It thus induces a linear map mL : A@E -+ E. Since mL(1 @e) = e, the map mL is surjective. Now any f E A@E can be written (in a non unique way) as f = Li ai @ ei for homogeneous ai E A and ei E E. But then we have, using the bilinearity of @, f = Li I . ai @ ei = I @ Li aiei. It follows that m L is also injective, i.e., m L : A@ E -+ E is an isomorphism. In the same way, right multiplication mR : E x A -+ E induces an isomorphism mR : E @ A -+ E.
Chapter 1.
22
2t-graded
commutative linear algebra
In the sequel we will always identify A Q9 E and E Q9 A with E by the isomorphisms m L and mR'
4.11 NotationIDefinition. If E l , ... , Ek are Qt-graded A-modules, we will usually denote their tensor product El Q9 .•• Q9 Ek as ®~=l E i . Implicit in this notation is the order: ®7=k Ei will denote the tensor product Ek Q9 ..• Q9 E l . If all Ei coincide with E, the k-fold tensor product is denoted as ®k E, for which the above mentioned order problem does not exist. Obviously, if k = 1, we do not take a tensor product and ®l E = E. We also formally define ®o E = A. With these definitions the equality (®k E) Q9 (®£ E) = ®k+£ E holds for all k, C E N (for k or C = 0, use the isomorphism ms [4.10], see also [5.8]).
4.12 Construction. For any two Qt-graded A-modules we let R : E x F ----., F the map defined by
R(e, f) =
L
Q9
E be
(-1)(1f3) ff3 Q9 e .
,f3E21
We leave it to the reader to check that R is an even bilinear map. The sign, which is in agreement with our guiding principle [1.21], will be crucial. It thus induces a linear map 9't : E Q9 F ----., F Q9 E. With a slight abuse of notation, denoting the analogous map from F Q9 E ----., E Q9 F also by 9't, it is immediate that 9't 0 9't applied to e Q9 f yields e Q9 f, and thus 9't 0 9't = id, proving that it is an isomorphism. This canonical isomorphism 9't : E Q9 F ----., F Q9 E is called the interchanging map of E and F. For homogeneous elements e and f it has the property
4.13 Discussion. We have already said that we will denote the two interchanging maps
E Q9 F ----., F Q9 E and F Q9 E ----., E Q9 F both by 9't. However, we will employ the symbol 9't in an even wider context. In a multiple tensor product G Q9 E Q9 F Q9 H it is easy to construct an isomorphism onto G Q9 F Q9 E Q9 H such that an element 9 Q9 e Q9 f Q9 h is mapped to 9 Q9 9't( e Q9 f) Q9 h. By abuse of notation we will denote this isomorphism also by 9't; we will say that it is the (interchanging) map that interchanges the neighbors E and F in such a multiple tensor product. Let us denote by 9't(ii+l) the map that interchanges the i-th and i + I-st place in a multiple tensor product El Q9 • •• Q9 Ek. It is well known that if (J' E fh is a permutation of k elements, we can write it as a product of neighbor interchanges (ii + 1) (which permutes the elements i and i + 1). Taking the corresponding product of isomorphisms 9't(ii+l) gives us an isomorphism 9't0' : El Q9 ••• Q9 Ek ----., EO'-l(l) Q9 •.• Q9 EO'-l(k)' Obviously the target space EO'-l(l) Q9 ••. Q9 EO'-l(k) is completely determined by (J' and does not depend upon the way we write (J' as a product of neighbor interchanges. However, the map 9't0'
§S. Exterior powers
23
itself, which is a product of maps 9't(ii+l)' might quite well depend upon the way we write as a product of neighbor interchanges. We are thus faced with the problem: how do we investigate whether 9't" does or does not depend upon the way we write (J as a product of neighbor interchanges? Another way to pose this problem is to ask whether the maps 9't(ii+l) generate an action of6k. Phrased this way the problem is solved in the theory ofCoxeter groups [Bo, Ch IV, §1-2J. Since 6k is a Coxeter group, the isomorphisms 9't(ii+l), 1 :::; i < k generate an action of6k if and only if they satisfy the relations
(J
(9't(ii+1) (9't(ii+l)
0
0
9't(ii+ 1) =
id
1:::; i < k ,
9't(jJ+l)) 2 =
id
l:::;i oD'tO' = (-1)0'cI>, where (_1)0' denotes the signature of the permutation 0'. If we replace this relation by cI> = cI> 0 D't0" the map is called Qi-graded symmetric. However, the subject of Qi-graded symmetric maps will not be pursued in this book.
5.2 Construction (k-th exterior power). Let E be an Qi-graded A-module, then we define the Qi-graded submodule N k of ®k E as being generated by the subset Tk defined as
§5. Exterior powers
25
Since Tk satisfies the assumptions of [1.25], Nk = Span(Tk) is an S2t-graded submodule of®k E. With these preparations we define theS2t-graded A-module I\k E as the quotient
N E = ®k E / Nk It is called the k-th exterior power of the S2t-graded A-module E. Associated to this exterior power we define an even k-linear map w : Ek ---., I\k E by w = 7r 0 X, i.e., as the composition of the tensor product map X : Ek ---., ®k E with the canonical projection 7r:
®k E ---.,
N E.
5.3 Nota Bene. The above definitions and constructions have no direct meaning if k < 2. For k = 1 we let the definition ofS2t-graded skew-symmetry be an empty condition, i.e., we define Mapsk(El; F) = MaPs(E; F). This is compatible with the definition of skew-symmetric maps as satisfying o9't" = (-1)"cI> for (J E 6k, because 61 = {id} and thus the condition of skew-symmetry reduces to the empty condition = cI>. We E = E and also define Nl = {O} which, together with ®1 E = E [4.11], implies w
= id(E) : E1
---.,
N E.
N
N
For k = 0 we define No = {O}, and thus E = A, because ®o E = A [4.11]. We also define formally MaPsk (EO; F) = MaPs(A; F), but no natural justification for this definition can be found and neither do we define a map w : EO ---., 1\0 E. We thus have by definition the equalities MaPsk(EO; F) = Maps(N E; F) as well as Homsk(EO; F) = Homs(N E; F).
5.4 Proposition. Let E be an S2t-graded A-module and k :2: 1, then w : Ek ---., I\k E is S2t-graded skew-symmetric. Moreover, given any S2t-graded A-module F and any map ¢ E MaPsk (Ek; F), there exists a unique map cI> E Maps (I\k E; F) such that ¢ = cI> 0 w. If ¢ has parity a, then so has cI>. Proof Let n : Hom(®k E; I\k E)o be the by w induced linear map. In order to show the S2t-graded skew-symmetry, we have to show that no (id + 9't(jj+1)) = O. But n is the canonical projection 7r : ®k E ---., ®k E/Nk = I\k E. We thus have to show that id + 9't(jj+1) maps ®k E into Nk. But this is immediate from the definition of the generating subset Tk . If ¢ : Ek ---., F is an S2t-graded skew-symmetric map, then the induced linear map 2 : ®k E ---., F satisfies the relations 2 0 (id+ 9't(jj +1)) = O. Butthis says that 2 vanishes
=
on Tk, hence on Nk and hence induces a unique linear map cI> : I\k E ®k E / Nk ---., F. Since both operations ¢ to 2 to cI> preserve parity, the parity claim follows immediately. IqEDI
5.5 Corollary. Given S2t-graded A-modules E and F and k :2: 1, we have a bijection !----> ¢ 0 w from Maps (N E; F) to Mapsk (Ek; F). Restricted to Horns (N E; F) it is an isomorphism onto Homsk(Ek; F). For k = 0 we refer to [5.3J.
¢
Chapter 1. 2t-graded commutative linear algebra
26
5.6 Lemma. Given an Qi-graded A-module E and k ::::: 1, then, up to isomorphism, the Qi-graded A-module I\k E is the unique Qi-graded A-module enjoying the property of
[5.41. Proof The proof is an exact copy of the proof of [4.4].
5.7 Notation. According to standard usage, one uses the wedge product symbol 1\ to replace the map w in the following way. Given k elements ei E V, one writes
It follows that the Qi-graded A-module I\k E is generated by the elements el 1\ ... 1\ ek = 1l"(el Q9 ..• Q9 ek), where the ei run through E (one might even assume the ei to be homogeneous [4.7]).
5.8 Construction. For kC > 0 we define the map cI> : ®k E x ®£ E ---+ ®k+£ E by (K, L) I--> K Q9 L. If either k or C is 0, we let cI> be left/right multiplication by elements of A (remember that ®o E = A). In this way is defined for all k, C E N. Note that the induced linear map [4.3] is the canonical map which identifies (®k E) Q9 (®£ E) with ®k+£ E (and thus the maps ms are the special cases k£ = 0). We now note that both (Nk, ®£ E) and (®k E, N£) are contained in NkH. Hence there exists a unique map 1\ : I\k Ex 1\£ E ---+ I\k+£ E, called the wedge product and also denoted by a wedge, such that the following diagram is commutative:
NExNE We leave it to the reader to verify that this map is even and bilinear (but beware: the notion of Qi-graded skew-symmetry does not apply). Because of the associativity of the tensor product and, more precisely, because we may use the same tensor product symbol throughout, we are justified in the use of the wedge product symbol throughout:
just because the wedge product is induced by the tensor product. In the particular cases k or Cis 0 the definition of together with the fact that the projection 1l" : ®o E ---+ 1\0 E is the identity map id : A ---+ A immediately gives, for a E A, the equalities
§S. Exterior powers
27
5.9 Proposition. Given an Qi-graded A-module E, k, C E N, and K E I\k E and L E 1\£ E homogeneous, then
Proof Since the wedge products I\k Ex
N
N
E ----+ I\k+£ E and Ex I\k E ----+ I\k+£ E are in particular bi-additive, it suffices to show this equality for elements of the form K = eli'" . ·/\ek and L = h /\ ... /\ h, where the ei, fj E E are homogeneous. The result
now follows immediately if we realize that c:(K) = 2::7=1 c:( ei), that c:(L) = 2::~=1 c:(Jj)' and that interchanging two neighboring homogeneous elements from ei /\ fj to fj /\ ei IqEDI introduces the sign (_1)1+(c(e i )lc(fj)).
5.10 Definition. The exterior algebra 1\ E of an Qi-graded A-module E is defined as the direct sum 1\ E = EB~=o N E. We equip 1\ E with a Z x Qi-grading by defining for (k,a) E Z x Qi: (N E)a, if k ~ 0 ( 1\ E) k a = { (, ) {O} , otherwise. We also extend the wedge product to a map /\ : 1\ E x 1\ E ----+ 1\ E, as the unique even E, k, C E N, reproduces the already bilinear map which, when restricted to I\k E x defined wedge product. Since taking wedge products is associative (taking tensor products is), 1\ E equipped with the wedge product as multiplication becomes a ring. We finally define the symmetric bi-additive map U J : (Z x Qi) x (Z x Qi) ----+ Z2 (abuse of notation because the same symbol is used for the map Qi x Qi ----+ Z2) by
N
((k,a)I(C,,6))
= (kC
mod 2)
+ (al,6)
.
It is now an immediate consequence of [5.9] that 1\ E is a Z x Qi-graded commutative ring. Anticipating definition [6.1] of an Qi-graded A-algebra, the fact that 1\ E also is an Qi-graded A-module turns it into a Z x Qi-graded commutative A-algebra, explaining the algebra part in the name exterior algebra.
5.11 Remarks .• In terms of the more general definition [1.9] ofQi-graded commutativity, the symmetric bi-additive map on Z x Qi becomes the product of the functions p for Z, given by p( k, C) = (-ll£, and for Qi, given by p( a,,6) = (-1) (alP). In other words, the function p for a product grading is the product of the separate functions p. • In the particular case Qi = Z2 (which is the case we will use exclusively starting in chapter II), some authors define a (single) Z2-grading on the exterior algebra 1\ E by
(k mod 2)+p=a
(k mod 2)+p+1=a
However, such a Z2-grading is not compatible with [5.9] in the sense that there does not exist a symmetric bi-additive map on Z2 reproducing the sign (_I)(c(K)lc(L))+k£.
28
6.
Chapter 1. 2t-graded commutative linear algebra
ALGEBRAS AND DERrV ATIONS
In the previous sections we have introduced constructions of new Qi-graded A-modules out of given Qi-graded A-modules. In this section we will introduce different structures on Qi-graded A-modules. More precisely, we will introduce the notions of associative Qi-graded A-algebra and Qi-graded A-Lie algebra. Associated to the notion of an algebra is the notion ofa derivation. It is shown that there exists a natural way to identify E* as a collection of derivations ofthe exterior algebra 1\ E. This identification is the algebraic version of the contraction of a vector field with a k10rm, employed systematically in differential geometry.
6.1 Definition. Let 9 be an Qi-graded A-module and m : 9 x 9 -+ 9 an even bilinear map. The couple (g, m) is called an (associative) Qi-graded A-algebra ifm seen as multiplication is associative, i.e., if (g, +, m) is a ring. It is called an Qi-graded commutative A-algebra ifm is Qi-graded symmetric. The couple (g, m) is called an Qi-graded A-Lie algebra if the map m is Qi-graded skew-symmetric and satisfies the Qi-graded Jacobi identity, i.e., for all homogeneous e, f, g E g : [e, [f,g]] = [[e,f],g] + (_I)(c(e)lc(f)) [f, [e,g]], where we have written [e,f] for m(e,f), as we will always do for Qi-graded A-Lie algebras. This will cause no problems concerning leftlright linearity because m is even. Using the Qi-graded skew-symmetry [e,f] = -(-I)(c(e)lc(f))[f,e], this relation can also be written in the more symmetric form
(-1) (c(g)lc(e)) [e, [f, g ]]
+ (-1) (c(e)lc(f)) [f, [g, e ]] + (-1) (c(f)lc(g)) [g, [ e, f]]
= 0.
For Qi-graded A-Lie algebras one usually calls the element [ e, f] == m( e, f) the bracket of e and f. An Qi-graded submodule F of an Qi-graded A-Lie algebra E is called an Qi-graded sub A-Lie algebra of E (or a subalgebra for short) if F is stable under the bracket operation [_, _], i.e., Ve,f E F: [e,f] E F.
6.2 Examples.• The ring A itself is a (rather trivial) Qi-graded commutative A-algebra. • If E is an Qi-graded A-module, the Qi-graded A-module Ends(E) becomes an Qi-graded A-algebra when we take composition of maps as multiplication. • For any Qi-graded A-module E, the exterior algebra 1\ E is a Z x Qi-graded commutative A-algebra, where we give A a Z x Qi-grading by Ao,a = Aa and An,a = {O} whenever n =1= O. • If (g, m) is an (associative) Qi-graded A-algebra, we can introduce an even bilinear Qi-graded skew-symmetric commutator map [ _, _] : 9 x 9 -+ 9 by its action on homogeneous elements e, f E 9 :
[e,f] =m(e,f) - (-I)(c(e)lc(f))m(f,e).
§6. Algebras and derivations
29
Ifwe identify the bilinear maps m and [ _, -l with the associated linear maps 9 Q9 9 -+ g, the definition of the commutator [ _, -l can be written as [ _, -l = m - m 0 D't. We leave it to the reader to verify that (g, [ _, -l) is an Qt-graded A-Lie algebra. In particular, (g, m) is Qt-graded commutative if and only if the commutator map [ _, -l is identically zero.
6.3 Definition. Applying the construction of the last example in [6.2] to the algebra 9 = (EndR(E), 0) gives us a commutator [_, _lR on EndR(E), turning the set of right linear endomorphisms of an Qt-graded A-module into an Qt-graded A-Lie algebra. In the same way 9 = (End L (E), 0) becomes an Qt-graded A-Lie algebra with bracket [ _, _lL given on homogeneous elements by [ ¢, 'ljJ lL
= ¢ 0 'ljJ -
(-1) (c(¢)lc(1/») 'ljJ
= _(_l)(c(¢)lc(1/»)
0
¢
= 'ljJ
0
¢ - (-1) (c(¢)lc(1/») ¢ 0 'ljJ
(¢o'ljJ _ (_l)(c(¢)lc(1/»)'ljJo¢) .
If no confusion is possible, we will omit the subscripts Land R in these brackets and simply write [ _, -l.
6.4 Lemma. For¢,'ljJ E EndL(E) wehave'.r([¢,'ljJlL)
= -['.r¢,'.r'ljJJR.
6.5 Remark. The above lemma is in agreement with the usual interpretation of the transpose of an endomorphism and the commutator. Moreover, written this way, no additional sign is involved according to our guiding principle [1.21], because the elements ¢ and 'ljJ are not interchanged.
6.6 Definition. Let E be an Qt-graded A-module and m an even bilinear map E x E -+ E. A right derivation of the couple (E, m) is a right linear endomorphism of E whose homogeneous parts ¢ satisfy for all homogeneous elements e, fEE the relations
¢(m(e, f)) = m(¢(e), f)
+ (_l)(c(e)lc(¢))m(e, ¢(I))
.
For a left derivation, ¢ has to be left linear and the relation has to be replaced by ((m(e,f)II¢))
= m(e, ((III¢))) + (-l)(c(¢)lc(f))m(((ell¢)),f)
.
The set of all right derivations of(E, m) is called DerR(E, m) C EndR(E), or DerR(E) if the bilinear map m is understood. Similarly the left derivations are denoted by DerL (E, m) or DerL(E).
Chapter 1. 2t-graded commutative linear algebra
30
6.7 Lemma. The set Ders(E, m) is a subalgebra ofEnds(E) when the latter is equipped with the commutator as bracket; in particular Ders(E, m) is an Qi-graded submodule of Ends(E). Moreover, 'r¢ is a right derivation if and only if ¢ is a left derivation. Proof We leave it to the reader to verify that Ders(E, m) is an Qi-graded submodule of Ends(E). Let us show that if ¢, 'ljJ are two homogeneous right derivations, then [¢, 'ljJ] is a homogeneous right derivation as well. Thus let e, lEE be homogeneous and compute:
[¢,'ljJ]([e,/]) = (¢o'ljJ - (-1)(c(cp)lc(1/»)'ljJo¢)([e,/D
= ¢([ 'ljJ( e), I] + (-1) (c(e)lc(1/») [e, 'ljJ(f) ]) - (_1)(c(cp)lc(1/») 'ljJ([ ¢(e), I] = [¢('ljJ(e)), I]
+ (_1)(c(cp)lc(1/>(e)))
+ (_1)(c(e)lc(1/»)
+ (_1)(c(e)lc(cp)) [e, ¢(f)])
['ljJ(e), ¢(f)]
[¢(e), 'ljJ(f)]
+ (_1)(c(e)lc([cp,1/>]))
[e, ¢('ljJ(f))]
- (_1)(c(cp)lc(1/») ['ljJ(¢(e)), 1]- (_1)(c(e)lc(1/») [¢(e), 'ljJ(f)] - (_1)(c(cp)lc(1/>(e))) ['ljJ(e), ¢(f) 1 - (_1)(c(cp)lc(1/»)+(c(e)lc([cp,1/>])) [e, 'ljJ(¢(f))]
= [[¢,'ljJ](e),/] + (-1) (c(e)lc([cp,1/>J)) [e,[¢,'ljJ](f)].
IqEDI
6.8 Discussion. In ungraded Lie algebras, the auto commutator [ e, e] of an element e E 9 is automatically zero, just because of the skew-symmetry. However, for Qi-graded A-Lie algebras this is no longer true. Indeed, for a homogeneous element e, Qi-graded skewsymmetry gives us [e,e] = _(_1)(c(e)lc(e))[e,e], and thus [e,e] necessarily equals 0 only if (c:(e)Ic:(e)) = O. This phenomenon becomes clearer in the context of derivations. If ¢ E Ends(E) is a derivation, its square ¢2 = ¢ 0 ¢ is, in general, no longer a derivation. But if (c:( ¢) Ic:( ¢)) = 1, it follows from the definition of the commutator that ¢2 = ¢, ¢] is again a derivation (see [V.1.23] for an explicit example).
![
6.9 Definitions . • Let (gl, [_, _ h) and (g2, [_, -lz) be two Qi-graded A-Lie algebras and let ¢ : 91 -+ g2 be linear. The map ¢ is said to be a morphism of Qi-graded A-Lie algebras ifitis even and preserves brackets, i.e., Ve, IE 91 : ¢([ e, Ih) = [¢(e), ¢(f) lz· • A left-representation of an Qi-graded A-Lie algebra (g, [ _, _]) on an Qi-graded A-module E is a morphism of Qi-graded A-Lie algebras ¢ : 9 -+ EudL(E). Rightrepresentations are defined similarly . • For any Qi-graded A-Lie algebra (g, [ _, _ ]) we define maps ads: 9 -+ Ends(g) by
adR(e):
I
I--->
[e,/]
&
adL(e):
I
I--->
[/,e].
The fact that the bracket [ _, _ ] is even and bilinear immediately shows that ads is a well defined even morphism of Qi-graded A-modules. The Qi-graded Jacobi identity tells us
§6. Algebras and derivations
31
that ad R is a right-representation of (g, [_, _ Dand at the same time that all adR(e) are right derivations of (g, [ _, _ D. The Jacobi identity plus the Q(-graded skew-symmetry tell us the same things for ad L : it is a left-representation and im(adL) C DerL(g). These two representations are called the (left- and right-) adjoint representations of the Q(-graded A-Lie algebra (g, [ _, _ D.
6.10 Lemma. For any Q(-graded A-Lie algebra g, the left and right adjoint representations are related by '.r 0 ad L = - ad R. Proof This is a direct consequence of the Q(-graded skew-symmetry and [6.4].
IQEDI
6.11 Remark. For generic Q(-graded A-Lie algebras, the condition that a morphism of Q(-graded A-Lie algebras preserves the brackets "implies" that it must be even, just by counting parities: ¢( [ e, f h) = [¢( e), ¢(f) h "implies" that we must have the equality c(¢) + c(e) + c(J) = c(¢) + c(e) + c(¢) + c(J). However, this argument is not valid whenever the zero element is involved, e.g., ifboth brackets are identically zero.
6.12 Remark. We have restricted our attention to even bilinear maps m : 9 x 9 ---., g, because we will need no others. However, in the literature one also finds non-even homogeneous maps m, especially in the context ofQ(-graded A-Lie algebras. Let us give the precise definition. An Q(-graded A-Lie algebra of parity a is an Q(-graded A-module 9 together with a bi-additive map [ _ Iml -1 : 9 x 9 ---., g, (e, f) I--Y [e Iml f 1satisfying:
D
(i) for homogeneous e, f E 9 : c( [ e Iml f = c( e) + c(J) + a (the bracket has parity a), (ii) for all A, f.L E A and e, f E g: [Ae Iml f f.L 1 = A [ e Iml f 1f.L (left linearity in the
first argument and right linearity in the second), (iii) for homogeneous e,f E g: [elmlfl = -(-l)(c(e)+lc(f)+ [5.1]. It is fairly easy to see that this is the case if ¢ is even. We conclude that for an even linear map ¢ : E ---. F there exists an induced even linear map I\k ¢ : I\k E ---. I\k F such that for ei E E one has
N
N
E = E [5.3], we quite naturally find that ¢ = ¢. For k = 0 we formally Since defined N E = A; for maps we now formally define N ¢ = id(A). An immediate consequence of these definitions and the definition of the exterior product [5.8] is that for A E I\k E and BEN E we have
It should be obvious that if ¢ is bijective, i.e., an isomorphism, then I\k ¢ also is bijective with inverse I\k (c/J-l). In categorical language one would say that I\k is a functor, but one which we only apply to even morphisms.
7.16 Construction (dual of exterior powers). Let E be an Ql-graded A-module. We will construct identifications J : I\k (E*) ---. (N E)* and J : I\k *E ---. *(I\k E). There are two ways to construct this isomorphism, a fast way and a pedestrian way. We start with the pedestrian way for the right linear case. We denote by J 0 the identification J 0 :~®k(E*) ---. (®k E)* of [7.12], and we introduce the modified permutation operators D'to" = D'tTO"T [4.13], where T E 6k is the fixed permutation T( i) = k + 1 - i (the one which reverses the order of k elements). For homogeneous elements ei this implies
where the sign is determined by the permutation (J and the parities of the ei. We now claim that we have the following equality of maps from ®k E to A : (7.17) Since the neighbor interchanges generate 6k, it suffices to check this for (J of the form (J = (ii + 1). The essential computation to verify is the case k = 2, which is left to the reader. We now define the skew-symmetrization operator A. by the formula
§7. Identifications
41
t
where (-1 denotes the sign of the permutation 0'. This skew-symmetrization operator has the important property that for any permutation 0' we have (7.18)
With these preparations we can construct the identification J. We first define the map'ljJ = J 0 oi;h : ®k E* ---., (®k E)*. Using (7.17) we obtain for any cI> E ®k E* the equality 'ljJ(cI» = J 0 (cI» oA. It then follows from (7.18) that 'ljJ(cI» : ®k E ---., A is Qt-graded skew-symmetric [5.1], and thus i~duces a map w(cI» E (I\k E)*. On the other hand, it also follows from (7.18) that W(D'tO'-l (cI») = (_1)0' . w(cI», i.e., the map W : ®k E* ---., (I\k E)* is Qt-graded skew-symmetric. We thus obtain an induced map J : I\k E* ---., (I\k E) *. This identification is given explicitly by the formula (7.19)
J(¢k /\ ... /\ ¢d(el /\ ... /\ ek)
L
=
(_1)0'. J 0 (¢k @ ... @¢I)(D'tO'(el @.··@ek)) .
O'E6 k
So far the pedestrian way to define J. The fast way uses the concept of derivations. Recall [6.16] that for ¢i E E* we have defined a derivation ~(¢) of bi-degree (-1, c:( ¢)) on the exterior algebra 1\ E. In particular the composite X( ¢I, ... ,¢k) = ~(¢I) 0 • • • 0 ~(¢k) restricts to a map I\k E ---., N E = A. It follows from [6.16] and [6.18] ([6.19]) that X : (E*)k ---., (I\k E)* is k-linear and Qt-graded skew-symmetric. We thus have an induced map J : E* ---., (I\k E)*. In [7.21] we will prove that these two definitions coincide, i.e., that we are allowed to use the same symbol. That we obtain equality of both definitions, and not an equality up to a sign, is due to the reversed order in the tensor product in [7.12]. In the left-linear case the construction is similar. The pedestrian way uses the identification J 0 : ®k(*E) ---., *(®k E) [7.12] and the fast way uses the left-derivations equivalent of [6.16] (which gives a map *E ---., Derdl\ E)).
N'
7.20 Example. Let ¢I, ¢2 E E* and el, e2 E E be homogeneous, then the identification between 1\2 E* and (1\2 E)* gives us:
J(¢2/\ ¢t)(el /\ e2) = J 0 (¢2 @¢I)(el @ e2) - J 0 (¢2 @ ¢dD't(12)h @e2) = (-I)(c(¢2)lc(¢1(e Il ))¢I( e l)¢2(e2)(-I)(c(¢1)lc(e 1))¢2(el)¢I(e2)
.
Apart from the additional signs, this formula can be seen as the ordinary determinant ofthe matrix ( ::
i::i :~i:~U· More generally, the value ofJ(¢I/\· .. /\ ¢k) (ek /\ ... /\ el) can be
seen as a generalization of Det( ¢i (ej)). However, this generalization of the determinant should not be confused with the Berezinian or graded determinant to be defined in [11.5.16]. The latter is related to the group Aut (E), whereas the former bears no natural relation to this group.
Chapter 1. 2t-graded commutative linear algebra
42
7.21 Proposition. Let ¢I, ... ,¢k E E* be arbitrary, then as maps from I\k E to A we have the identity J(¢I/\OO'/\¢k) =~(¢I)o", o~(¢k)'
where the
~(¢i)
are the derivations defined in [6.16J.
Proof Since both expressions are left k-linear in the ¢i, it suffices to check the identity for homogeneous ¢i. Since elements el /\ ... /\ ek with ei E E homogeneous generate I\k E, it again suffices to check this identity on elements of this form. Now using the derivation property of ~(¢), it is easy to check the following identity: k
~(¢)(el/\"'/\ ek) =
I)-I)
(i-I)+ I:(c(ei)lc(ej))
j
¢(ei)el'" /\ ei-l /\ ei+I/\"'/\ ek .
i=1 Using the map idk-I Q9 ¢ : ®k E -+ A Q9 ®k-I E ~ ®k-I E and the canonical projection 7r : ®k E -+ I\k E, where we introduced the abbreviation id k for the identity map on ®k E, we can write this identity as
(i-I)+ I:(c(ei)lc(ej))
k
(7ro(idk-IQ9¢))(I)-I)
j~(e)(l) = e, then we can interpret JLR(e) as the dual map ~(e)* : E* ...., A. Here is how it works. Let e E E and ¢ E E* be arbitrary, then we have the equality ((¢IIcI>~(e)*)) = ¢o~(e) E A* and
(((¢IIcI>~(e)*»)(l) = ¢(e) = ~¢IIJLR(en = cI>~(~¢IIJLR(e))))(l) and thus ((¢IIcI>~(e)*)) = ~(((¢IIJLR(e)))). By abuse of notation, forgetting about the isomorphisms cI>~ and cI>~, we thus find J LR (e) = e*, where we have made the identificationsE ~ HomR(A; E) and HomL(E*; A*) ~ HomL(E*; A) == *(E*).
46
Chapter 1. 2t-graded commutative linear algebra
7.28 Construction (sums of tensor products). Let E be an Qt-graded A-module and let (Fi)iEI be a family of Qt-graded A-modules. We want to show that E @ (EBiEI Fi ) and EBiEI (E @ Fi ) are isomorphic. First recall that 7ri and Si denote the canonical projections and injections between Fi and EBiEI Fi [3.1]. The map E x (EBiEI Fi ) -+ EBiEI(E@Fi ) given by (e, f) !----> ffii (e @ 7ri(J)) is even and bilinear, and thus gives rise to an even linear map E @ (EBiEI Fi ) -+ EBiEI(E @ Fi). On the other hand we have the even linear map I:iEI id@Si : EBiE1(E@Fi ) -+ E@ (EB iE1 Fi). Let us show by computation that these two maps are inverse to each other.
iEI
iEI
and
iEI
iEI
where we used that trj 0 Si = 0 whenever i =1= j. We conclude that we have constructed an isomorphism between E@ (EB iE1 Fi ) and EBiE1(E@ Fi). In the same way we construct an isomorphism between (EBiEI Fi ) @ E and EBiEI(Fi @ E).
8.
ISOMORPHISMS
In §7 we have defined a number of identifications. In this section some technical proofs are given to show sufficient conditions for these identifications to be isomorphisms. It turns out that this is the case if the Qt-graded A-modules are finitely generated and projective. The condition finitely generated and projective for (Qt-graded) modules is equivalent to the condition finite dimensionalfor vector spaces. At the end of this section a summary of the more interesting identifications can befound.
8.1 Definitions. • A subset G of an Qt-graded A-module E is called a set of generators for E iffor each e E E there exist 91, ... ,9k E G (a finite number!) and a 1 , ... , a k E A such that e = I:i ai9i. The 9i and ai are not supposed to be unique. • An Qt-graded A-module E is called finitely generated, or offinite type, if there exists a finite set of generators G. • A subset B of an Qt-graded A-module E is called a set of independent elements if for all e1, ... ,ek E B (a finite number!) and for all a 1, ... ,a k E A one has the implication: I:i aiei = 0 =} \Ii : a i = O. In words: any (linear) relation between elements of B is necessarily trivial. • A subset B of an Qt-graded A-module E is called a basis for E if the elements of B are at the same time independent and generating. For vector spaces it is well known
47
§S. Isomorphisms
that there always exists a basis. However, for S2t-graded A-modules the existence of a basis is no longer guaranteed. It is immediate that an S2t-graded A-module admits a homogeneous basis ifand only ifitis (isomorphic to) afreeS2t-gradedA-moduleF(G,s) on homogeneous generators G [3.4] . • An S2t-graded A-module E is called projective if there exists an S2t-graded A-module E' such that EEBE' admits a homogeneous basis G, i.e., E EB E' = F(G, s). An S2t-graded A-module that is both finitely generated and projective will be calledfg.p.
8.2 Remark. We have defined the notions of independence of vectors, generating sets and bases with respect to the left module structure, completely ignoring that S2t-graded A-modules also have a (compatible) right module structure. Of course we could have developed the right module case parallel to the left module case. However, this is hardly necessary. If E is an S2t-graded A-module, gEE homogeneous and a E A, we have the relation ag = 9 . I:a:E=21( -l)(c(g)la:)aa:' This shows that a (left) generating set of homogeneous elements is also generating for the right module structure. And if a set of homogeneous elements is (left) independent, it is also independent for the right module structure. Hence for homogeneous sets, there is no difference between the notions of generating and independence for the left or right module structures. Moreover, by splitting into homogeneous components, any generating set can be made homogeneous. The only possible difference between left and right module structures thus lies in non-homogeneous independent sets. Since we will not use these, we will not see any difference between our left module structure definitions and the corresponding right module equivalents.
if and only if there exists an S2t-graded A-module E' such that EEBE' is afree S2t-graded A-module on afinite set of homogeneous generators. 8.3 Lemma. An S2t-graded A-module E is fg.p
Proof If E EB E' = F(G,s) with G a finite set, E is projective and 1fE(G) is a finite set of homogeneous generators for E, proving the if part. To prove the only if part, assume that E is projective and admits a finite number of generators. Since E is projective, there exists E' such that E EB E' = F(G', s') for some set of homogeneous generators G'. Our problem is that we do not know whether we can take G' to be finite. Let G = {gl,"" gn} be a finite set of generators for E. By splitting these generators into their homogeneous parts, we may assume that all gi are homogeneous. We thus obtain an even surjective map ¢ : F( G, s) -+ E defined simply by ¢(gi) = gi' Define the even map 'ljJ : E EB E' = F(G', s') -+ E as 'ljJ = id(E) + Q [7.2], where Q: E' -+ E is identically zero. Define also a map X : F(G', s') -+ F( G, s) as follows. For any g' E G' choose an element f' E F( G, s) such that ¢(f') = 'ljJ(g'). This is possible because ¢ is surjective. Since both ¢ and 'ljJ are even, we may assume that f' and g' have the same parity. Now define X by the formula
x:
L 9'EG'
Ag,g'l-->
L g'EG'
Ag'l'
.
Chapter 1. 2t-graded commutative linear algebra
48
This is a well defined even linear map satisfying the relati on 'ljJ = c/J 0 x. Restriction of X to the Qt-graded submodule E gives us the relation id( E) = c/J 0 (xl E). By using the equality x = (x - (XIE)(c/J(X))) + (XIE)(c/J(X)) it follows easily that the Qt-graded submodules ker(c/J) and im(xIE) are supplements. Hence by [3.9] we obtain E EB ker(c/J) = F(G, c:). IQEDI
8.4 Lemma. If an Qt-graded A-module E isfg.p, then so are E* and *E. Proof If E is a free Qt-graded A-module on a single homogeneous generator eo, it follows immediately that E* is a free Qt-graded A-module on the single homogeneous generator c/Jo defined by c/Jo( eo . A) = A. If E is f.g.p, there exists an Qt-graded Amodule E' such that E EB E' = EB~=1 E i , where all the Ei are free Qt-graded A-modules on a single homogeneous generator [3.4]. Using [7.3] we thus obtain an isomorphism E* EB (E')* ~ EB~=l Ei- Since the last one is a free Qt-graded A-module on n generators, the result follows. The left linear case is analogous. IQEDI
8.5 Proposition. If E is fg.p, the identification J : E
---+
*E* [7.26J is an isomorphism.
Proof Consider the special case of*(E*).If E is a free Qt-graded A-module on a single homogeneous generator, the result follows immediately from the proof of [8.4]. If E 1 , •.• , En are Qt-graded A-modules, we have identifications J i : Ei ---+ * (En and JE& : EB~=1 Ei ---+ *((EB~=1 Ei)*). Applying the isomorphism [7.2] twice gives us the following diagram: n EB .=1 *(E*) EB~=1 Ei •
I1~=1
Ji
II EB~=1 Ei
J(j)
--->
is. *((EB~=1 Ei)*) .
We leave it to the reader to verify that this diagram is actually a commutative diagram. Now let E' be an Qt-graded A-module such that E EB E' = EB~=1 Ei where the Ei are free Qt-graded A-modules on a single homogeneous generator. Applying the commutative diagram to the family of two: E and E', shows that JE& is an isomorphism if and only if both J and J' are isomorphisms [7.8]. Applying the commutative diagram to the finite family Ei proves that JE& is an isomorphism because for free Qt-graded A-modules on a single homogeneous generator the J i are isomorphisms. It follows that J is an isomorphism. IQEDI
8.6 Proposition. If E isfinitely generated or if the index set I isfinite, then the identifications J : EBiEI Homs(E; F i ) ---+ Homs(E; EBiEI F i ) [7.6J are isomorphisms. Proof Since injectivity is automatic, we have to assure surjectivity. For an arbitrary c/J in Homs(E; EBiEI F i ) we define the maps c/Ji E Homs(E; Fi) by c/Ji = 7ri 0 c/J, where
49
§S. Isomorphisms
7ri : EBi Fi ---., Fi denotes the canonical projection. It is immediate that J(EBi¢i) = ¢, except for the fact that we do not know whether (¢i)iEI lies in EBiFI Homs(E; Fi), i.e., whether only finitely many ¢i are non-zero. If the index set I is finite, this is obvious and we may conclude that in that case J is an isomorphism. So suppose that el) ... ) en are generators for E. Since ¢( ej) lies in EBi Fi , it follows that only finitely many ¢i (ej) = 7r i( ¢( ej)) are non-zero. Since there are only finitely many ej, which generate E, it follows that only finitely many ¢i are non-zero. Hence (¢i)iEI E EBiEI Homs(E; Fi), proving that J is an isomorphism. IQEDI
8.7 Proposition. Given three Qi-graded A-modules E, F, and G, we want to investigate the following two identifications: J : F @ HomR(E; G) ---., HomR(E; F @ G) and
J: HomL(E; G)
@
F ---., HomL(E; G@ F).
(i) ifF is projective, then the J's are injective. (ii) ifF isfg.p, then the J's are isomorphisms. (iii) If E is fg.p, then the J's are isomorphisms. Proof We only treat the right linear case, the left linear case being similar. Let us start with the special case in which F is a free Qi-graded A-module on a single homogeneous generator fa. By definition, the map p : A ---., F, a I--> fa . a is a right linear bijection of the same parity as fa. Using [7.14] one can show that C : HomR(E; F @ G) ---., HomR(E; A @ G) defined by C( ¢) = (p-l @ id) 0 ¢ is a right linear bijection. We thus obtain a map Co J 0 (p @ id) : A @ HomR(E; G) ---., HomR(E; A @ G). Identifying A @ H with H for any Qi-graded A-module H, it is an elementary verification that this map is the identity on HomR(E; G). Since C and (p@ id) are bijective, we deduce that J is bijective. We conclude that J is an isomorphism in case F is a free Qi-graded A-module on a single homogeneous generator. To prove (i) and (ii), let us fix E and G, and let us consider a family Fi , i E I of Qi-graded A-modules. We thus have identifications J i : Fi @ HomR(E; G) ---., HomR(E; Fi @ G) andJE&: (EBiEIFi)@HomR(E;G)---.,HomR(E;(EBiEIFi)@G). According to [7.28] there exists an isomorphism (EBiEI Fi ) @ HomR(E; G) ---., EBiEI(Fi @ HomR(E; G)). According to [7.28] we also have an isomorphism (EBiEI Fi)@G ---., EBiE1(Fi@G). This last isomorphism combined with the identification [7.6] gives us an injective identification Jh : EBiEI HomR(E; Fi @ G) ---., HomR(E; (EBiEI Fi) @ G). We thus obtain the following diagram of maps:
"'" 1
[7.28J
(EBiEI Fi ) @ HomR(E; G) We leave it to the reader to check that this diagram is commutative. Now take a family of two: F and F'. Since the index set consists of two elements, the map J h is a bijection [7.6]. Using [7.8] we conclude that JE& is injectivelbijective if and
50
Chapter 1. 2l-graded commutative linear algebra
only ifboth 'J and 'J' are injectiveibijective. Ifwe suppose that F is projective, we can take F' such that F EEl F' is a free SZl-graded A-module on a set of homogeneous generators, i.e., FEEl F' = EBiEI Fi where each Fi is a free SZl-graded A-module on a single homogeneous generator. For these we know that all 'J i are isomorphisms. However, for this family we only know that 'J h is injective. We conclude that 'JEll is injective. This proves (i). If F is also finitely generated, it follows that the family Fi can be taken finite, in which case 'J h becomes bijective, hence 'JEll is bijective, and thus 'J is bijective. This proves (ii). To prove (iii) we proceed in the same way. If E is a free SZl-graded A-module on a single homogeneous generator, one can easily establish the existence of homogeneous linear bijections between Hand HomR(E; H) for any SZl-graded A-module H. Using these bijections one then proves that 'J is an isomorphism if E is a free SZl-graded A-module on a single homogeneous generator. Now let us fix F and G and let us take a finite family (Ei)f=l. We thus have identifications 'J i : F 0 HomR(E i ; G) -+ HomR(E i ; F 0 G) and 'JEll : F 0 HomR(EB~=l E i ; G) -+ HomR(EB~=l E i ; F 0 G). According to [7.2] we have an isomorphism EB~=l HomR(E i ; G) -+ HomR(EB~=l E i ; G). Taking the tensor product of this isomorphism with the identity on F (see [7.14]) and composing it with the isomorphism EB~=l (F 0 HomR(E i ; G)) -+ F 0 (EB~l HomR(E i ; G)) gives us an isomorphism 'J v : EB~l (F 0 HomR(E i ; G)) -+ F 0 HomR(EB~=l E i ; G). We thus obtain a diagram of maps:
~ -------+ J(j)
1
[7.2J
HomR(EB~=l E i ; F 0 G) ,
As before, it is left to the reader to check that this diagram is commutative. If we now suppose that E is f.g.p, then there exists an SZl-graded A-module E' such that E EEl E' = EB~=l E i , where each Ei is afree SZl-graded A-module on a single homogeneous generator. Applying the commutative diagram to the finite family of two: E and E', shows that 'JEll is bijective if and only ifboth 'J and 'J' are bijective [7.8]. Applying it to the finite family Ei shows that 'JEll is bijective if and only if all 'J i are bijective. Since the Ei are free on a single homogeneous generator, the 'J i are isomorphisms, and we conclude that both 'J and 'J' are bijective. IQEDI
8.8 Corollary. Let E and F be SZl-graded A-modules. ifF is projective, the identifications *E 0 F -+ HomdE; F) and F 0 E* -+ HomR(E; F) [7.llJ are injective. Ifeither E or F isjg.p, these identifications are isomorphisms.
8.9 Lemma. If Ei = F(G i , ci), i = 1,2 arefree SZl-graded A-modules on homogeneous generators G i , then El 0 E2 is afree SZl-graded A-module on homogeneous generators G 1 x G 2 with parity mapc(91,92) = Cl(91) --'-c2(92).
51
§S. Isomorphisms
Proof Choosing left coordinates for E1 and right coordinates for E 2, we define a map 'I/J : E1 x E2 -- F(G 1 X G2, c) by 'I/J(2::i Ai . 91i, 2:: j 92j . /-Lj) I----' 2::i,j Ai . (91i, 92j) . /-Lj. We leave it to the reader to verify that 'I/J is even and (right) bilinear, and thus induces an identification E1 &; E2 -- F( G 1 X G2, c). It is easily seen that the inverse is given by the map 2::i,j Vij . (91i, 92j) I----' 2::i,j Vij . 91i &; 92j, proving the lemma. IQEDI
8.10 Corollary. If E and F are projective '21.-graded A-modules, then E If they arefinitely generated, then so is E &; F.
&; F
is projective.
Proof The elements e &; j, e E E, j E F generate E &; F. If E is generated by (ei)iEI and F by (Ii) j E J, then E &; F is generated by (ei &; Ii) iE I,j E J. We conclude that if both E and F are finitely generated, so is E &; F. If E EEl E' and FEEl F' are free '21.-graded A-modules on homogeneous generators, then (E EEl E') &; (F EEl F') is a free '21.-graded A-module on homogeneous generators. But by [7.28] this is isomorphic to (E that E
&;
&;
F) EEl ((E
&;
F') EEl (E'
&;
F) EEl (E'
F is projective.
&;
F I ) ) , showing IQEDI
8.11 Discussion. If E and F are '21.-graded A-modules, we have defined the operation of right dual map, which is an even linear map * : HomR(E; F) __ HomL(F*; E*). We also have identifications F &; E* -- HomR(E; F), *(F*) &; E* -- HomL(F*; E*), and F -- * (F*). We leave it to the reader to verify that these identifications fit together in a commutative diagram
F
&;
E*
1
------?
*(F*)
&;
E*
1
If E and F are f.g.p, we know that the three unlabeled arrows are isomorphisms, and thus thatthe map * : HomR(E; F) -- HomL(F*; E*) is an isomorphism as well. Obviously a similar result is true for left dual maps.
8.12 Proposition. Let Ei and F i , i = 1, ... , n be two families of '21.-graded A-modules and let'J : ®t=n Homs(Ei; F i ) -- Homs(®~=l Ei ; ®~=1 Fi ) be the identification given in [7.12J. (i) If all Ei arefg.p then 'J is an isomorphism. (ii) If all Ei and Fi arefg.p, with the possible exception ofa single pair (Ei, F i ), then 'J is an isomorphism.
52
Chapter 1. Qt-graded commutative linear algebra
Proof We give the proof in the left linear case, the right linear case being similar. We create the following isomorphisms: ~
~ ~ ~
HomL(E2; F2) 0 HomL(E 1; Fd HomL(E2; F2 0 HomL(E 1; Fd) HomL(E 2;HomL(E1;Fd 0F2) HomL(E2; HomL(E 1; F1 0 F2)) HomL(E1 0 E 2; F1 0 F2)
by [8.7] with E2 f.g.p using that 9t is an isomorphism by [8.7] with E1 or F2 f.g.p by [7.10]
We leave it to the reader to trace these isomorphisms and to show that the final result is indeed the identification given in [7.12]. It follows that we have proven the proposition for n = 2, where we used that either E1 and E2 are f.g.p, or E2 and F2 are f.g.p. The general result follows by induction. In case (i) one uses [8.10]. In case (ii) one uses that all Ei and Fi are f.g.p for i = 1, ... , n - 1. Since permuting factors in a tensor product is an isomorphism, we may indeed assume that it is the last couple (En, Fn) that is not IQEDI f.g.p.
8.13 Corollary. Let E 1, ... , En be '21.-graded A-modules. If all but one are fg.p, the identification E~ 0··· 0 Ei -- (E1 0··· 0 En)*, defined on homogeneous elements by
L: (e( 1>i) le(1)j (ej ») (rPn 0···0 rP1)(e1 0··· 0 en) = (-l)i>j
. rP1(ed'" rPn(e n ) ,
is an isomorphism, as is the identification *En 0 ... 0 *E1 -- * (E1 0 ... 0 En), defined on homogeneous elements by ~(e1 0··· 0
en)(rPn 0··· 0
L: (e(1>i)le(1>j(ej))) rPd = (-l)i<j
. ~(e1)rP1'" ~(en)rPn .
8.14 Proposition. If E = F(G, c) is afree '21.-graded A-module on homogeneous generators G, then /\k E is a free '21.-graded A-module. Proof From [8.9] we know that the set B = {91 0 ... 0 9k I 9i E G} is a basis for ®k E. We now choose a total order on G and we define the subsets B g1 ... 9k C B for increasing sequences 91 :::; 92 :::; ... :::; 9k by
B g1 ... 9k = {90-(1) 0··· 090-(k)
I
(J
E
6k } .
B g1 ... 9k form a partition of B in disjunct subsets. We now introduce the modified sets B~1 ... 9k defined as
It should be obvious that the
It is easy to see that the set {91 0 ... 0 k
sub module of ® E as B g1 ... 9k •
9k} U B~1 ... 9k generates the same '21.-graded
53
§S. Isomorphisms
We now recall the construction of /\k E = ®k E / Nk as given in [5.2]. By k-linearity, it is immediate that the szt-graded submodule Nk C ®k E is also generated by the set
T£ = {91 0 ... 09k
+ 91(jj+1) (91 0··· 09k) 11:::; j < k,
9i E G} .
An elementary but slightly tedious verification then shows that Nk is also generated by the union of all sets B~l" .gk (and recall that the index set 91 .. . 9k is an increasing one). Since the B g1 ... gk form a partition of the basis B, it follows that we have found a basis for a supplement of Nk in the form of the set of all 910' .. 09k subject to two conditions (i) that 91 :::; ... :::; 9k and (ii) that it does not belong to the szt-graded submodule generated by B~1 ... 9k' The projection of this basis to /\k E = ®k E / Nk then is a homogeneous basis for /\k E, proving that E is a free szt-graded A-module. IQEDI
N
8.15 Discussion. It can be shown that the condition that 91 0 ... 0 9k belongs to the szt-graded submodule generated by B~1 ... 9k is equivalent to the existence of a permutation (J E 6k such that -(-1)0"910"(91 0··· 09k) = 91 0 .,. 09k. This in turn can be shown to be equivalent to the existence of an index i < k such that 9i = 9i+1 and such that (_I)(o(gi)lo(g,)) = 1. It follows that a basis of /\k E is given by the vectors 91 1\ ... 1\ 9k where the 9i form an increasing sequence such that a 9i is not repeated if (_I)(o(gi)lo(g,)) = 1.
8.16 Proposition. If E is fg.p, the identifications J : N (E*) ~ (/\k E)* and J : N (*E) ~ *(N E) [7.16J are isomorphisms. Proof As usual we treat the right linear case, the left linear case being similar. We denote by n : ®k E ~ /\k E and by n' : ®k E* ~ /\k E* the canonical projections. By [2.21] the dual map n* : (/\k E)* ~ (®k E)* is injective. It then follows from the pedestrian way of constructing J [7.16] that we have the equalities 'I/J = J® 0 A, 'I/J = n* 0 and W = Jon'. We thus obtain the equality n* 0 Jon' = J® 0 A, i.e., the commutative diagram ®k E* (®k E)*
w,
~
J0 ok.
~'
1
NE*
~
J
r~* (N E)* .
The essence of the proof will be "running around this diagram." If E is f.g.p, it follows from [8.13] that J 0 is bijective, i.e., we may write A = J;S/ 0 n* 0 Jon'. Since n' is szt-graded skew-symmetric, we obviously have the equality n' 0 A = (k!) . n'. We now start running:
(k!)· n'
= n' oA = n'
0
(J;s;I on* oJ on') = (n' oJ;S;l on* oJ) on' .
Since n' is sUljective, this implies that we have the equality n' 0 J;S;l 0 n* 0 J = (k!) . id, and thus J must be injective. On the other hand, since each e E (/\k E)* represents an
54
Chapter 1. Qt-graded commutative linear algebra
szt-graded skew-symmetric map on Ek, it follows from (7.17) that we have the equality A 0J,s/ 01l'* = (k!) . J,s/ 01l'*. We now start running again:
(k!) .J,s;101l'*
= AoJ,s;l 01l'* = p,s;101l'*oJ01l")oJ,s;101l'* = J,s;l 01l'* 0 (J01l" oJ,s;l 01l'*) .
Since J,s; 1 01l'* is injective, we deduce the equality J 01l" 0 J,s; 1 01l'* = (k!) . id, and thus J must be surjective. IQEDI
8.17 Corollary. If E is jg.p, there is a natural identification Hornik(Ek, A) ~ /\k *E.
8.18 Summary. The following table summarizes some of the more interesting identifications, as well as where one can find sufficient conditions for the identification to be an isomorphism (a ~ indicates that it is always an isomorphism, and a '----+ indicates that it is always injective). In any case, if all szt-graded A-modules involved are f.g.p, and if the index sets are finite, all identifications are isomorphisms. identification
E&J (EBiEIFi) EBiEI Horns (Ei; F) EBiEI(Ei) EBiEI Horns(E; Fi ) ®i=n Horns (Ei; Fi ) Ei &J ... &J E~
N (E*) E F&JE* *E&JF
definition - isomorphism
'='"
EBiE/(E &J Fi )
[7.28]
'----+
Horns (EBiEI Ei ; F)
[7.2]
- [7.2]
'----+
Ei)* Horns(E; EBiEI Fi )
[7.2]
- [7.3]
[7.6]
- [8.6]
Horns(®~=l Ei ; ®~1 Fi )
[7.12] - [8.12]
(En &J ... &J Ed*
[7.12] - [8.l3]
(N E)*
[7.16] - [8.16]
'----+
-------
(EBiEI
*(E*)
~
(*E)*
[7.28]
[7.26] - [8.5]
HornR(E; F)
[7.11] - [8.8]
HornL(E;F)
[7.11] - [8.8]
Chapter II
Linear algebra of free graded A-modules
In chapter I we have studied general '21.-graded commutative algebra for an arbitrary abelian group '21. and an arbitrary '21.-graded commutative ring A. Starting this chapter we specialize to the case '21. = Z2 and we will abbreviate Z2-graded to simply graded. In order to get "close" to ordinary linear algebra, we also impose two conditions on the graded commutative ring A: it should contain the real numbers R as a sub ring and the nilpotent elements in A should form a supplement to RcA. This allows us to define the notions of basis and graded dimension for free graded A-modules. Except for subspaces and quotients (where one has to be a bit careful), these notions behave exactly as one would expect from ordinary linear algebra. Special attention has to be paid to matrices associated to linear maps. Whereas in usual linear algebra there is a single natural way to associate a matrix to a linear map when a basis has been given, in graded linear algebra there are three natural ways to do so. Each of these three ways has its own advantages and disadvantages. For instance, one of them is particularly adapted to express the graded trace of a linear map. The same representation by matrices is useful to define the graded determinant, also called Berezinian, ofa linear map. However, the proofthat the graded determinant is a group homomorphism is a bit cumbersome. Graded linear algebra is sometimes understood as meaning linear algebra of vector spaces (over R) that split as a direct sum oftwo subspaces: the even and odd parts. In our context that would be the special case in which one takes A to be equal to the real numbers R. However, it only requires a very small step to relate the general case to the special case A = R. This small step is the introduction ofthe notion ofan equivalence class ofbases. This idea allows us to reduce all discussions about free graded A-modules to discussions about bases. And then the only difference between a generic A and the special case A = R isjust the choice ofcoordinates with respect to a basis. 55
Chapter II. Linear algebra of free graded A-modules
56
1.
OUR KIND OF Z2-GRADED ALGEBRA
A
In this section we introduce the additional conditions we will imposefrom now on on the ring A: it should be a special kind of Z2-graded commutative R-algebra. The basic example ofsuch a ring is the exterior algebra ofan infinite dimensional real vector space [/.2J. The most important result of this section is the lemma that says that any finite number of nilpotent elements in A can be "killed" multiplicatively by a non-zero nilpotent element. We also introduce matrices with entries in A and derive some useful properties concerning nilpotent and invertible matrices.
1.1 Definition. Starting this chapter, we will be concerned mostly with Z2-gradings. This being the case, we will abbreviate Z2-graded to simply graded. As usual elements of parity 0 will be called even, but now elements of parity 1 will be called odd. When speaking of graded commutativity, we will always use ordinary multiplication in Z2 for the bilinear symmetric map (_1_) : Z2 x Z2 ~ Z2' For any (Z2-)graded commutative ring A we define the set of nilpotent elements Nby N = {a E A 1 :3k EN: ak = o}.
1.2 Examples. • The real line R is a graded commutative ring if we take AD = R and Al = {o} . • The complex line C is a graded commutative ring if we define AD = C and Al = {a}. Had we defined AD = R and Al = iR, the ring C would have been Z2-graded, but not graded commutative . • Let X be a real vector space (finite or infinite dimensional), and let A be the exterior algebra A = /\ X = EB~=o /\k X. It is a graded commutative ring if we define AD = EB~=o Nk X, Al = EB~=o N k+1 X and the wedge product as multiplication operation. We have 1 E N X ~ Rand N = EB~=I N X.
1.3 Lemma.
N = (N n AD)
EEl Al and
N is an ideal of A.
Proof The proof naturally splits in four steps. (i)If a E Al then the graded commutativity implies a· a = -a· a and hence a 2 = 0, i.e., Al C N. (ii) If n E N then we have 0= (no + nl)k = n~ + kn~-lnl (because of the graded commutativity and (i». Hence ngk = (-kn~-lnd2 = Pn~k-2ni = O. Thus n EN=} no EN. This proves the first assertion. (iii) If n, mEN then (n + m)k = (no + mo)k + k(no + mo)k-l(nl + ml). Since we know that no and mo are also nilpotent, it follows from the binomial formula that (n + m)k = 0 for sufficiently large k. Thus n, mEN =} n + mEN. (iv) If a E A and n EN then an = anD + anI. Since no is even, we have (ano)k = akn~ and thus anD is nilpotent. Since (anl)2 = anI anI = a (aD - al) ni = 0, anI is nilpotent too. Thus a E A, n EN=} an E N. A similar argument applies to na. This proves the second assertion. IQEDI
§1. Our kind of Z2-graded algebra A
57
1.4 Lemma. If nl, ... ,nN is a finite number of nilpotent elements in A, then there exists a non-zero homogeneous nilpotent n such that Vi : nni = O. Proof Without loss of generality we may assume that all ni are non-zero. Moreover, by considering the homogeneous parts of the ni separately, we may assume also that all ni are homogeneous. Define k i E N such that n7 i f- 0 and n7 i + 1 = O. Now define AD = 1 (N ota Bene. This is just formal, A need not have a unit) and Ai = Ai-l n7 i if Ai-l n7 i f- 0, Ai = Ai-l otherwise. Taking n = AN and using the graded commutativity, we find that the product nni can be written as Ainia for some a E A and hence by definition of ki we IQEDI have nni = O.
1.5 Definition. If A is a graded commutative ring, we denote by B the canonical projection B : A ~ A/N and call it the body map. Since N is an ideal in A, A/N is a ring and B a ring homomorphism. If a graded commutative ring A is an R-algebra, then we have in particular that AD, Al,N, and A/N are vector spaces over R and that B : A ~ A/N is a linear map between vector spaces over R. From now on, A will always denote a graded commutative R-algebra with unit such that A/N is isomorphic (as ring) to R. It follows that the body map B : A ~ A/N ~ R has a canonical section R ~ A given by r f--+ r·l (which is well defined because A is a vector space over R). By abuse of notation we will always identify r E R with r·l E A and write a = r + n, when B(a) = r E R, and n EN.
1.6 Examples .• The real line R itself verifies the conditions on A given in [1.5] . • The complex line Cwith Co = C and C 1 = {O} is a graded commutative R-algebra, but it does not verify the conditions of [1.5] . • If X is a vector space over R, its exterior algebra /\ X is a graded commutative R-algebraA verifying the condition of [ 1.5].
1.7 Remark. We restricted the symbol A to denote a graded commutative R-algebra with unit such that A/N ~ R. Although the choice of the field of real numbers is important in the next chapters, the results of this chapter remain valid if one replaces R by any other field of characteristic 0, e.g., C. Moreover, most of the analytic results of the subsequent chapters also remain valid when replacing R by C.
1.8 Discussion (Geometric interpretation of [1.4]). If A is the exterior algebra of a vector space X as in [1.2], [1.6], we can give a geometric interpretation of the element n of [104]. Let ni be a non-zero nilpotent element in A, then by definition of the exterior product as the direct sum of exterior powers of X, ni is a finite sum of monomes, each of which is a finite wedge product of vectors in X. Since there is also a finite number of ni's, we find a finite number of vectors in X that are involved in the definition of the ni's.
58
Chapter II. Linear algebra of free graded A-modules
These vectors span a finite dimensional subspace of X; the wedge product of vectors that form a basis of this finite dimensional subspace is an element n E A which satisfies the requirements of the lemma.
1.9 Definitions. The set M(m x n, A) denotes the set of all matrices of size m x n (m being the row dimension and n the column dimension) with entries in A. The usual matrix multiplication M(m x n, A) x M(n x r, A) __ M(m x r, A) still makes sense on these sets. The body map B extends in a natural way to these matrices: B: M(m x n, A) -- M(m x n, R); it is surjective and preserves matrix multiplication. The set M(n x n, A) equipped with matrix multiplication is a ring with unit In (in terms of matrix elements (In); = S} with S} the Kronecker delta) for which the body map B: M(n x n,A) -- M(n x n,R) is a surjective ring homomorphism. Fora E M(m x n, A) we define the rank ofa, denoted as rank(a), as the rank of its body Ba E M(m x n, R) : rank(a) = rank(Ba), i.e., as the number of independent rows or columns in Ba.
1.10 Lemma. Let a E M (n x n, A) such that B (a)
= 0,
then a is a nilpotent matrix.
Proof The condition B(a) = 0 implies that each entry of the matrix a lies in the kernel of the body map, i.e., is nilpotent. It follows that the homogeneous parts of the entries are also nilpotent. Since there are 2n 2 homogeneous entries (i.e., a finite number), there exists a number N such that aN = 0 for all homogeneous entries a of a. Now the entries of the matrix a k are sums of terms, each of which is a k-fold product of homogeneous IQEDI entries of a. Hence for k > (N - 1)· 2n 2 we have a k = o.
1.11 Lemma. An element a E M(n x n,A) is invertible ifandonlYlfDet(B(a))
=I-
o.
Proof If b is the inverse of a in M(n x n, A), then B(b) is an inverse for B(a) in M(n x n, R) and hence Det(B(a)) =I- O. Conversely, suppose Det(B(a)) =I- 0 and write a = B(a) + nwith B(n) = O. Writing x for the inverse ofB(a) in M(n x n, R), we define b = L:~=o( -xn)kx . This is actually a finite sum because of [1.10]; moreover, one IQEDI easily verifies that b is an inverse to a in M(n x n, A).
2.
FREE GRADED A-MODULES
In this section we define the graded dimension ofafree gradedA-module. It consists of two integers: the number of even respectively odd vectors in a homogeneous basis. We show that this is an invariant ofa (finite dimensional)free graded A-module. More precisely, up to isomorphism there exists only onefree graded A-module of graded dimension plq. These results depend crucially on the conditions we imposed on A in [1.5].
§2. Free graded A-modules
59
2.1 Definition. In this section and the subsequent ones we will be mainly interested in free graded A-modules F( G, c) (on homogeneous generators G). We recall that all free graded A-modules admit a homogeneous basis (e.g., G). We will call a free graded A-module finite dimensional if it admits a finite (homogeneous) basis. A homogeneous basis el, ... , en of a finite dimensional free graded A-module is called ordered if all even vectors come first, i.e., ei even and ej odd implies i < j. A subset F of a free graded A-module E is called a graded subspace if it is a graded submodule of E that in itself is (isomorphic to) a free graded A-module.
2.2 Nota Bene. Any finite dimensional free graded A-module is in particular f.g.p, and thus the identifications given in [1.8.18] apply.
2.3 Remark. Since A is in particular an R-algebra with unit, it follows immediately that any A-module, and thus any free graded A-module, is also a vector space over R. Anticipating the fact that the dimension is an invariant of a free graded A-module, it is immediate that the dimension of a free graded A-module E as vector space over R equals the product of the dimension of E asfree graded A-module and the dimension of A as vector space over R. Since our main example for A is the exterior algebra of an infinite dimensional vector space, any free graded A-module (over this A) is infinite dimensional as vector space over R.
2.4 Example. We know already that A is (trivially) a graded A-module. The unit 1 E A is both generating and independent, hence A admits a basis. Since moreover 1 is even, A is a free graded A-module. Note however that not every non-zero element is a basis for A: if a =I- 0 is nilpotent, then a is not generating (l can not be written as ba with b E A), nor is it independent (if an =I- 0, a n +1 = 0 then an a = 0 is a relation with a non-zero coefficient) . More generally, the graded A-module An is a finite dimensional free graded A-module because the vectors (1,0, ... ,0), (0,1,0, ... ,0), ... , (0, ... ,0,1) are generating, independent and even.
2.5 Lemma. Letel, ... , en, iI, ... , 1m be elementsofagradedA-module E, and suppose that there exist ai j E A such that Ii = 2: j ai j ej. If the iI, ... ,1m are independent, then
rank (Baij){;i: ...... :::. = m. In particular m :::; n. Proof If rank(Baij) < m, then there must exist a relation among the rows of Baij, i.e., there exist ri E R, not all of them zero, such that for all j : 2:i riB(aij) = O. We then define iJ,i = 2:i riaij E A and note that B(iJ,i) = 0, i.e., that the iJ,i are nilpotent. Thus, by [104], there exists a non-zero no E A such that Vj : no f.1j = O. But then 2:i(n O ri)Ii = 2:i,i no(riaij) ej = 2: j no f.1 j ej = 0, contradicting the independence of the k We conclude thatthe rank ofBaij is m and hence m :::; n. IQEDI
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Chapter II. Linear algebra of free graded A-modules
2.6 Proposition. Let el, ... , en be a basis of a graded A-module E.
(i) If iI, ... , f m is another basis of E, m = n. (ii) All other bases {iI, ... , f n} are classified by invertible matrices a E M (n x n, A) with fi = 2:: j aij ej. (iii) If iI, ... , fn is either generating or independent, it is a basis. Proof • Since the fi and ej form a generating subset, there exist ai j , bj i E A such that fi = 2:: j aij ej and ej = 2::i b/ k Since the fi and ej are independent, it follows from [2.5] that n :s; m and m :s; n . • If iI, ... , f n is a basis, by the preceding argument ::Jai j E A such that fi = 2:: j aij ej and rank(Baij) = n. Thus by [1.11] aij is invertible. On the other hand, if fi = 2:: j aijej with an invertible ai j , then it follows easily that the fi form a basis . • If the fi are independent, there exist aij E A such that fi = 2:: j aijej. By [2.5] we know that rank(BaJ) = n, and thus by [1.11] aij is invertible. By the preceding result the fi form a basis. If the fi are generating, there exist b/ E A such that ej = 2::i b/ k As before b/ must be invertible and thus (by applying the inverse of b/) the fi form a IQEDI basis.
2.7 Counter example. To show that not all graded A-modules are free graded A-modules, consider the free graded A-module A2 and choose a non-zero even nilpotent element /-L E Ao. We then define the subset F c A2 by F = Span( {(I, 0), (0, /-L)}), which is a graded submodule by [1.1.25]. We will show that F does not admit a basis, showing at the same time that not every graded A-module is a free graded A-module, and that not every graded submodule of a free graded A-module is again a free graded A-module. Clearly { (1, 0), (0, /-L) } is a generating set and hence a basis of F (if it exists) comprises at most 2 elements. Now let us first suppose that a single element { (a, b/-L) } is a basis, then there exists K, E A with (1,0) = K,( a, b/-L) = (K,a, K,b/-L) and hence K, must be invertible. It follows from [2.6-ii] that (l,0) is a basis, which is clearly false (it is not generating). Let us suppose next that { (a, b/-L), (c, d/-L) } is a basis, then there exist K" A E A with (1, 0) = K, ( a, b/-L) + A(c, d/-L) = (K,a + Ac, K,b/-L + Ad/-L). Since 1 is not nilpotent, it follows that not both a and c are nilpotent. Hence by applying [2.6-ii] with a suitable choice of an invertible matrix in M (2 x 2, A) we may assume without loss of generality that a = 1 and c = O. But then /-Lk-l (c, d/-L) = (0, d/-Lk) = (0,0), where k is chosen such that /-Lk-l -=f- 0 and /-L k = O. Since this contradicts the independence, we conclude that F does not admit a basis in the sense of a subset that is both generating and independent.
2.8 Lemma. Let E be a graded A-module that admits a basis consisting offinitely many elements. Then E admits a homogeneous basis, i.e., E is a finite dimensional free graded A-module. Proof Let {iI, ... , f n} be a basis of E and fi = fiD + fil the decomposition in homogeneous parts. Since (Ii) is a basis, there exist matrices a, b E M(n x n, A) such that
61
§2. Free graded A-modules
j j iiD = L: j ai Ii and iiI = L: j bi Ii· Since the ii are independent we have a j j j Decomposing ai = ai 0 + ai 1 into homogeneous parts we find iiD
= "2)ai
j
oijo
+ b = In.
+ ai j dj!)
j
(the other two terms necessarily cancel because iiD E Eo). Resubstituting the expressions . k . k . k . for iiD and iil we fmd L:k ai ik = L:jk(a/Oaj + ail1bj )ik. or (smce the ii are independent) in terms of matrices:
If we realize that B (aij) = B (ai j 0) and B (ai j 1) = 0, we deduce from this equation B(a) = B(a)B(a) i.e., B(a) is a projection. Standard linear algebra then shows the existence of a real matrix X such that XB(a)X-1 = (I; ~) with :s; p :s; n. Since X is invertible,. g; = L: j Xij Ii is also a basis of E. Defining the matrix c = X aX-I, j n1 we have giO = L: j Ci gj and B(c) = (I; ~). It follows that c = (Ip:3 ~:), where the ni are matrices of the appropriate size with nilpotent entries. We finally define the j vectors ei by ei = giO if i :s; P and ei = gil if i > p. It follows that ei = L: j di gj with d = (I~+nl I n-p n~ n4 ). Since B(d) = In, d is invertible and (ei) is a basis, which n3 obviously consists of homogeneous elements. IQEDI
°
2.9 Proposition. The number of even vectors in a homogeneous basis ofafinite dimensional free graded A-module E is an invariant of E.
Proof Let { e1, ... , en } and {e1' ... , en } be two ordered homogeneous bases with p and peven elements respectively. These two are related by an invertible matrix a with ei
=
n Laijej j=l
=
p (Laijoej j=l
+
n j L ai 1e j) j=p+1
+
p j (L a i 1e j j=l
+
n L aijOej). j=p+1
Independence of the (ei) and homogeneity of (ei) and (ei) then show that ai j is even if i :s; Pand j :s; p or if i > pand j > p; it is odd otherwise. It follows that B (a) = (~ ~ ) where b is a real matrix of size p x p. Since such a real n x n matrix has necessarily determinant zero if p =I- p, it follows from [1.11] that p = p. IQEDI
2.10 Definition. The graded dimension of a finite dimensional free graded A-module E is a pair of integers (p, q) where p, called the even dimension ofE, is the number of even vectors in a basis for E and q, called the odd dimension ofE, the number of odd vectors in (the same) basis. We usually denote this as dim(E) = plq; in particular dim(An) = niO. If the graded dimension of E is plq, we define the total dimension of E as the sum n = p + q, also denoted as dim(E) = n. In general, when we use the word dimension, we will mean the graded dimension, unless it is clear from the context that we mean the total version (which is notably the case when we talk about finite dimensional spaces).
62
Chapter II. Linear algebra of free graded A-modules
2.11 Proposition. Two finite dimensional free graded A-modules E and F are isomorphic ifand only if they have the same graded dimension.
Proof Let 1> : E -- F be an isomorphism and B a homogeneous basis of E. Then 1>( B) is a homogeneous basis for F. Since the parity is not changed by the even map 1>, it follows that dim(E) = dim(F). On the other hand, suppose dim(E) = dim(F) and let (ei) and (fi) be ordered homogeneous bases of E and F respectively. Then the map 1> defined by (( L:i aiei 111>) = L:i ai Ii is an even bijective (left) linear map, i.e., an isomorphism IQEDI between E and F.
2.12 Remark. If one extends the definition of graded dimension to arbitrary free graded A-modules as yielding a pair of cardinals, then [2.11] remains true; even the proof only needs cosmetic changes. Note that our definition of an isomorphism as an even invertible map is crucial in the invariance of the graded dimension. Had we defined an isomorphism as just any invertible map, the graded dimension would not have been an invariant.
2.13 Definition. On a graded A-module E = Eo tB El we define an involution
2.14 Lemma. The involution
2.15 Remark. We could have defined an involution
rE
2 ,
where (J
§5. The graded trace and the graded determinant
75
5.3 Construction (graded trace). Let E be a finite dimensional free graded A-module and consider the space *E 0 E. As we know, it is isomorphic to EnddE). To this space we cannot apply directly the contraction of E with *E, but after interchanging the factors, we can. We thus obtain an even linear map gtrL = Er 0 91: EnddE) ~ A- For homogeneous elements e E E and 1> E *E this map is given by
Another way to obtain this map is as the linear map induced by the even bilinear map *E x E ~ A defined on homogeneous elements by (1), e) 1--7 (-1) (e(4» lee e)) (( e 111> )). The map gtr L is usually called the (left) graded trace of a left linear endomorphism. The (right) graded trace gtrR is defined similarly: gtrR = rE 091: EndR(E) ~ AFor homogeneous elements e, 1> it is given as gtrR(e 01» = (_l)(e(e)le(4»)1>(e).
5.4 Proposition. Foranyfinitedimensionalfree graded A-module E we have the identities gtrL = rE 0 ('I'E 0 id(E)) and gtrR = Er o (id(E) 0 't::-J).
Proof Let (ei)f=l be a homogeneous basis of E. The first identity follows from the fact that both maps send the basis element ie 0 ej to (-1) (e(ei)le(ej)) 53 = (-1) (e(ei)le(ei)) 53' The second identity follows from a similar argument.
IQEDI
5.5 Corollary. gtrR o'tEndL(E) = gtrL' 5.6 Lemma. For a finite dimensional free graded A-module E, a homogeneous basis (ei)~l of E and 1>s E Ends(E) we have
Proof We prove the formula for gtrd1>d, the case gtrR(1)R) being analogous. First note that 1> L = L:i,j ie 0 !VJl (1) L)i j . ej = L:i,j ie 0 ej . ([.e(ej) (!VJl (1) L)i j ), and thus:
i,j
= 2)_1)(e Ce)le(e j ))5; ([.e(ej) (!VJl(1>L)i j ) ij
5.7 Discussion. It will be clear from the formula; for the graded trace that in both the left and right case the operation on the corresponding matrix is the same: a sum over diagonal
76
Chapter II. Linear algebra of free graded A-modules
elements, adding a sign and applying the involution
gtrs(1)s) = tr(As) - is even, both gtr s apply. It follows immediately from [5.5] that gtr R (1)) = gtr L (1)). But the same result also follows from the facts that (i) for even 1> the submatrices A and D have only even entries, (ii) the ordinary trace is invariant under ordinary transposition, and (iii) formula (4.10). For even 1> the formula for the graded trace reduces to the formula gtrs(1)) = tr(A) - tr(D).
5.8 Counter example. One might be tempted to think that gtr Land gtr R give the same result when a map is both left and right linear. The following example shows that this need not be the case. Suppose A = /\ X, where X is a finite dimensional vector space over R. Let (xi)f=l be a basis of X and denote 0 -=f- b = Xl 1\ ... 1\ Xn E /\n X. Consider furthermore the free graded A-module E of dimension 011 with odd basis vector el. Finally define the map 1> : E ~ E by
This map is obviously right linear, but is also left linear! To see this, let A E A be odd, then 1>(Aela) = 1>( -eIAa) = -elbAa = 0, because for any odd A E A the product bA = O. Similarly A1>(ela) = O. And thus 1> is also left linear. However, ifn is odd, elb = -bel, and thus we find gtr R (1)) = -
5.9 Lemma. If E and F are finite dimensional free graded A-modules, then we have for homogeneous 1> E Homs(E; F) and 1fJ E Homs(F; E) the equality
IfE
= F this
equality can be written as gtrs([1>, 1fJ]s)
= o.
Proof In order to prove the right linear case (the left linear case being similar), we consider the free graded A-module HomR(E; F)0HomR(F; E) 2'! F0E*0E0F*. Theequality we have to prove amounts to proving that the maps gtr R 0 r E and gtr R 0 r F 091(23) defined on this tensor product with values in A are the same. But this follows from the fact that both map a basis vector fi 0 e j 0 ek 0 fe to the same value
where the last equality follows from the Kronecker 8's.
§5. The graded trace and the graded determinant
77
5.10 Remark. Let E be a free graded A-module of dimension plq. It is an elementary exercise to prove that gtrs : Ends(E) -- A is the unique even linear map such that (i) gtrs(id(E)) = p - q, and (ii) gtrs([1>, 1/J]s) = 0 for all 1>, 1/J E Ends(E).
5.11 Definition. The ordinary determinant, defined for square matrices with real coefficients, is a polynomial function in the matrix entries. It follows that we can extend the determinant to square matrices with coefficients in any commutative ring, yielding a value in this ring. Since Ao is a commutative ring, we thus can extend the determinant function to M(n x n, Ao). We note that the determinant so extended preserves most of its properties, in particular that it is a ring homomorphism and that it can be calculated by expansion according to a row or column.
5.12 Lemma. Let A E M (n x n, Ao) be an invertible matrix, and let V E M (n xl, Ao), WE M(l x n,Ao) anda E Ao = M(l x 1,Ao) be arbitrary. Then
Det
Proof
(it
~) = (a -
WA-IV). Det(A) .
(~~) = (W~-l ~)(~~) andDet(W~_l~) = Det
e; a_W~-lv)·IQEDI
5.13 Lemma. For V E M(n x 1, Ad and WE M(l x n, AI) we have Det(In + VW) = (1 + WV)-I . Proof First note that VW E M(n x n, Ao) and hence that the determinant function can be applied. The actual proof is by induction on n. For n = 1 we have Det(l + VW) = 1 + VW = 1 - WV = (1 + WV)-I (because V 2 = W 2 = 0). For n = + 1 we define V and W to be the column vector consisting of the first entries of V respectively the row vector consisting of the first entries of W With these notations we have
n
n
In
n
+ VW = (I; + VW VnW
VWn ) . 1 + VnWn
We then can apply the previous lemma and the induction hypothesis; we obtain
Det(In + VW) = (1
+ VnWn - VnW(I;+ VW)-IVWn )· Det(I;+ VW) 00
= (1 + VnWn - VnW(2:)-ll(VW)k)VWn)' (1 + WV)-I 00
= (1
+ Vn(2:)-ll(WVl)Wn)·
(1
+ WV)-I
+ WV)-IWnVn ) . (1 + WV)-I = (1 + (1 + WV)-IWnVn)-I. (1 + WV)-I = (1 + Wv + WnVn)-1 = (1 + WV)-I .
=
(1 - (1
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Chapter II. Linear algebra of free graded A-modules
5.14 Corollary. Let A E M (n x n, AD) be an invertible matrix, and let V E M (n xl, AI) andW E M(l x n, AI) be arbitrary. ThenDet(A+ VW) = Det(A). (1 + W A-1V)-1.
5.15 Lemma. For X
E
M(p x q, AI) and Y Det(Ip
E
M(q x p, Al)we have
+ XY) . Det(Iq + Y X) = 1 .
Proof The proof proceeds by induction on p; for p = 1 it is [5.13]. Ifp = p+ 1 we define X E M(px q,A 1), Y E M(q x p,A 1), V E M(q x 1,A 1) and WE M(l x q,A 1)
such that X = ( ~) and Y = (Y V). With these definitions we have
Ip
+ XY =
( I~+WYXY ~
p
XV) l+WV
Iq
and
+ YX = (Iq + YX) + VW .
Using [5.12] we find
Det(Ip
+ XY) = Det(Ip+ XY) . (1 + WV - WY(Ip+ Xy)-l XV) = Det(I~+ XY) . (1 + W(Iq + YX)-l V) . p
Using [5.14] we find Det(Iq+ Y X) = Det(Iq + YX). (1 + W(Iq + YX)-l V)-l. Since AD is commutative the result now follows. IQEDI
5.16 Definition. We define the graded determinant gDet : Gl(piq, A) -- AD (also called the Berezinian) by the following procedure. For X = (~ ~) E Gl(piq, A) we know that the entries of A and D are even and those of Band C odd. This plus [1.11] imply that both A and D are invertible matrices. gDet(X) is then defined as
gDet(X)
= Det(A - BD-1C) . (Det D)-I.
5.17 Proposition. The map gDet : Gl(piq, A) __ AD is a homomorphism. Proof Given two such matrices X
gDet(XX)
= (~
D) and X = ( ~ B), we compute: B
~
~
~
= Det(AA + BC - (AB + BD)(CB + DD)-l(CA + DC)) x Det(CB + DD)-l
= Det(AA + BC - (ABD- 1 + B)(D-1CBD- 1 + Iq)-1(D-1CA + C)) x Det(D)-l . Det(D)-l . Det(D-1CBD- 1 + Iq)-l .
79
§5. The graded trace and the graded determinant
We introduce the matrices Z
= D-1C and Y = jjD-l
and compute:
(AjjD- 1 + B)(D-1CjjD- 1 + Iq)-l(D-lCA + C)
= (AY + B)(ZY + Iq)-l(ZA + C) = AY(ZY + Iq)-l ZA + B(ZY + Iq)-l ZA + AY(ZY + Iq)-lC + B(ZY + Iq)-lC 00
= A(2) _l)k(YZ)k+l)A + BZ(YZ + Ip)-l A + A(YZ + Ip)-lyC k=O + B(ZY + Iq)-lC and thus:
AA + BC - (AjjD- 1 + B)(D-1CjjD- 1 + Iq)-l(D-lCA + C) 00
= A(Ip - 2)-ll(YZl+1)A - BZ(YZ + Ip)-l A - A(YZ + Ip)-lyC k=O 00
+B(Iq - 2)-ll(zY)k)C k=O = A(YZ + Ip)-l(A - YC) - BZ(YZ + Ip)-l A + BZ(YZ + Ip)-lyC
= (A - BD-1C)(YZ + Ip)-l(A - jjD-1C) We thus finally find
gDet(XX) = Det(A - BD-1C). Det(YZ + Ip)-l. Det(A - jjD-1C) x Det(D)-l . Det(D)-l . Det(Iq + Zy)-l
= gDet(X) . gDet(X) . (Det(YZ + Ip) . Det(Iq + Zy))-l = gDet(X) . gDet(X) because of [5.15]
5.18 Remark. Using the decomposition (~ ~) = ( ~ ~) [5.15], it is elementary to show that we have the equalities
A gDet ( C
1
. (D!1C A-I B),
[5.17], and
B) _ Det(A - BD-1C) Det(A) D = Det(D) = Det(D - CA-IB) .
5.19 Definition. Let E be a finite dimensional free graded A-module of dimension plq, then we extend the definition of graded determinant to Aut(E) by the following procedure. Choose an ordered homogeneous basis (ei) for E and define
gDet(¢) = gDet(Nrs(¢)) .
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Chapter II. Linear algebra of free graded A-modules
5.20 Proposition. The value of gDet( ¢) does not depend upon the choice of NIL or NI'R> nor upon the choice of the basis (e;).
),
Proof Let (ei) be an ordered homogeneous basis of E, then, using the block matrix form NI (¢) = (~= ~= we have gDet(NIL(¢)) = Det(AL - BLDr;ICL). Det(DL)-1 and
s
gDet(NIR(¢))
=
Det(AR - BRDi/CR) . Det(DR)-I. But the equality of these two quantities is immediate if we use (4.9), realize that the ordinary determinant is invariant under transposition, and realize also that C Lt(DL t)-l BL t = -(BLDr;ICL)t because Band C have odd entries that produce a minus sign when interchanging. The fact that gDet (¢) does not depend upon the choice of the basis is an immediate consequence of [5.17], [4.15] and [4.17]. IQEDI
5.21 Corollary. The map gDet : Aut(E) morphism.
~
Ao is a well defined multiplicative homo-
5.22 Remark. The graded determinant gDet is a generalization of the ordinary determinant in the sense that it is the integrated form on the A-Lie group Aut(E) of the graded trace on the A-Lie algebra Ends(E) (see [III.3.14] and [VI.2.1S]), just as the ordinary determinant is the integrated form of the usual trace.
6.
THE BODY OF A FREE GRADED A-MODULE
In this section we discuss in detail the relation between a generic A and the special case A = R. We define, in analogy with the case of the ring A, the set ofnilpotent vectors in a free graded A-module. This allows us to extend the body map B tofree graded A-modules, yielding free graded R-modules (i. e., a direct sum of two ordinary vector spaces over R). We then introduce the notion ofequivalent bases and we define an A-vector space to be a free graded A-module together with an equivalence class ofbases. Restricting the notion ofa subspace slightly, we then can prove that there is "no" difference between a generic A and the special case A = R. All is encoded in terms ofa basis, and the only difference is what kind of coefficients one puts in front of basis vectors.
6.1 Discussion. As has been said before, the real numbers perfectly fit the conditions imposed on A in [1.5], so let us startthis section with a discussion what happens if we use A = R. It will be obvious that a graded R-module is a vector space over R in the usual sense, but ... there is more to a graded R-module than that. A graded R-module splits into an even and an odd part, both of them vector spaces over R. More precisely, if X is a graded R-module, then it defines two subspaces Xo and Xl such that X = Xo EEl Xl. On
§6. The body of a free graded A-module
81
the other hand, the reader can convince himself easily that, given a pair (Xo, Xd of vector spaces over R, the vector space X = X 0 EEl Xl is a graded R-module when we define X a to be the part of parity cx. We conclude that graded R-modules are nothing more nor less than vector spaces over R together with a splitting into a direct sum of two subspaces. In particular, a graded R-module is always a free graded R-module. However, instead of taking A = R, we could have used any A to obtain R, just by taking the body map B : A ~ R. A natural question now is, can we obtain graded R-modules in a similar way from any graded A-module? As we will see, the answer is positive when we restrict our attention to free graded A-modules.
6.2 Definition. For a free graded A-module E we define the set NE of nilpotent vectors by N E = {x EEl :3a E A : a # 0 & ax = 0 }.
6.3 Lemma. NE consists of those elements in E that have nilpotent coefficients with respect to any basis of E. Moreover, NE is a subspace (over R) of E, ANE e NE and
NE eNE . Proof Let GeE be a basis for E and let x E E, then there exists eI, ... , en E G and aI, ... , an E A such that x = 2:~ 1 a i ei. If all the a i are nilpotent, it follows from [1.4] that x ENE. On the other hand suppose x E NE,i.e., there exists an a E A,non-zero such that ax = O. It follows from the properties of a basis that then Vi = 1, ... ,n : aa i = O. If a i is not nilpotent, it is invertible and we deduce that a = 0, in contradiction with the IQEDI hypothesis. This proves the first part of the lemma; the rest follows easily.
6.4 Counter Example. Of course the definition of nilpotent vectors can be given for any graded A-module. And although the results of [6.3] remain valid for any graded submodule of a free graded A-module, they are not valid in the more general setting of arbitrary graded A-modules. To appreciate the problem, consider the graded commutative ring A with Ao = R and Al = X, where X is a vector space over R of dimension at least 2, and where the multiplication in Al is trivial, i.e., AI' Al = {O}. This A verifies the conditions given in [1.5]. In X we choose two independent vectors x and y, and we consider the sub module F ofA2 generated by the two vectors (x, 0) and (0, y). Since these vectors are homogeneous (they are odd), F is a graded sub module of A2, and thus the quotient E = A2 / F is a graded A-module. We claim that for this E the subsetNE is not a subspace (over R) of E. If7r denotes the canonical projection 7r : A2 ~ E, the vectors 7r(l,O) and 7r(0, 1) belong to NE because X· 7r(1, 0) = 7r(x, 0) = 0 and y. 7r(0, 1) = 7r(0, y) = O. However, we claim that 7r(1, 1) does not belong to NE. For suppose it did, then there should exist 0# a E A such that a . 7r(1, 1) = 7r(a, a) = 0, i.e., (a, a) E F. This implies that a must be a multiple of both x and y (remember that multiplication in Al = X is trivial), and thus must be zero. Since this contradicts the assumption a # 0, we conclude that 7r(1, 1) does indeed not belong to NE.
Chapter II. Linear algebra of free graded A-modules
82
6.5 Definition. We extend the notion of the body map B to any free graded A-module E as being the canonical projection (in terms ofreal vector spaces) B : E -- E/NE. It maps the free graded A-module E into a vector space over R, called the body of E. Note that the definition ofB for the free graded A-module A coincides with the original definition of body map just because of our requirement A = REB N, i.e., NA = N.
6.6 Lemma. Let E be a free graded A-module, thenfor all a E A andfor all x E Ewe have B(ax) = (Ba)(Bx). Moreover, ifG is a set of homogeneous elements in E, then G is an independent set in E ifand only if BG is an independent set (over R) in BE.
= (Ba)(Bx) = B((Ba)x) (the last equality because B is linear over R), we have to show that ax - (Ba)x = (a - Ba)x ENE. But a - Ba E A is nilpotent, from which we deduce that indeed ax - (Ba)x ENE. For the second assertion, let us first assume that G is independent and let us suppose that there exist r1, ... , rn E Rand e1, ... , en E G with 2:7=1 ri(Bei) = 0, or equivalently 2:7=1 riei ENE. Combining [3.3] and [3.7], we may assume thate1, ... , en are elements in a homogeneous basis of E. It thus follows from [6.3] that the coefficients ri are nilpotent. Since they are also real by hypothesis, they must be zero. In the other direction, suppose BG is independent over R and suppose there are aI, ... , an E A and e1, ... , en E G with 2:i aiei = O. Applying B and the independence of BG shows a i E N. Now let B be a homogeneous basis for E, then there exist II, ... , f m E Band bi ] E A such that ei = 2:] bi ] Ii. Again applying B and using the independence of BG and BB (just proved!) then shows that m 2 n and that the matrix Bb i ] contains an invertible n x n submatrix. Without loss of generality we may assume that it is the submatrix (Bbi])~]=l' From 0 = 2:i aiei = 2:i,] a i bi ] Ii and the independence of B we deduce that Vj : 2:i a i bi ] = O. But by [1.11] the matrix bi ], j :::; n is invertible, and hence a i = 0, finally proving that G is an independent set. IQEDI Proof To show the relation B(ax)
= EoEBE1 isafreegradedA-module, then BE = (BEo)EB(BE1) is a (free) graded R-module of the same graded dimension; in particular B(Eo.) = (BE)o..
6.7 Corollary. If E
6.8
Lemma. Given any (left or right) linear map 1> : E -- F between two free graded A-modules, there exists a unique linear map of (free) graded R-modules B1> : BE __ BF making the following diagram commutative:
E
~
F
BE
-----4
BF .
B.p
If X : F -- G is a linear map, then B (X 0 1» parity a then B1> is too.
=
(BX)
0
(B1». If 1> is homogeneous of
§6. The body of a free graded A-module
83
Proof If B¢ exists, it must be defined by (B¢)(Bx) = B(¢(x)). From this formula, linearity and uniqueness follow immediately. It thus remains to show that ¢ exists. For that it suffices to show that B(¢(x)) = 0 whenever Bx = O. To see this, choose a basis for E. The vector x has nilpotent coefficients with respect to this basis, and thus by linearity of ¢ its image is a sum of terms, each a product of a vector of F and a nilpotent coefficient. From this it follows that ¢( x) is a nilpotent vector in F. The equality B (X 0 ¢) = (BX) 0 (B¢) follows from the defining equation for B on maps. The last statement follows immediately IQEDI from the fact that (BE)o = B(Eo) (and idem for F).
6.9 Remark. The above result can be rephrased as saying that the map B is a functor from the category of free graded A-modules with its linear maps (left or right) to the category of (free) graded R-modules with its linear maps (for which the distinction left and right does not exist).
6.10 Remark. If E is a finite dimensional free graded A-module, Homs(E; F) is also a free graded A-module. In that case we have two different definitions of B¢ for a linear map ¢ : E ~ F: the one of [6.8], but also the one as given by application of the map B to the free graded A-module Homs(E; F). In the first case B¢ lies in Homs(B(E); B(F)) and in the second case in B(Homs(E; F)). Declaring these two definitions to coincide gives us an isomorphism between these two spaces. Actually, this is a special case of the more general fact that the body map B "preserves" all our constructions offree graded A-modules, e.g., 0 k (BE) is isomorphic to B( 0 k E), where the first tensor product is that of graded R-modules and the second that of free graded A-modules. The proof (using homogeneous bases) of these and similar isomorphisms are left to the reader (see also [6.26] and [6.27]).
6.11 Discussion. Since any free graded A-module E is a vector space over R, it follows from [6.6] thatB : E ~ BE is a linear map between vector spaces over R whose kernel is the linear subspace N E . However, no canonical choice for a supplement (over R) to NE presents itself. The purpose of the next definition is to create such a supplement.
6.12 Definition. Two bases (ei)iEI and (fj )jEJ of a given free graded A-module E are said to be equivalent if they are related to each other by real coefficients, i.e., if there exist aij E R such that ei = L: j aij Ii. Reflexivity and transitivity are obvious. For symmetry, suppose Jj = L:i bj iei with bj i E A. From Ii = L:i k b/a/ Jk it follows that L:i a/ = 5j. Taking bodies, we obtain the equation L:JBbj i)a/ = 5j. Defining
b/
Jj = L:i(Bb/)ei, we find Jj = L:i,k(Bb/)a/ Jk follows that the bj i are necessarily real.
= L:k 51!k = h
From this it
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Chapter II. Linear algebra of free graded A-modules
6.13 Discussion. Specifying a basis B for a free graded .A-module E automatically gives a supplement RE for NE (over R) defined by RE = SpanR(B). It will be obvious that an equivalent basis defines the same supplement RE. We thus obtain a map from the set of equivalence classes of bases to the set of supplements for N E . This map is injective but in general not surjective [6.15].
6.14 Lemma. If E is a finite dimensional free graded .A-module, then any supplement
RE forJVE defines an equivalence class of bases of which RE is the real span. Proof By definition of a supplement, B : RE __ BE is an isomorphism of vector spaces over R. Let B be a homogeneous basis for the R-vector space BE and denote by jj C RE its image under B- 1 . Imitating the proof of [6.6], we see that jj is independent in E (using that is independent but not needing that jj is homogeneous in E). Combining [6.7], [2.6-iii], and the finite dimensionality, we conclude that jj is a basis whose real span is RE. IQEDI
BB
6.15 Counter example. To show that the condition of finite dimensionality is not void, let us consider the infinite dimensional free graded .A-module E of [3.9] with its basis {ei liE N} and the vectors fi = ei - niei+1. Since the ni are nilpotent, we have Bei = Bfi, Since the Bei form a basis of BE, it follows that the fi generate a supplement (over R) for NE. However, the fi do not form a basis for E as shown in [3.9].
6.16 Corollary. For finite dimensional free graded A-modules equivalence classes of bases can be identified with supplements RE (over R)for N E.
6.17 Discussion. So far in talking about supplements for NE we have interpreted E as an ordinary vector space over R. However, we can interpret E as a graded R-module because Eo andE1 are subspaces of E (over R). Moreover,NE is a graded subspace of the graded R-module E, i.e., NE = (NE n Eo) EEl (NE n Ed. It thus is reasonable to require that a supplement RE to NE is a graded subspace. If that is the case, the map B : RE -- BE is an isomorphism of graded R-modules. A little reflection will show that a supplement RE generated by an equivalence class of bases is a graded subspace (of E, seen as a graded R-module) if and only if the equivalence class contains a homogeneous basis.
6.18 Counter example. To show that an equivalence class need not always contain a homogeneous basis, consider the free graded .A-module E of dimension 111 with homogeneous basis {e, fl. Let MEND be non-zero, then the vectors = e + Mf and = f + Me form a basis for E whose equivalence class does not contain a homogeneous basis.
e
f
§6. The body of a free graded A-module
85
6.19 Definition. By an A-vector space E we will mean afinite dimensional free graded A-module together with an equivalence class of bases whose associated supplement RE to NE is a graded supplement (in other words, the equivalence class should contain a homogeneous basis). Moreover, when using a basis for E, it will always be a (homogeneous) basis within the given equivalence class. If E is an A-vector space, the supplement RE is part of the A-vector space structure. We thus can use the isomorphism B : RE -- BE to identify BE with RE c E and to forego the notation RE. This means in particular that we see the map B as a projection B : E -- E whose kernel is NE and whose image is BE == RE c E. The image BE can be described as those points of E that have real coordinates with respect to a basis in the equivalence class.
6.20 Remark. There is no special reason to require A-vector spaces to be finite dimensional. However, since we do not need infinite dimensional ones, our definition avoids having to say all the time that they are supposed to be finite dimensional.
6.21 Nota Bene. In case A = R we obtain the definition of an R-vector space. The reader should not confuse this with the notion of a vector space over R. Since for a graded R-module there exists only one equivalence class of bases (there are no nilpotent vectors and all linear maps are smooth), an R-vector space is exactly the same as a graded R-module. We conclude that an R-vector space is the direct sum of two vector spaces over R.
6.22 Definition. Given two A-vector spaces E and F, a linear map 1> : E __ F will be called smooth if and only if 1>(BE) c BF c F. Using one of the matrix representations of [4.1], 1> is smooth if and only if all its matrix elements are real. This is most easily seen by noting that basis vector~iin E belong to BE and that the matrix elements of 1> are (up to conjugation) the components of 1>(ei). Using [6.19] this can be stated as 1> E B Horns (E; F) or equivalently as B1> = 1> (see also [6.10]).
6.23 DiscussionlDefinition. With the above definitions, we have created a new category: the category of A-vector spaces, together with the smooth linear maps. In this category, we can still perform our constructions of new A-vector spaces. In fact, for direct sums, tensor products, exterior powers and endomorphism spaces we have given a homogeneous basis of the new free graded A-module in terms of homogeneous bases for the old free graded A-modules. The equivalence class of bases for the new A-vector space then is generated by the basis constructed from bases for the old A-vector spaces that lie within the given equivalence classes. We leave it to the reader to check that changing the original bases wi thin their equivalence classes does not alter the equivalence class of bases of the
86
Chapter II. Linear algebra of free graded A-modules
newly created A-vector space. For the A-vector space F(G, c) generated by the graded symbols G the obvious choice for a homogeneous basis is G itself. For graded subspaces (and thus quotients) the question is more delicate since there is no obvious way to induce a basis on a graded subspace once a basis for the total A-vector space is given, nor is there an obvious way to induce a basis on a quotient. We therefore restrict the notion of a graded subspace in our new category as being a graded subspace in the old sense with the additional restriction that there exists a homogeneous basisfor the total A-vector space within its equivalence class such that a subsetforms a basisfor the graded subspace in question. The equivalence class induced by this subset defines the equivalence class for the graded subspace (again, different bases within the equivalence class on the ambient A-vector space induce the same equivalence class on the graded subspace). With this definition of a graded subspace, a quotient space is well defined in our new category. Ifwe have a homogeneous basis within the equivalence class, a subset of which defines the graded subspace, the complement induces a homogeneous basis on the quotient. And as before, changing the basis on the original space (with all the restrictions specified above) does not change the equivalence class on the quotient.
6.24 Lemma. Let 1> E Homs(E; F) be a smooth homogeneous linear map between A-vector spaces. Then ker( 1» and im( 1» are graded subspaces in the sense of[6.23]. Proof Let P c BE and Q c BF be homogeneous bases in the corresponding equivalence classes. Left and right linear being similar, we consider the right linear case and we start with the even case. Since 1> is smooth, the matrix representation [4.1] of 1> (all are equal) is of the form 1> ~ ( ~ ~) with A and D matrices with real entries. In particular for any e E Po, 1>( e) is a linear combination with real coefficients (the matrix elements of A) of elements of Qo. Elementary linear algebra over R then tells us that we can change the basis vectors in Po and in Qo by real coefficients in such a way that there exist subsets PfJ C Po and Q~ C Qo such that PIJ is a basis for ker( 1>1 BEo), such that Q~ is a basis for im(1)I BEo), and such that 1> is a bijection between Po \ PfJ and Q~. A similar result holds for the odd basis vectors in Hand Ql. We conclude that there exists homogeneous bases P and Q within the corresponding equivalence classes and subsets pi C P and Q' C Q such that P' is a basis for ker(1)IBE), that Q' is a basis for im(1)IBE), and that 1> is a bijection between P \ pi and Q'. It follows easily that Span( Pi) = ker( 1» and that Span( Q') = im( 1». The result then follows from [3.3]. If 1> is odd, the matrix representation [4.1] of 1> is of the form 1> ~ (~ ~) with Band C matrices with real entries. In particular for any e E Po, 1>( e) is a linear combination with real coefficients (the matrix elements of C) of elements of Ql. Elementary linear algebra over R then tells us that we can change the basis vectors in Po and in Ql by real coefficients in such a way that there exist subsets PIJ C Po and Qi C Ql such that PIJ is a basis for ker( 1>IBEo)' such that Qi is a basis for im( 1>I BEo), and such that 1> is a bijection between Po \ PIJ and Qi. A similar result holds for the basis vectors in PI and Qo. The rest of the proof is as for the even case. IQEDI
§6. The body of a free graded A-module
87
6.25 Counter example. In [3.12] we have seen that the condition that 1> be homogeneous is not superfluous. To show that smoothness is also not superfluous, we consider the map 1> : A2 ~ A2 defmed by 1>(a, b) = (a + /-Lb, /-La + /-L2b) for some /-L E Ao, /-L (j. R. This 1> is not smooth because /-L (j. R, ker phi is generated by (- /-L, 1) and im 1> is generated by (1, /-L). Since /-L is not real, ker phi and im 1> are not graded subspaces in the sense of [6.23]; they are however graded subspaces in the sense of [2.1].
6.26 Remark. Let E be afinitedimensional A-vector space with basis (ei)f=l and F an A-vector space with basis (Ii )jEJ (homogeneous, in the appropriate equivalence classes). It follows that the elements Ii 0 e i define the equivalence class of bases for the A-vector space HomR(E; F). A right linear map 1> : E ~ F is smooth if and only if its matrix elements are real numbers, i.e., its coefficients withrespectto the basis (Ii 0ei)f=1,jEJ are real. The analogous result holds for left linear maps. We conclude that 1> E Horns (E; F) is smooth if and only if 1> E B(Homs(E; F)) c Homs(E; F), i.e., the smooth linear maps E ~ F are just the maps belonging to the body of Homs(E; F). Now recall that for amap 1> E Horns (E : F) wehave B1> = 0 if and only if its matrix elements are nilpotent (see the proof of [6.8]). This can be stated intrinsically either as 1>(BE) C NF or as 1> E NHoms(E;F), descriptions that should be compared with the definition of smoothness: 1>(BE) c BF or 1> E B(Homs(E; F)). With these preparations we can describe the isomorphism between Homs(BE; BF) and B(Homs(E; F)) (see [6.10]). The kernel of B : Homs(E; F) ~ Homs(BE; BF) defined in [6.8] is the space NHoms(E;F), which is a supplementto B(Homs(E; F)). It follows that B : B(Homs(E; F)) ~ Homs(BE; BF) must be an isomorphism.
6.27 Discussion. On the category of A-vector spaces and smooth maps we can still apply the body map B as a functor to the category of R-vector spaces. The upshot of what we will do in the remainder of this section is that we will construct a functor G from the category of R-vector spaces to our new category, which is "inverse" to B in the sense that it allows to show that these two categories are equivalent. However, we do not insist on this categorical language, we will merely point out some key constructions and properties that will be useful in the future. Readers versed in the language of categories will find it easy to complete the proof that these two categories are equivalent. The fact that these two categories are equivalent immediately implies that the constructions of direct sums, free modules, and tensor products coincide, being solutions to universal problems. But we have more: our definition of graded subspace in the category of A-vector spaces [6.23] is such that they correspond exactly with the graded subspaces in the category of R-vector spaces.
6.28 Remark. Even with the additional data of an equivalence class of bases and requiring maps to be smooth, the only invariant of a free graded A-module remains its graded
88
Chapter II. Linear algebra of free graded A-modules
dimension. More precisely, if E and F are two A-vector spaces, then they have the same graded dimension if and only if there exists a smooth isomorphism between E and F.
6.29 ConstructionIDiscussion. If E is an A-vector space, one can ask whether E is completely determined by the R-vector space BE. We will show that the answer is positive. For an R-vector space X = Xo EEl Xl we define GX as
If we introduce left multiplication by A as a· (b 0 x) = (ab) 0 x, and the grading (GX)a = (Ao 0R Xa) EEl (AI 0R Xl-a), then GX becomes a free graded A-module. If (ei)iEI is a basis of X, (1 0 ei)iEI is a basis of the free graded A-module GX (use [1.7.28]). Since the equivalence class of this basis does not depend upon the choice of the original basis (ei)iEI, we conclude that we can turn GX into an A-vector space of the same graded dimension as the R-vector space X. Using [6.3] it is elementary to show that B(GX) c GX is just the subset of elements of the form 10x, i.e., we have a canonical identification between X and B(GX). Coming back to our original question, if B is a basis for E (homogeneous, within the specified equivalence class), then BB = B (sic!) is a basis of BE, and thus 1 0 B is a basis of G(BE). We thus can identify canonically G(BE) with E, thus answering the question in the positive. Strictly speaking the spaces B(GX) and X are not the same, nor are G(BE) and E. The correct terminology is that they are canonically isomorphic. However, in the sequel we will forget such subtleties and we will write B(GX) = X and G(BE) = E, pretending that Band G are really inverse to each other.
6.30 Lemma. Let E and F be A-vector spaces and let 1> : BE
-+ BF be a linear map of R-vector spaces. Then there exists a unique smooth (left or right) linear map ofA-vector spaces G1> : E -+ F making the following diagram commutative:
E
~
F
BE
-------+
BF .
Moreover,
if1> is homogeneous of parity Ct, then G1> is too.
Proof G1> is determined by its values on a basis. Since a basis of E is contained in BE (by definition of an A-vector space), proving existence and uniqueness. Smoothness
follows because this G1> obviously satisfies (G1»(BE)
c BF.
IQEDI
§6. The body of a free graded A-module
89
6.31 Corollary. B (G1» = 1> and B is an isomorphismfrom smooth linear maps between A-vector spaces to (smooth) linear maps between R-vector spaces.
Proof The equality B(G1» = 1> follows from the uniqueness in [6.8]; that B is an IQEDI isomorphism follows from the uniqueness in [6.30].
6.32 Discussion. We can summarize the above results by saying that there is no real difference between the category of A-vector spaces (with smooth linear maps) and the category of R-vector spaces. This fact is most easily seen when we think in terms of bases: the operations Band G do not change the basis, only the set of scalars one puts in front of a basis element. From this observation it follows immediately that all constructions that are performed using (homogeneous) bases are preserved by the functors Band G. As said before, for B we have to be careful with graded subspaces and quotients, but for G there are no problems since for vector spaces over R one can always complete an independent set to a basis.
6.33 Nota Bene. Having claimed that there is no real difference between the categories of A-vector spaces and R-vector spaces (more precisely, we claim that G and B are isomorphisms of categories), we immediately have to warn the reader for a pitfall: for this statement to be true, one has to take as morphisms in the category of A-vector spaces either left or right linear smooth maps, but not both at the same time! Whenever it is useful to consider both at the same time (and it often is), Band G are no longer isomorphisms (e.g., for every odd linear map between R-vector spaces, there exist two smooth odd maps between A-vector spaces: a right and a left linear one).
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Chapter III
Smooth functions and A-manifolds
For those readers who skipped or have already forgotten the first two chapters, we suggest that they imagine that the ring A is the exterior algebra A = /\ X of an infinite dimensional vector space X over R, which we usually split into an even and odd part: AD = EB~=o X and Al = EB~=o + I X. We will never need these details about A, but it fixes the ideas. Moreover, an A-vector space E can be thought of as being a direct sum of two ordinary vector spaces over R (the even and odd parts) in which the coefficients with respect to a basis are replaced by elements ofA. In this interpretation the even pan Eo consists of those vectors which have even coefficients with respect to a basis of the even part and odd coefficients with respect to a basis of the odd part, while for the odd part EI the parities are reversed. The only thing one should remember is thatfor the basis vectors of the odd part it makes a difference whether one puts the coefficients on the left or on the right. Finally, we will always assume that our A-vector spaces are finite dimensional, and that a basis is ordered in the sense that we first put the basis vectorsfor the even part and then thosefor the odd part. The main subject of this chapter is the notion of an A-manifold, a generalization of the notion of an ordinary manifold in which R is replaced by a graded commutative ring A and in which Rn is replaced by the even part Eo ofan A-vector space E of dimension plq. Since the standard approach to smooth functions cannot be copied to this more general setting, we provide an alternative definition which works in both cases. We prove that smooth functions on Eo can be identified with ordinary smooth functions on RP times skew-symmetric polynomials on Rq, i.e., cOO(Eo) ~ COO (RP) 0/\ Rq. Once we know what smooth functions are, we define their derivatives, something which is not automatically included in our approach to smooth functions. This requires a condition on A, but this condition is satisfied by the basic example for A given above. It then is elementary to generalize the implicit and inverse function theorems; even partitions of unity pose no problems. With these ingredients we then just copy the definition of ordinary manifolds in terms of charts and transitionfunctions between charts to obtain
Nk
Nk
91
Chapter III. Smooth functions and A-manifolds
92
A-manifolds. Product manifolds are defined easily, but the construction ofa submanifold requires some care. Due to our definition of smooth functions, one cannot always construct submanifolds of lower dimension passing through an arbitrary point of the ambient manifold. We finish this chapter by extending the body map B to A-manifolds and their smooth maps. We prove that the topology ofan A-manifold M is completely determined by the topology of its body B M (which is a manifold in the ordinary sense of the word, or, in our terminology, an R-manifold). This extended body map preserves all constructions, e.g., B(M x N) = BM x BN. In terms of coordinates, the body map is just the projection onto their realpart (i.e., the projection A = /\ X __ R = /\0 X in terms ofour example of A).
1.
TOPOLOGY AND SMOOTH FUNCTIONS
In this section we provide our alternative approach to smooth functions which avoids the use of limits of difference quotients. We show that the set Coo (U; A) of A-valued smooth functions on an open set U c Eo is a graded R-algebra and that the set Coo (U; F) of F-valued smoothfunctions (F an A-vector space) is a free graded Coo (U; A)-module of the same graded dimension as F. Using the parity reversal operator, we show that the restriction to the even part of an A-vector space is in reality not a restriction at all.
1.1 Discussion. Just as smooth functions on open sets in RP are the basic ingredients for ordinary manifolds, we want smooth functions on open sets of ... to be the basic ingredients for A-manifolds. Three questions arise: what are the open sets, what are the smooth functions and, most important for the moment, what kind of spaces to put on the dots? In trying to find reasonable answers to these questions, we will be guided by the idea that R-manifolds, i.e., those obtained by taking A = R, should be the same as the ordinary well known manifolds. The most naive answer to the space question would be to take any A-vector space, but an R-vector space X is a direct sum of two ordinary real vector spaces Xo EEl Xl and for ordinary manifolds there is no subdivision of the (local) coordinates. A less naive answer, and the one we will adopt, is to take the even part Eo of an A-vector space E. If {el' ... ,ep , h, ... ,iq} is a homogeneous basis of E with the ei even and the ii odd, then a point in Eo has even coordinates with respect to the ei and odd coordinates with respect to the Ii. The number of even coordinates gives us the even dimension of E and the number of odd coordinates gives us the odd dimension of E. Moreover, if we take the special case A = R, all odd coordinates are zero, and what remains is essentially RP ,just as we wanted. We note that we thus have a profound distinction between a generic A (with Al #- {O}) and the special case A = R. In the first case we have non trivial odd coordinates and
§1. Topology and smooth functions
93
thus retain information about q (the odd dimension), whereas in the second case we lose all information about q. Another way to see this difference is to consider the body map B : E -> BE, which provides the passage from the generic case to the special case A = R. We have seen that BE contains enough information to reconstruct E as G(BE). On the other hand, BED does not contain enough information to reconstruct Eo (and certainly not the whole of E): the odd dimension of E is missing. It is this difference that gives A-manifolds their extra flavor.
1.2 DefinitionIDiscussion. Let E be an A-vector space of dimension plq. We define a topology on E as the coarsest topology for which the body map B : E -> BE £:! Rp+q is continuous, i.e., U c E is open if and only if U = B- 1 (0) for some open subset 0 of BE. Said differently, U is open if and only if BU is open and U = B- 1 (BU). Yet another way to say the same is to state that the map B : U -> BU induces a bijection between the open sets of E and the open sets of B E ~ Rp+q. This topology is usually called the DeWitt topology. All subsets of E, and in particular Eo, will be equipped with the relative topology. Since BE = E/NE , it follows that open sets U c E are saturated with nilpotent vectors, i.e., U open implies U + NE = U. For R-vector spaces the body map B is the identity, so the DeWitt topology on X is the standard euclidean topology on X. In order to have a closer look at the DeWitt topology in the general case, let us choose a basis (ei)f=l of E. With respect to this basis we consider left coordinates x = 2:i Xi ei E E to identify E with An as sets. This gives us an identification of BE with Rn such that the body map is given by B (xl, ... , xn) = (BXl, ... , Bx n ). Using this identification, a point (Xl, ... , xn) will be in the open set U if and only if (Bxl, ... ,Bxn) lies in the open set 0 = BU c Rn, i.e., the topology is completely determined by the body parts of the coordinates. If we assume furthermore that the homogeneous basis is an ordered one, a point (Xl, ... , xn) lies in Eo if and only if xl, ... ,xP E AD and x p + l , ... ,xn E AI. We deduce that a set U c Eo is open if and only if there exists an open set 0 C RP such that
Ai
c Eo A particular consequence is that the induced topology on a fiber {( Xl, ... , x P )} x is the indiscrete topology (either all or nothing). Another consequence is that for the topology on Eo we have the same characterization as for the topology on E: U C Eo is open if and only if BU is open in BED ~ RP and U = B-l(BU), where we now see B as the restriction B : Eo -> BED. Or again, B induces a bijection between open sets of Eo and open sets of BED ~ RP.
1.3 Corollary. The DeWitt topology on an A-vector space E and on its even part Eo are locally connected (any neighborhood oj a point contains an open connected neighborhood ojthat point, which is equivalent to the connected components ojopen sets being open [Du, theorem V.4.2]).
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Chapter III. Smooth functions and A-manifolds
1.4 Lemma. Let E be an A-vector space, then: (i) the product topology of Eo EEl El coincides with the topology of E; IfF and F are supplementary graded subspaces of E, then the product topology of F EEl F coincides with the topology of E; (ii) the A-vector space operations of addition (E x E ~ E) and multiplication by elements ofA (A x E ~ E and E x A ~ E) are continuous; (iii) left linear maps to A are continuous, and in particular the coordinate projections
je : x
= 2:i Xi ei
f--+
xj
= ~ x II
je )).
1.5 Notation. According to standard abuse of notation, we will often denote the (left) coordinate projections je : (xl, ... ,xPH ) f--+ x j by xj.
1.6 Discussion. So far in discussing the DeWitt topology we have treated the spaces E and BE as different. However, we have identified BE with those points in E that have real coordinates [11.6.19], i.e., we have interpreted B as a projection map B : E ~ E. For an arbitrary subset U c E we thus interpret BU as a subset of E. Moreover, the inclusion BU c U is equivalent to the equality BU = Un BE, but neither condition need be true. However, if U is open in E, and thus in particular U = B-l(BU), we have BU c U, and thus BU = Un BE c U. In other words, for open sets U the body BU consists of those points in U that have real coordinates.
1.7 Discussion. Once we have a topology, we can speak of continuous functions, and in particular we can speak of functions on open sets U of the even part of an A-vector space E with values in an A-vector space F. As usual we will denote these sets by CO(U; F). However, we want to speak of differentiable functions, and there we encounter a problem. The standard way to define a differentiable function is to say that the derivative should exist. Such a derivative is (usually) defined as the limit of a difference quotient. But in our context, we can not always write such a difference quotient because of the existence of nilpotent elements. Even disregarding the nilpotent elements, the DeWitt topology is not Hausdorff, so a limit need not be unique. In order to circumvent these problems, we will use an alternative approach to differentiable functions which does not use limits nor difference quotients. The essential idea of this approach is expressed in [1.8].
1.8 Proposition. Let 0 C RP be open, and let f : 0 ~ RN be afunction. Then f is of classC k+ l ijandonly If there exists an open coverU = {Ua I a E I} of 0 andforall a E I functions 9ai : U; ~ R N , 1 :::; i :::; p of class C k such that for all x, y E Ua : P
(1.9)
f(x) - f(y) =
L 9ai(X, y) . (xi i=l
yi) .
§ 1. Topology and smooth functions
95
Proof If f is of class C k + 1 , we take any cover of 0 consisting of convex open sets (for instance open balls). For each convex Ua in this cover we define the functions 9ai by
Jor
1
(LlO)
9ai(X,y)
=
of oxi(sx+(l-s)y)ds.
The convexity of Ua guarantees that this integral makes sense, and since the of / ox i are of class C k , the 9ai are of class C k as well. Computing L,i 9ai(X, y) . (Xi - yi) immediately gives f(x) - f(y)· If on the other hand the condition is satisfied, we compute on each open Ua the partial derivatives of f in x E Ua by (Lll)
of ux'
.
~(x) = hm
h->O
f(x
+ hei) h
f(x)
.
= hm
h->O
9ai(X + hei, x) = 9ai(X, x) ,
where ei = (0, ... , 0, 1, 0, ... , 0) denotes the i-th basis vector in RP. Since the 9ai are of class C k , this shows that the partial derivatives of f exist everywhere and are of class C k , i.e., f is of class C k + 1 . IQEDI
1.12 Remark. One can prove a stronger version of [1.S] which does not need a cover and which states that f is of class C k + 1 if and only if there exist functions 9i : 0 2 ~ R N of class C k satisfying (1.9) for all x, yEO. The reason we did not give this stronger result in [1.S] is that [1.S] as it is, remains true if we replace the field of real numbers R by the field of complex numbers C (and then it says that f is holomorphic if and only if the 9ai are continuous). On the other hand, if we replace [1.S] by the stronger result without the cover, then we no longer can go over to the complex case. The simple reason is that the stronger result needs a partition of unity argument. In fact, domains in CP for which no cover is needed occur naturally in complex analysis: by a classical corollary ofH. Cartan's theorem B, this is the case for any domain ofholomorphy (== a pseudo convex domain == a Stein domain), see [Ra, corollary 6.26]. Let us now give the proof ofthe stronger result. Since the if part is proved as in [l.S], we attack the only ifpart. We choose a cover V = {Va I a E I} of Ox o having the following property: for V E V we have either (i) :JC convex open set in 0 such that V = C x C, or (ii) :Jj (1 :s; j :s; n) such that V(x, y) E V : x j -=f- yj. The existence of such covers follows easily from the fact that RP is Hausdorff. Given such a cover, we define on each Va E V functions 9ai : Va ~ R N as follows. In case (i) we define 9ai by (1.10); in case (ii) we define 9aj(X, y) = (f(x) - f(y))/(x j - yj) and for i -=f- j we define 9ai = O. In both cases the functions 9ai are of class C k and satisfy (1.9) for all (x, y) E Va. Now let Pa be a partition of unity of class C k subordinated to the cover V (see [5.1S] for an exact definition), anddefinethefunctions 9i : 0 x 0 ~ RN by 9i(X, y) = L,a Pa(x, y) 9ai(X, y). Then for any (x, y) E 0 x 0 wehave Pa(X, y) -=f- 0 =} (x, y) E Va, and thus L,i(X i _yi) 9i(X, y) = L,a Pa(x, y) L,i(X i - yi) 9ai(X, y) = L,a Pa(x, y) (f(x) - f(y)) = f(x) - f(y). This finishes the proof.
96
Chapter III. Smooth functions and A-manifolds
1.13 Discussion. If f : 0 C RP ~ RN is smooth (of class COO), application of [1.8] gives us an open cover {Ua} of 0 and again smooth functions f~~) : U; ~ RN satisfying (1.9). We then can apply [1.8] again to each of the functions f~~), to obtain an open cover
{Uab } of Ua and smooth functions fgl)(bj) : U;b ~ RN satisfying a condition analogous to (1.9). Continuing this procedure, we get an infinite tree of open covers {Ual a2 ... ar } (i.e., U ar Ual a2 ... ar = Ual ... ar-l' and the index family for a r might depend upon the sequence and an infinite tree of smooth functions j 1:S; ij:S; p2 - 1 satisfying for allx,y E Ual ... arb:
al ... ar-l)
f«r)) ( '): alt1 ... artr
U;l ... ar ~ RN,
If on the other hand we have such a tree of covers and functions (without any differentiability assumption) satisfying (1.14), we can argue as follows. If for a fixed value of r all functions fg~ill ... (arq are continuous, then by [1.8] all functions f(~0~j ... (ar_lir_l) are of class C 1 . Continuing this argument we conclude that the initial function f == f(O) is of class Varyingthevalueofrproves [1.15].
cr.
1.15 Proposition. Let f : 0 C RP ~ R N be a function. Then f is smooth if and only if there exists an infinite tree of open covers {Ual ... ar } and continuous functions
fg~ill ... (arq as described in [1.13 J satisfying (1.14).
1.16 Definitions. With the above preparations concerning smooth functions on RP, we now return to the question how to define smooth functions on open subsets U c Eo of the even part of some A-vector space E. We start with some preparatory definitions . • Given a continuous function f : U ~ F with U c Eo open and E and FA-vector spaces, a smooth tree associated to f will be an infinite tree of open covers {Uala2 ... ar} and continuous functions fg~il) ... (arir) : U;l ... ar ~ F satisfying for all x, y E Ual ... arb: 2r dim(E)
(r) f (alill ... (a r ir) (X)
-
f(r)
(alil) ... (arq ( y ) --
(j x - Yj) . f(r+l) (alill ... (a rir)(bj) ( x, ) y .
'"'
L
j=l
• A symbol F is called a smooth system of F-valuedfunctions if for all E and for all U c Eo open, we have a set F(U; F) c CO(U; F) of continuous functions from U to F such that for any f E F(U; F) the following two conditions are satisfied: (Al) there exists afamily Ua C U of open subsets covering U and n = dim E functions 1>a,i E F(U;; F) such that n
\/x,y E Ua
f(x) - f(y) = L(x - y)i ·1>a,i(X, y) , i=l
97
§ 1. Topology and smooth functions
where the Xi are the coordinates of x with respect to some basis of E (homogeneous, in the specified equivalence class); (A2) f(BU) C BF.
1.17 Remark. In case A = R, condition (A2) is completely superfluous. We added it for exactly the same reason we added a similar condition for smooth linear maps in [11.6.22]: to be able to prove that the body map B is a bijection on certain sets of functions (see [2.16]).
1.18 Remark. We have restricted our attention to functions defined on open sets of the even part of an A-vector space. However, one can define as easily differentiable functions on open sets of the entire A-vector space. We also have restricted our attention to infinitely often differentiable functions, but functions of class C k can be defined in a similar way. For more details the reader is referred to [Tu2].
1.19 Example. Let f : U C Eo ~ F be a continuous function, and suppose there exists a smooth tree offunctions associated to f. We then can construct a smooth systemFf (of F-valued functions) as follows.
Ff(U;1 a2··· a; F) r
=
{f«r)) ( .) al1-l ... artr
11 ::; i J· ::; 2j - 1 dim(E)}
,
U;l
and if 0 is not of the form a2 ... a ' then F f (0; F) = ~, with the exception of the first r term of the tree Ff(U; F) = {f}. The definition of a smooth tree immediately proves that Ff is indeed a smooth system.
1.20 Definition. We can order smooth systems by inclusion, i.e., .1'1 ::; .1'2 if and only if for all U : F 1 (U; F) c F2 (U; F). It is also easy to prove that the union Fu of two smooth systems .1'1 and .1'2, defined by Fu (U; F) = .1'1 (U; F) U .1'2 (U; F), is again a smooth system. It thus makes sense to speak about the maximal smooth system. Which brings us to our final definition: Coo is the maximal smooth system of F-valuedfunctions. Elements f E Coo(U; F) will be called smoothfunctions (on U, with values in F). More generally, if D is an arbitrary subset of F, we define the sets Coo(U; D) by
Coo(U; D) = {f
E
Coo(U; F) I f(U) cD} .
The special cases D = Fa, CI'. E Z2 are also denoted as Coo(U; Fa) = Coo(U; F)a. Note that this definition is quite different from the definition of Map s (E; F) "" although both concern sets of maps.
98
Chapter III. Smooth functions and A-manifolds
1.21 Proposition. In the case A = R the above definition of COO (U; F) is equivalent to the usual definition of infinitely often differentiable functions.
Proof If f : U -- F is infinitely often differentiable, we can construct a smooth tree associated to f by [1.15]. As in [1.19] we then can construct a smooth system Ff . By maximality of Coo , F f is included in Coo, and thus f E Coo (U; F). On the other hand, if f E Coo(U; F), then by repeatedly applying property (AI) we can construct a smooth tree associated to f. And then by [1.15] the function f is infinitely often differentiable. IQEDI
1.22 Discussion. We have, intentionally, swept one slightly disturbing aspect of the proof of [1.21] under the rug. It concerns the number of functions to be used in (AI). In case A = R, an R-vector space X of dimension plq is a direct sum of two ordinary vector spaces X = Xo EEl Xl of dimensions p and q respectively. Functions of the even part Xo = Xo EEl {O} are obviously functions ofpreal coordinates, not ofn = p+q coordinates. Now in [1.15] the smooth tree is constructed with p functions at each stage, whereas (AI) requires n = p + q functions. This problem is easily solved by taking the zero functions for the missing ones. Since the additional q coordinates are identically zero, these terms do not contribute to the summation in (AI). On the other hand, when constructing the smooth tree from (AI), we get at each stage n = p + q functions, where we need only p. This is even simpler to solve: we just neglect the functions we do not need before we apply [1.15]. In view offormula (1.11), we will wantto define partial derivatives ad(x) as CPa,i (x, x), i.e., as the diagonal of CPa,ii. Although the functions CPa,i are not unique, we learn from (1.11) that the diagonal of CPa,i is independent of the choice for CPa,i' On the other hand, the above considerations tell us that for A = R, we can choose the functions CPa,i with i > p completely arbitrary, and thus the partial derivatives ad are also arbitrary for i > p. There is no contradiction, because (l.ll) was given only for i :s; p; and no one would try to define partial derivatives ad for i > p if the function f depends only upon p coordinates. But it shows that we have to be careful when we want to define partial derivatives for a more general A. And indeed, we will see that the structure of A is crucial in the definition of partial derivatives by means of (1.11) (see [3.1] and [V. loS]). But before we can address the question of partial derivatives, we need to analyze the consequences of our definition of smooth functions in more detail.
1.23 Lemma. The sets Coo(U; F) enjoy thefollowing properties: (a) All constant maps with real image belong to Coo(U; A), as do the coordinate projections ie. (b) For f E coo(U; A) and ex E Z2 define 9 : U -- A" c A by g(x) = (f(x))",. Then 9 E Coo(U; A)", c Coo(U; A). (c) If>' E Rand f E Coo(U; A) then>.· f E Coo(U; A). (d) Iff, 9 E Coo (U; A), then f + 9 E Coo(U; A). (e) If property (AI) holds for some homogeneous basis, it holds for all homogeneous
bases.
99
§ 1. Topology and smooth functions
(f) For f E Coo(U; A) and E' an A-vector space we define 9 : U x Eb -- A by g(x, x') = f(x). Then 9 E Coo(U x Eb; A). (g) Suppose E = E' EEl E"for two A-vector spaces E' and E" and suppose that
U' C Un Eb is open in Eb. Let x" E BE" n U and f If 9 : U' -- F is defined by g(x') = f(x', x"), then 9
E E
Coo (U; F) be arbitrary. coo(U'; F).
(h) If f, 9 E Coo(U; A) then f· 9 E Coo(U; A). (i) f E Coo (U; F) If and only lffor all j : jf 0 f E Coo(U; A), where jf are the (left-linear) coordinate functions on F. U) Let be an open set of F o, f E Coo(U; F) and 9 E Coo(D; G). If im(f) c then 9 0 f E Coo(U; G). (k) Let Ui CUbe a family of open sets covering U. If f : U -- F is such that Vi: flu; E COO (Ui ; F), then f E Coo(U; F).
D
D,
Proof In this proof we leave it to the reader to verify, whenever necessary, that condition (A2) is satisfied and that the constructed functions are continuous; the conditions in the lemma assure that there will be no problem. The basic line of proof will be the construction of a smooth tree of (continuous) functions satisfying (Al) by means of induction to the leveL As in [1.19] we then can form a smooth system, from which we can conclude that it must be included in Coo because the latter is maximal. We will abbreviate this procedure by saying that we use the recursion argument. • To prove (a), first note that constant functions and the je are continuous [104]. For any constant function f we can choose the functions 1>i = O. Hence by maximality all constant functions belong to Coo(U; F). For the je one can choose 1>j = 1, all others constant 0 (with respect to the same basis as used for the coordinate projections). Note that in both cases the cover {Ua} consists of the single element U itself. • To prove (b), note that the i-th coordinate of x E Eo has parity c(ei). Using the same cover {Ua} as for f, we define the functions Xa,i = (1)a,i)'''+E(ei). It follows that g(y) = g(x) + L,i(y - x)i . Xa,i(y, x). By [104] the Xa,i are continuous; the rest follows by the recursion argument. • For (c), multiply the 1>a,;'s by A and use the recursion argument. • For (d) and (e), let f and 9 be as in (d). First note that, by taking pairwise intersections of elements of the original covers for f and g, we may assume that the same cover {Ua} serves for both f and g. Denote by (ei) and 1>i the basis and functions in (Al) for f and denote by (fi) and Xi the same ingredients fog. Finally denote by zi the (left!) coordinates of the vector x - y with respect to the basis (ei) and by (i its coordinates with respect to the basis (fi). It follows that there exists a matrix Ai j E R such that (j = L,i zi . Ai j and hence we have f(x)
+ g(x) =
f(y)
+L
zi ·1>i(X, y)
+ g(y) + L
(j . Xj(x, y)
j
= f(y) + g(y) + L
zi . (1)i(X,y)
+ LAi j · Xj(x,y))
.
j
We conclude the proof of (d) by noting that Ai j . Xj (x, y) is smooth by (c) and by applying the recursion argument. The proof of (e) follows by taking f = O.
Chapter III. Smooth functions and A-manifolds
100
• To prove (f), note that
g(x, x') - g(y, y')
= f(x)
- f(y)
= 2)x -
y)i ·CPa,i(X, y)
=L..,,(x-y) ·Xa,i(X,x,y,y '""' i "), where the functions Xa,i : (Ua x Eb)2 ~ A are defined by Xa,i(X, x', y, y') = CPa,i(X, y). We then apply the recursion argument after having added zeros for the functions Xa,i for coordinates in Eb. • For (g), let (ei)~l be a basis for E' and let (ei)f=m+1 be a basis for E", making (ei)f=l a basis for E. If we define Xi(X', yl) = CPi( (x', x"), (y', x")), i :::; m on the subset U~ = U' n Ua, then obviously g(x ' ) = g(yl) + 2:::1 (x' - yl)i. Xi(X',yl). The result then follows by the recursion argument. • For (h), let CPa,i and Xa,i be the functions of property (Al) for the functions f and 9 respectively. As before, we may assume that these functions share the same cover {Ua}. By [104] the function f . 9 is continuous. By (b) and (d) we may assume that the images of f andg are homogeneous. Using the functions CPa,i and Xa,i we obtain n
(f. g)(x) - (f. g)(y)
=
2:(x - y)i. (CPa,i(X,y). g(y) i=l
+ (_l)(E(ei)IE(f(x))) . f(x)· Xa,i(X,y)) . The result now follows if we apply (c), (d), (f), and the recursion argument. • To prove (i), suppose first that f is smooth. Using the left-linearity of the jf we obtain (if 0 f)(x) = (jf 0 f)(y) + 2:i(X - y)i . (if 0 CPa,i)(X, y). The recursion argument then shows that the jf 0 f are smooth. For the other implication, suppose that the jf 0 f are smooth. We then have to prove that the function f = 2: j (if 0 f) . fj is smooth, where (fj) denotes the basis of F dual to the coordinate functions jf. By hypothesis each of the functions jf 0 f has locally defined functions CP~,i as in (Al). As before, by taking multiple intersections when necessary, we may assume that the domains of definition of these functions coincide. We then define the functions CPa,i : U; ~ F by CPa,i(X, y) = 2: j CP~,i(X, y) . fj~ By definition of the topology, these functions are continuous. Moreover, they satisfy CP~,i = jf 0 CPa,i. We then conclude by the recursion argument. • To prove 0), let CPa,i and Xb,j be the functions of property (Al) for the functions f and 9 respectively. By taking intersections with the sets g-l(Db) if necessary, we may assume that each f(U a ) is contained in some Db. We then compute:
(g 0 f)(x) = (g 0 f)(y)
+ 2:(f(x) - f(y))j . Xb,j (f(x), f(y)) j
= (gof)(y) + 2:(X-y)i. (2:Cf CPa,i)(X,y). Xb,j(f(x),f(y))) . O
i
j
The result then follows by applying (d), (h), (i), and the recursion argument. • Finally, for (k) let Ui,a be the families of open sets as defined by property (Al) for each Ui. Since the Ui cover U, it follows that Ui,aUi,a = U, proving that f is smooth. IQEDI
§ 1. Topology and smooth functions
101
1.24 Summary. We can reformulate most of{1.23J in the following more manageable
form. (i) Being a smoothfunction is a local property, stable under composition and fixing
ofvariables (to real values). (ii) The set Coo(U; A) = Coo(U; A)o EEl Coo(U; Ah is a graded commutative Ralgebra with unit under pointwise addition and multiplication offunctions. (iii) For any A-vector space F, Coo(U; F) = Coo(U; F)o EEl Coo(U; Fh is afree gradedCoo(U; A)-module of the same graded dimension as F. Proof The graded commutativity of Coo(U; A) follows from the graded commutativity of A and the fact that multiplication in Coo (U; A) is pointwise. That Coo (U; F) is a free module of the same dimension as F is an immediate consequence of the fact that we can write f = 2: j (jE 0 1) . Ej for an element f E Coo (U; F) and that we can interpret Ej as a IQEDI constant function with real image on U.
1.25 Discussion. If f : U ~ F is a smooth function, we get from property (Al) the functions 1>a,i : U; ~ F. We can combine these functions into a single function 1>a : U; ~ Hom£(E; F) defined by 1>a(x, y) = 2:i ie 01>a,i(X, y). It is an immediate consequence of [1.23] that 1> is smooth. It follows that for each function f E Coo(U; F) there exist functions 1>a E Coo (U;; Hom£( E; F)) such that
Vx,y
E
Ua : f(x) - f(y) = ((x - yll1>a(x,y))) .
In this way we obtain a definition of smooth functions which is obviously independent of a basis. It should be noted that we have made a choice here to take left linear endomorphisms, a choice already present in (Al) by using left coordinate functions and writing the coordinates (y - X)i to the left of the cPa,i. In analogy with ordinary real valued Coo functions (see (1.11)) we will (want to) define partial derivatives in our graded setting by the functions 1>a,i (x, x). It follows that we need that these functions are uniquely determined by the function f. We will see in [3.1] that this is not automatically true, but that it requires a condition on A.
1.26 Construction. The reader might have got the impression that only the even part of an A-vector space can be the domain of smooth functions (in the sense of [1.20]). However, any A-vector space E can be interpreted as the even part of an A-vector space E". Before we give the definition of E", let us have a quick look at what we need. A vector x E E is even only if it has homogeneous coordinates Xi of the correct parity with respect to a homogeneous basis (ei)f=l' But a generic vector does not have homogeneous coordinates. On the other hand, if we split the coordinates Xi into their homogeneous components Xi = (Xi)O + (xih, we get homogeneous coordinates, but twice as many and as many even as odd ones. The A-vector space Ea thus should have twice the dimension of E and with the same number of even and odd basis vectors.
102
Chapter III. Smooth functions and A-manifolds
To prepare the definition of E", we recall the parity shift operation introduced in [1.6.13]. For any ex E Z2 we define the A-vector space EDa in the following way: as a left A-module it is the same as E, but the Z2-grading is given by (EDa){3 = E{3-a. Since Z2 has only two elements, we have EDo, which is obviously the same as E, and EDI, which has its parities reversed with respectto E : (EDI)O = EI and (EDI h = Eo. In the literature the operation DI is also called the parity reversal operation. As a right A-module EDI is certainly not the same as E, but the (set theoretical) identity map {d : EDI ---? E is an odd left linear bijection. Using this bijection, we can transport the structure of an A-vector space from E to EDI : if( ei)i=l is a basis in the prescribed equivalence class for E, the vectors ei = {d- 1 (ei) define an equivalence class of bases on EDI, independent of the choice of the basis (ei) in its equivalence class. With these preparations, we define the A-vector space E~ as E~ = E EEl EDI. And then indeed E~ = Eo EEl E5 1 = Eo EEl EI ~ E, but the last identification is not an identification of A-vector spaces. In order to get a better idea of what this space E" is, we define the (left linear) projection 7r : E" ---? E as 7r = id + {d. We also define the map
O. For the homogeneity statement, note that f is homogeneous of degree ex if and only if f takes values in BFa. Since the n in [2.11] are even, the result ((Dk f)(Bx))(n, ... , n) remains in Fa. IQEDI
2.16 Proposition. Let E be an A-vector space without odd dimensions and U an open subset of Eo. Then B : Coo(Uj F) ~ COO (BU, BF) is an isomorphism. Proof Surjectivity is assured by [2.15]. To prove injectivity, let f E Coo(U; F) be such that Bf == 0, i.e., flBU is identically zero, and let Xo E BU c U. Now look at the proof
108
Chapter III. Smooth functions and A-manifolds
of [2.1] and suppose that g(k) is zero on BUk . It immediately follows that g(k+l) is zero on BUk+l, provided Byn+l =I- Byj. By continuity of g(k+l) and the fact that it takes real values on BUk+l, it follows that g(k+l) is zero on the whole ofBUk+l. This last conclusion would not have beenjustified, had we not known that g(k+l) takes real values on BUk+1 : the non-Hausdorff character allows for non-unique limits! We conclude that I(k) IBUk == 0 for all k. If there were only one even coordinate, the expansion of [2.1] would prove that I == O. For more than one even coordinate we apply induction on their number n. We fix xn = rn + an and apply the expansion of [2.1] with j = n. By the previous argument we have BI(k) == O. Since in the expansion of [2.1] the functions I(k) only appear with the real valued n-th coordinate rn, we may by [1.23-g] interpret the I(k) as smooth functions ofn -1 coordinates. By the (unstated) induction hypothesis we conclude that I(k) == 0, and thus I == o. IQEDI
2.17 Remark. In §I1.6 we showed that B is an isomorphism when applied to smooth linear maps. Here we show that B is an isomorphism when applied to arbitrary smooth functions on an open subset of the even part of an A-vector space without odd dimensions. Said differently, as long as the domain of definition only contains even coordinates, there is no difference between the general case and the special case A = R. The restriction to A-vector spaces without odd dimensions is necessary because we take as domain of definition not (an open subset of) an A-vector space, but only (an open subset of) its even part. The same phenomenon is also reflected in the fact that in reconstructing U from B U by means of G we only recover the part without odd dimensions, i.e., G(BU) = U only if E has no odd dimensions.
2.18 Notation. Up until now we have carefully distinguished an ordinary real valued smooth function of p real variables and its corresponding smooth function G I of p even variables. However, [2.16] has shown that there is no real distinction between these two objects. Therefore we will in the future use the same symbol for both objects. We thus might start with an ordinary real valued smooth function I and use even coordinates as arguments, meaning that we take G f. Or we might start with a smooth function of even variables, plug in real coordinates (a point in BE) and claim that we have an ordinary real valued function, this beingjustified by either taking B I or using (A2).
2.19 Discussion. Let E be an A-vector space of dimension plq and let U be an open subset of Eo. Combining [2.5] and [2.16], we see that we have a nearly complete control over the elements of Coo (U; F). Thefunctions ga,il ... ik E COO (Ua ; F) are only used with all odd coordinates being zero. This means that we only need their restrictions to U;:·o.d, but (by [2.16]) these are uniquely determined by ordinary real valued smooth functions Ia,il ... ik E Coo (BUa ; BF) as ga,iI ... ik (Xl, ... , x P , 0, ... ,0) = Ia,il ... ik (xl, ... ,xP ) (remember, we do not use the symbol G any longer). However, our control is not complete because (i) this description of I is only local on Ua, not on U, and (ii) it is not clear
§2. The structure of smooth functions
109
whether the function f determines the local functions fa,i1 ... i k uniquely. In order to make our control complete, we need to formulate conditions on A, conditions that will also be useful in defining the partial derivatives using the diagonal of the functions CPa,i as in (1.11).
2.20 Definition. Let A be a graded commutative R-algebra and let n E N* be a non-zero natural number. We say that A satisfies (C [nJ) if (C [nJ)
We will say that A satisfies (C [ooJ) if it satisfies (C [nJ) for all n E N*. Obviously (C [001) and (C [n+lJ) imply (C [nJ). In the standard example A = /\ X with X a vector space over R, (C [nJ) is satisfied if and only if dim X ::::: n, a condition valid also for n = 00.
2.21 Proposition. Let E be anA-vector space of dimension plq, let f be in Coo (U; F), and let f i 1... i • E Coo (BU, BF) be such that f(x) = 2:k 2:;. 1 ••• ~ik . f i 1... i k (Xl, ... , x P). J Then the functions fi 1... ik are unique Ifand only If A satisfies (C [kJ).
e
Proof We start by observing thatthe separate additive terms ~i1 .•. ~ik . f i 1... ik (Xl, ... , x P) are always uniquely determined by f: a simple induction argument with respect to k and taking all coordinates ~j = 0 whenever j r:J. {i l , ... , ik} suffices. Now suppose ~i1 ... ~ik . fi 1... ik (Xl, ... , x P ) to be identically zero and take the coordinates Xl, ... , x P to be real. It follows that ~i1 ... ~ik . fi 1... ik (xl, ... , x P ) = 0, which implies that either ~i1 ... ~ik or f i 1""k (Xl, ... , x P ) is zero (because the latter is real, and thus can be inverted if non-zero). If (C [kJ) is satisfied, we can choose ~i1 .•. Eik to be non-zero, implying that fi 1... ik must be zero. On the other hand, if (C [kJ) is not satisfied, the term ~i1 ... ~ik is IQEDI always zero, leaving the function f i 1 ... i k completely undetermined.
2.22 Corollary. Let E be an A-vector space of dimension plq and let U c Eo be open. Then COO(U; F) is in bijection with COO(BU; BF)d, i.e., with dcopies of COO(BU; BF), where d is determined as follows.
(i) If Al = {O}, then d = 1. (ii) If 1 :::; £ < q is such that A satisfies (C [eJ) but not (C [£+lJ), then d (iii) If A satisfies (C [qJ), then d = 2q.
= 2:~=o (3)'
Proof • If Al = {O}, it follows from [2.5] that f(x,O = ga, (x, 0), which proves that f(x,~) = f(x, 0), i.e., f is a function of the even coordinates only. And then the conclusion follows from [2.16] . • Let us now suppose that A satisfies (C [kJ). From [2.5] we obtain (%) local smooth functions ga,i1 ... i k «(k) = the number of increasing sequences 1 :::; i l < ... < ik :::; q). Since fa,i1 ... i k(X l , ... ,xP) = ga,i 1... ik(X l , ... ,xp ,0, ... ,0) depends only upon even coordinates, it follows that fa,i 1... ik E Coo (BU; BF). Butthen we can combine [2.5] and
Chapter III. Smooth functions and A-manifolds
110
[2.21] to conclude that the functions fa,il ... ik E Coo (BU; BF) are uniquely determined by f. Hence they must coincide on overlaps Ua nUb. It thus follows from [1.23-k] that we have global functions fil ... ik E COO(BU; BF) such that fa,il ... ik = fil ... ik Iua . Since (C [kJ) implies (C [k-lJ), the same conclusion holds for all lower order functions. On the other hand, if A does not satisfy (C [kJ), the terms ~il ... ~ik . 9a,il ... ik (x, 0) do not contribute to f. The conclusions of (ii) and (iii) follow. IQEDI
2.23 Discussion. With [2.22] we have complete control over smooth functions. If the A-vector space E has dimension plq and if U c Eo is open, then there is a one to one correspondence between smooth functions f E COO(U, F) and collections of functions fil ... ik E COO (BU, BF) given by (2.24) min(q,C)
L
f(xl, ... ,xp,e, ... ,~q)=
Cil ... ":,Cik .
":,
j.'l.l ... 'I,k. (X 1 , ... ,XP)
= {O}
where £ is the maximal value such that A satisfies (C [cJ) (if Al
,
we take £ = 0).
2.25 Remark. If one takes into account how the functions fil ... ik in (2.24) behave under a change of the odd basis vectors, then it is not hard to show that Coo (U; F) is bijective with min(q,C)
EB
COO(U;F)~COO(BU;BF)0(
N(Rq)).
k=O
It is not surprising that the exterior power /\k (Rq) appears because a k-fold product of the q odd variables ~q is k-linear and skew-symmetric in these variables. If £ ::::: q we have the identification Coo (U; F) ~ Coo (BU; BF) 0 /\ R q, which is sometimes taken as definition of smooth functions of p even variables and q odd variables. Since starting in [3.3] we will assume that A satisfies (C [ooJ) (and thus 00 = £ ::::: q), we obtain this identification as a consequence of a more basic definition of smooth functions.
e, ... ,
2.26 Examples . • According to [II. 1.11] and [1.2.9], the group Aut(E) is an open subset ofEnds(E)o. We claim that taking inverses Inv : Aut(E) -- Ends(E)o is smooth (see also [VI. 1.6]). To prove this, it suffices to show that Inv is of the form (2.24) when using left-coordinates eMs on Aut(E). On the other hand, calculations are much simpler when using the coordinates NIs because these preserve composition (see [11.4.2]), and in particular NIs (1)-1) = (NIs( 1» )-1. Writing tIs (1» = (~ ~ ), A and D are invertible matrices with even entries, and Band C have odd entries. We thus can write:
( AB)-l _ (A-l 0 ) [I CD
-
0
_ (A-
-
0
D- l
•
p
+q
-
(0 _BD- l )]-l _CA- l
00
l
0
D- l
)
•
'"' (
L
k=O
0
_CA- l
-BD0
0
k
l )
'
§2. The structure of smooth functions
111
where the infinite sum is actually finite because Band C are nilpotent (it breaks off after the 2pqth term). We now note that (fortunately) the left coordinates eMs (1)) only differ by a sign from the coordinates NIs (1)) (see [11.4.1]). We thus are allowed to use the entries of A, B, C, and D as coordinates on Aut ( E ). Since taking the inverse of a matrix with real entries is smooth, the maps A 1--7 A-I and D 1--7 D-I are smooth in the even coordinates of A and D. It thus follows that the matrix entries of NI s (1» -1 are smooth functions of the even coordinates and polynomials in the odd coordinates. In other words, Inv: Aut(E) -+ Aut(E) C Ends(E)o is smooth . • The ordinary determinant function is a polynomial in its entries. It thus follows immediately from the definition of the graded determinant and the arguments given in the previous example that gDet : Aut(E) -+ Ao is smooth.
2.27 Proposition. Let f E Coo(U; F) and 1 :::; j :::; n be arbitrary. Then there exist functions f(k) E Coo(U; F), f(O) = f such that Vx E U, Va E N n AE(ej) : 00
(2.28)
f( x 1 , ...
, x j - I ,xj
) + a, x j+I , ... , x n) = '"' L ak · f(k)( x. k=O
by means of (2.24) and [2.16]. If c( ej) = 1, i.e., if the j-th f2?ik = 0 whenever j E {i I , ... , id· If j r:J. {iI"'" id, we define f2.~.ik = ±hO ... jk' where {jo, ... ,jd = {j, iI, ... ,ik} and where the sign is
Proof We will define the
f(k)
coordinate is odd, we define
determined by the position of j within the (increasing) sequence il ... ik : if j is in m-th position, the sign is (_l)m-I. (In particular f~I) = fi,) For k > 1 we define f(k) == O. The result then follows from (2.24). If c( ej) = 0, we first note that by using [2.21] and (2.24) we can reduce the problem to functions of even coordinates only. For a fixed sequence iI, ... ,if we then define fi(,~~.ii by fL~~.ii = k!· (oxj)kfi, ... ii' This makes sense because functions of even coordinates only can be interpreted as ordinary infinitely often differentiable functions. And then a IQEDI simple but tedious computation using definition (2.12) completes the proof.
2.29 Remark. [2.27] is similar to [2.1], but stronger in that the f(k) are defined on the whole of U (and do not depend on the point x o ). We will see in [3.7] that if A satisfies (C [HIJ), then we can interpret the functions k! . f(k) as the k-th order partial derivatives of f with respect to the j-th coordinate. In this way we have a convergent Taylor series expansion for nilpotent increments. The obvious question which comes to mind when seeing this result is: couldn't we prove [2.27] directly, without the intermediate result [2.1]7 The answer is no, and the reason is the presence of a cover in property (Al) of smooth functions. The argument given in [ 1.12] can be used to show that in the real case this cover is not needed, in which case one could prove [2.27] directly. We did not follow this approach because it would not be valid in a complexified setting. In the approach we used here, we are sure that [2.27] is
112
Chapter III. Smooth functions and A-manifolds
valid in a complexified setting as well (i.e., a setting in which we consider commutative C-algebra such that A/N ~ C, see §II.l).
A as a graded
2.30 Discussion. In [1.27] we raised the question whether a linear map is determined by its restriction to the even part of an A-vector space. We now are able to prove that under suitable circumstances the restriction of a smooth family of linear maps to the even part of an A-vector space does indeed determine this family completely.
c Go be open, and let be a smooth map. Suppose furthermore that A satisfies (C [HI]) with q the odd dimension of G. If \/g E U : 'I/J(g)IEo = 0, then \/g E U : 'I/J(g) = 0. In other words, 'I/J(g)IEo uniquely determines 'I/J(g). 2.31 Proposition. Let E, F, and G be A-vector spaces, let U
'I/J: U ~ HOills(E; F)
Proof We treat the right linear case, the left linear case being similar. Let (ei) and (Ii) be bases of E and F respectively (in their equivalence classes !). Since the map HOillR(E;F) x E ~ F, ('I/J,e) f---+ 'I/J(e) is smooth, as are the maps Ji, we conclude that the maps 'l/Jj : U x E ~ A, (g,e) f---+ Ji('I/J(g)(e)) are smooth. Evaluating 'l/Jj on a basis vector ei gives us the middle matrix element 'l/Jj (g) (ei) = !VJR ('I/J(g))j i. Knowledge of these matrix elements completely determines 'I/J. If ei is an even basis vector, our assumption implies that 'l/Jj (g) (ei) = !VJR ('I/J(g))j i = 0, i.e., all the corresponding matrix elements are zero. Now suppose that ei is an odd basis vector. By [1.23-g] the map 1> : U X Al ~ A, (g,ry) f---+ 'l/Jj(g)(eiry) is smooth. By assumption it is identically zero and by linearity it is given by 1>(g, ry) = 'l/Jj (g)( ei) . ry. Since U x Al contains q + 1 odd coordinates and since A satisfies (C [HI]), we deduce from [2.21] that 'l/Jj (g) (ei) is zero. We conclude that all matrix elements of'I/J(g) are zero, i.e., 'I/J(g) is zero. IQEDI
3.
DERIVATIVES AND THE INVERSE FUNCTION THEOREM
The first part of this section is devoted to the definition of the (partial) derivative( s) of a smooth function. This requires a condition on A slightly stronger than the one mentioned in §2. The second part is devoted to the classical theorems ofdifferential calculus: the inverse function theorem, the implicit function theorem, invariance of graded dimension, and the canonical form of a smooth function of maximal rank.
3.1 Proposition. Let E be an A-vector space of dimension plq, let j belong to Coo (U; F), let1>a,i be as in property (Al) and let gi E Coo(U;F) be defined by gi(X) = 1>a,i(X,X). For i :::; p, gi is uniquely determined by j, independent of the index a; for i > p it is uniquely determined if and only if A satisfies (C [HI]).
§3. Derivatives and the inverse function theorem
113
Proof Fix i and let CPa,i and CPb,i be two possibilities. Let x, Y E Ua n Ub be such that x j -=f- yj =} j = i, i.e., x and y differ only in the i-th coordinate. It follows that (yi _ Xi). (CPa,i(y,X) -CPb,i(Y,X)) = O. Denoting by h the difference h = yi - Xi, it follows that the function CPa,i(Y, x) - CPb,i(Y, x) is a function b..(x, h) of the p + q + 1 variables (x, h). We thus are given the equality h· b..(x, h) = 0, while we want to prove that b..(x, 0) = O. Once we have that, smoothness of 9i on U follows from the fact that it is smooth on all Ua and [1.23-k] Consider first the case i :::; p, i.e., h is an even coordinate. According to expansion (2.24) there exist smooth functions (of even variables only) Xil, ... ,ik defined in a neighborhood of(x,O) suchthat
b..(x, h)
'""' i~ i = '""' LL~ k. (Xil ...I ik)(X , ... ,xP,h). 1 .••
k
ij
Taking real values for Xl, ... , x P and h, we deduce (as in [2.21]) from h· b..(x, h) = 0 that ~il ... k . h. Xil .. ,ik (xl, ... , x P, h) must be zero. Since the coordinates (~il, ... , ~ik) and (xl, ... ,xP, h) are independent, it follows that either ~il ... ~ik or h·Xil ... ik (xl, ... ,xP, h) must be identically zero. If the former, the term ~il ... ~ik . Xi1 ... ik (Xl, ... , x P, h) does not contribute to b..(x, h). If the latter, continuity ofXil ... ik (and the fact that it is real valued) implies that h·Xil ... ik (Xl, ... ,xP, h) is identically zero if and only if Xil ... ik (xl, ... ,xP, h) is identically zero. We conclude that b..(x, h) = 0 and in particular b..(x,O) = O. This shows that 9i is uniquely defined by f. Next let i > p, i.e., h is an odd coordinate. Again using expansion (2.24), there exist smooth functions Xil, ... ,ik and '0i 1 , ... ,ik defined in a neighborhood of (x, 0) such that
e
b..(x, h) =
L L e ek . ((Xi1 ... ik )(x\ ... , x P) + h· ('0il ... ik)(XI, ... , x P)) 1
k
.. ,
ij
Sinceh 2 = 0, h·b..(x, h) = 0 implies that eitherh.~il '" ~ik or Xil ... ik(X I , ... ,xP) must be identically zero. !f(C [HIJ) holds, the first is impossible, so Xi1 ... ik is identically zero, proving that b..(x, 0) = 0, i.e., that 9i is uniquely determined by f. On the other hand, if (C [HIJ) does not hold, we can change CPa,i by adding ~k, where k is chosen such that this product is not identically zero, but any k + l-fold product of odd elements is zero (in particular if k = 0, i.e., Al = {O}, we add the constant function 1). By hypothesis such a k :::; q exists. This does not invalidate (Al) but changes 9i also by this non-zero IQEDI amount, showing that 9i is not uniquely determined by f.
e ...
3.2 Remark. As in the proof of [2.16], the fact that we can reduce functions of even variables to real valued functions of real variables is essential. Otherwise, the nonHausdorff character of the DeWitt topology would invalidate the continuity argument.
3.3 Definition/Convention. As suggested already several times, we want to define the partial derivatives of a smooth function f by the diagonal of the functions CPa,i from
114
Chapter III. Smooth functions and A-manifolds
(AI), i.e., by the functions gi from [3.1]. In analogy with ordinary smooth functions and knowing that gi depends only upon f, we will denote these partial derivatives by gi == ad == af / axi, 1 :::; i :::; n = p + q. If we want to make an explicit distinction between even and odd coordinates we will use the notation axi f or af / axi for i :::; p and af.i-p f or af / a~i-p for i > p. According to [3.1] the partial derivatives ad are always defined for i :::; p, i.e., for the even directions. On the other hand, the partial derivatives in the odd directions exist only if A satisfies (C [q+lJ). This immediately raises the question what happens for A = R (or more generally when Al = {O}), because then (C[q+lJ) is certainly not satisfied. Of course this is no problem if there are no odd coordinates present, i.e., when E is without odd dimensions. But in all other cases it poses a problem. Thinking of the case A = R, in which we know that there ought to be only p partial derivatives, not p + q, we can "solve" this dilemma by just ignoring the undefined partial derivatives in the odd directions. That this is reasonable to do is confirmed in [v. loS], where we show that if Al = {O}, then we do not lose anything by ignoring the ad for i > p. For all other cases, i.e., when q > 0, Al =I {O}, and (C [q+IJ) not satisfied, we do have a problem. For that reason we will assume throughout the rest ofthis book that A satisfies (C [=J). Most of the time this is stronger than strictly needed, but it avoids changing (the condition on) A every time we change the odd dimension. It also guarantees that a smooth function f is always represented in (2.24) by the maximal number (only depending upon the odd dimension) of ordinary smooth functions fil ... ik. Of course, with this convention we do not cover the case A = R. However, most of the statements, including their proofs, remain valid for A = R. It is only occasionally that there will be a difference in treatment, but these we will point out in separate remarks.
3.4 Discussion. Let us now for the last time distinguish between an ordinary smooth (vector valued) function f of p real coordinates, and the (smooth) function Gf of p even coordinates. It follows immediately from the proof of [1.21] that aiGf = G( ad), which justifies at the same time our use of the symbol i for this operation, as well as our identification of f with Gf. With this knowledge, the reader should be able to convince himself of the correctness of the following result: if a function f E C= (U; F) is explicitly given by 2q functions fil ... ik E C=(BU, BF) (expansion (2.24)), then the partial derivatives are given by
a
(3.5)
af q ae(x) = L k=O
k
L L
In the last formula, the b~j is the Kronecker delta, always zero except if i j = £ when it
§3. Derivatives and the inverse function theorem
115
is 1. The sign (-l)j-l in the second formula appears when we put the coordinate ~ij in front of the rest.
3.6 Proposition. The maps they commute: [ai, aj ] = O.
ai belong to DerR(CCXl(U; A)), they have parity c(ei), and
Proof We will use the proof of [1.23] to prove the frrst two statements. From part (b) it follows that the parity of i is the same as that of ei. From parts (c,d) it follows that it is linear over R. (Nota Bene. CCXl(U; A) is an R-algebra and as such right and left linear are the same, the difference becomes apparent in the derivation property.) For homogeneous f, part (h) shows that ai(f . g) = (ad) . 9 + (-1) (e(ei)le(f)) . f . (aig), proving that ai is a right-linear derivation of parity c(ei). For the last statement we use (3.5). If Xi and x j are both even coordinates, the equality ai 0 aj = aj 0 ai follows from the same equality for ordinary real valued smooth functions and (3.5). If Xi and x j have different parities, the equality ai 0 aj = aj 0 ai follows immediately from (3.5). Finally, if Xi and x j are both odd coordinates, the equality [ai, aj] == ai 0 aj + aj 0 ai = 0 follows from (3.5) by an elementary calculation. IQEDI
a
3.7 Discussion. In §2 we have given two variants of an expansion that looked like a convergent Taylor series with a nilpotent increment. Let us show that it indeed is a Taylor series (up to factors k!). And remember, our convention that A satisfies (C [CXll) guarantees that the partial derivatives exist for any odd dimension. Looking at the proof of [2.27] and using that for an even coordinate x j we have ajGf = G(ajf), we see immediately that the functions k! . f(k) in [2.27] are exactly the k-th order partial derivatives with respect to an even xj. The same result follows from the proof of [2.27] for odd coordinates, once one realizes that for odd coordinates (a j )2 = 0 [3.6]. We thus are allowed to write (finally) for homogeneous nilpotent a of the correct parity: CXl k CXl k ak a ((aj )kf)( X) -_ ""' a (axj)k f (X) . f( X1 , ... ,x j-l ,Xj +a,x j+l , ... ,xn) -_ ""' L."kf· L."kf· k=O
k=O
In case there are no odd coordinates and if we apply this expansion repeatedly to all (even) coordinates, we obviously recover definition (2.12) of Gf. But also in the case of odd coordinates have we already obtained such an expansion! Using (3.5) we have an alternative way to describe the functions fil ... ik appearing in the expansion (2.24): they are given as partial derivatives of f with respect to odd variables:
It is obviously a circular argument to think that we could define the functions fil ... ik in terms of derivatives. We needed them in order to be able to show that (partial) derivatives
116
Chapter III. Smooth functions and A-manifolds
exist. On the other hand, the expansion (2.24) now can be written in the form of a Taylor expansion: q
I(x,~)
=
L k=O l~i, 1l"F 0 h 0 s.
1.23 Lemma. Let p : B -+ M be a fiber bundle, U an open set in M and S E fu(B) a local smooth section. Then the image s(U) is a submanifold ofB and p : s(U) -+ U is a diffeomorphism
Proof Since being a submanifold is a local property, we may assume without loss of generality that U is a trivializing chart. If'l/J: 1l"-l(U) -+ U x F is a trivializing diffeomorphism, it transforms the image s(U) into the graph Gr(s1j;). The result then follows from [111.5.13]. IQEDI
1.24 Construction. Let p : B -+ M be a bundle with typical fiber F and structure group G, and let U be a trivializing atlas. It is easy to show, using the pseudo effectiveness of the action, that the collection of maps 'l/Jba has the following properties:
(1.25)
'l/Jaa (m) = ide the identity element in G , Ua nUb n Uc :::} 'l/Jcb(m)· 'l/Jba(m) = 'l/Jca(m) .
Vm E Ua mE
What we will do is show that the collection {Ua, 'l/Jba} completely determines the fiber bundle B. To make this more precise, let M and F be A-manifolds and let G be an A-Lie group with a smooth action on F. Suppose also that we have an atlas U = { 'Pa : Ua -+ Ga } of M and smooth maps 'l/Jba : Ua n Ub -+ G satisfying (1.25). With these data, we claim, we can (re)construct a bundle p : B -+ M with typical fiber F and structure group G such that the maps 'l/Jba appear as described above. To define the A-manifold B and the surjection p : B -+ M we will use [111.4.9]. The ingredients for this construction are the open sets {ja = Ga xF, the subsets {jab = Gab xF and the transition functions rpba defined by rpba(X,f) = ('Pba(X),'l/Jba('P;;l(x))(f)). To realize that these ingredients satisfy (111.4.10), remember first that Gab = 'Pa(Ua nUb) and that 'Pba = 'Pb 0 'P;;l. It follows immediately that the maps rpba are smooth; that they also satisfy (111.4.10) follows from (1.25). According to [111.4.9] we thus may conclude that B is a well defined A-manifold. The projection p : B -+ M we define in local charts by p : (x, f) E {ja = Ga x F I--> X E Ga. It should be obvious from the definition of the maps rpba that this is a well defined, smooth sUljective map as required.
Chapter IV. Bundles
150
It thus remains to exhibit the existence of an atlas oftrivializing charts satisfying (FB 1) and (FB2). We claim that the atlas {(Ua, 'l/Ja) I a E I} with 'l/Ja = (cp;;-l X id(F)) 0 i{5a will do. But this should be obvious from the commutative diagram
p-l(Ua) = Ua
pi Ua
-----'Pa
Oa = Oa
X
F
'1';;1 xid(F) )
1
1
KU a
KOa
-----'Pa
Oa
Ua x F
-1 'Pa
------
Ua
and the definition of the transition functions i{5ba for the A-manifold B (and recall that the charts CPa and i{5a are recovered/defined as in [111.4.9]). We thus have shown that the data {Ua, 'l/Jba} determine a bundle p : B -+ M with typical fiber F and structure group G. To show that this bundle is "unique," we proceed as follows. Suppose p : B -+ M and p: -+ M are two bundles with typical fiber F and structure group G. Suppose furthermore that {(Ua, 'l/Ja)} is a trivializing atlas for B and that {(Ua, -J;a)} is one for i.e., the same charts on M but different trivializing maps (by taking intersections we can always accomplish this). Ifwe suppose that the transition functions'l/Jba and -J;ba are the same (i.e., that 'l/Jb 0 'l/J;;l = -J;b 0 -J;;;l as diffeomorphisms of (Ua nUb) x F), then we can define an isomorphism of fiber bundles h : B -+ by hl p -1(Ua ) = -J;;;l 0 'l/Ja. The fact that the transit~on functions are the same guarantees that
ii
ii,
ii
this is well defined. We conclude that Band B are isomorphic fiber bundles. And thus we have proven that the data {Ua, 'l/Jba} determine the bundle p : B -+ M uniquely up to isomorphisms.
1.26 Remark. Attentive readers will object that the above construction is faulty because we cannot apply [111.4.9]. They are right because F is not an open set in the even part of some A-vector space. However, taking an atlas for the A-manifold F and replacing the open sets Oa by direct products of a chart Oa and a chart for F will make the construction a valid one. The details are left to the reader.
1.27 DiscussionlDefinition. What we have shown in the above construction is that giving a bundle B over an A-manifold with typical fiber F and structure group G is completely equivalent to giving an atlas U and transition functions 'l/Jba : Ua n Ub -+ G satisfying (1.25). It thus follows that such a set oftransition functions defines the bundle p : B -+ M. It is this way that we will define and/or construct most of our bundles.
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§2. Constructions offiber bundles
2. CONSTRUCTIONS OF FIBER BUNDLES In this section we discuss various constructions of new fiber bundles out of old ones: associated bundles, pull-back bundles, and product bundles. We also define the notion of a principal fiber bundle and we show that all fiber bundles can be seen as associated to a principal fiber bundle.
2.1 Construction (associated bundles). Let p : B -+ M be a bundle with typical fiber F and structure group G. Let furthermore H be another A-Lie group with a pseudo effective smooth action on an A-manifold E. Now suppose we have an A-Lie group morphism p : G -+ H. Associated to p and B we will construct a new bundle over M with typical fiber E and structure group H, which we will denote as pP : BP,E -+ M, and which is called an associated bundle, associated to B by the "representation" p. To that end, let U = {(Ua,'l/Ja) I a E J} be a trivializing atlas for B with the associated set of transition functions'l/Jba' It is easy to verify that the functions po 'l/Jba satisfy the requirements of [1.24]. We thus obtain a bundle pP : BP,E -+ M with typical fiber E and structure group H defined by these po 'l/Jba. Moreover, the construction is such that we automatically have a trivializing atlas U' = {(Ua , 'l/J~) I a E I} for the new bundle BP,E with the same trivializing charts Ua as U and whose associated transition functions are the po 'l/Jba. (Note that this implies that for each trivializing chart (U, 'l/J) for B we have a corresponding trivializing chart (U, 'l/J') for BP,E, just by adding (U, 'l/J) to U.) There are two ways to assure that the result does not depend upon the choice of a trivializing atlas. The first is to take the whole fiber bundle structure as trivializing atlas (the biggest possible). The second is to show that a different choice oftrivializing atlas leads to an isomorphic fiber bundle. The details of this are left to the reader.
2.2 Construction (pull-back bundle). Let q : 0 -+ N be a fiber bundle with typical fiber -+ N be a smooth map. We will construct a fiber bundle p == g*q : B == g*O -+ M with typical fiber F and structure group G, as well as a fiber bundle map !J : g* 0 -+ 0 inducing g. We thus will have a commutative diagram
F and structure group G and let 9 : M
B == g*O
9
------+
p=g'q 1 M
0
lq ------+
N.
9
Moreover, the construction will be such that the restriction!J : p-l(m) -+ q-l(g(m)) is a diffeomorphism. This fiber bundle will be called the pull-back of Cover g. Let us start with the abstract set-theoretic definition. Consider the direct product M x C and the subset g*O = {(m, c) EM x 0 I g(m) = q(c)}. We claim that the surjection p == g* q = 7r M : g* 0 c M x 0 -+ M is the fiber bundle we are looking for and that the map iJ = 7rG : g*O c M x 0 -+ 0 is the bundle map. What remains to be shown is that
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g*C is a fiber bundle over M in the sense of A-manifolds and that all maps involved are smooth. Let V = {(Va, Xa) I a E J} be a trivializing atlas for C and letU = {Ua I a E J} be an atlas for M such that for each Ua E U there exists Vg(a) E V such that g(Ua ) C Vg(a) (we use the same symbol 9 to denote the map on indices). We now define the map 'l/Ja : p-l(Ua ) C M x C ~ Ua x F by 'l/Ja: (m,c)
f---4
(m,7rp(X g(a)(c))) ,
where Xg(a) : q-l(Vg(a)) ~ Vg(a) x Fis the local trivialization. It is elementary to check that 'l/Ja is a bijection. Moreover, one also can check that we have
where Xa{3 denotes the transition function with respect to the atlas V. In other words, B == g*C is a fiber bundle with transition functions 'l/Jba = Xg(b)g(a) 0 9 (use [1.24]). Finally, there are two ways to show that g is smooth. The first is to note that g*C is a submanifold of M x C and then applying [111.5.7]. The second is to note that on the open set p-l(Ua ) the map g is defined by
g = X;(~) 0 (g x id(F)) o'l/Ja
.
In other words, in the trivializations (u a , 'l/Ja) and (Vg(a), Xg(a)) the map g is given as (m, f) f---4 (g(m), f). The result then follows from [111.4.18]. This also shows that g is a diffeomorphism when restricted to a fiber: in the given local trivializations it is the identity onF.
2.3 Example. Let p : B ~ M be a fiber bundle and N a submanifold of M with canonical injection i : N ~ M. It is immediate from the definition of a pull-back bundle that i* B is isomorphic to the restricted bundle BIN = p-l(N). and that { : i* B ~ B corresponds to the canonical injection of BIN in B. In other words, restriction ofa bundle to a submanifold is a particular case of a pull-back bundle.
2.4 Proposition. Let h : B ~ C be a fiber bundle map between the fiber bundles p : B ~ M and q : C ~ N, inducing the map 9 : M ~ N. Then there exists a unique fiber bundle map hind: B ~ g*C such that h = hind 0 g, i.e., we have a commutative diagram h
B
------>
g*C
------>
pI M
Ig*q ------> id(M)
C
9
hind
M
Iq
------> 9
N.
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§2. Constructions offiber bundles
Proof Let us start with uniqueness. If b E B then, by our definition of fiber bundle map, hind(b) must lie in (g*q)-l(p(b)). Since gis a bijection from (g*q)-l(p(b))to q-l (g(p(b))), the condition h = hind 0 !J implies the sought for uniqueness. To show the existence 0 f hind, we define the map hind : B -+ g*O by the formula hind(b) = (p(b), h(b)) E g*O c M x O. The image lies within g*O because h is a fiber bundle map. To verify that hind is smooth one uses the local charts defined in the IQEDI construction of the smooth structure of g*O; this is left to the reader.
2.5 Corollary. The map h I--> hind is a 1-1 correspondence between fiber bundle maps h : B -+ 0 that induce a given map 9 : M -+ N andfiber bundle maps hind: B -+ g*O (which induce by definition the identity on M).
2.6 Discussion. The above corollary shows that we can always transform a question concerning a fiber bundle map between two fiber bundles over different spaces into a question concerning fiber bundle maps over the same base space. This is another justification not to consider very extensively fiber bundle maps between fiber bundles with the same base space that do not induce the identity.
2.7 Construction (product bundles). Let p : B
-+ M and q : 0 -+ M be two bundles over M with typical fiber F and structure group G respectively E and H. Let { (Ua, 'l/Jba) } respectively { (Ua, Xba) } be the atlas with the transition functions defining the bundle B respectively C. Indeed, by taking, if necessary, pairwise intersections, we may assume that both atlases have the same set of charts Ua • We now define the smooth maps 'l9 ba : Ua n Ub -+ G x H by 'l9 ba (m) = ('l/Jba(m), Xba(m)); they obviously satisfy (1.25). Moreover, there exists a natural smooth action W of G x H on F x E defined by W(g, h)(f, e) = ( (~, 1], ~1]a) is a right linear vector bundle morphism. But ker( h) is not a subbundle. Even more, except for ~1] = 0, ker(hIB(E,ry)) is not a graded subspace of B(f.,1)) (though it is a graded submodule).
4.
CONSTRUCTIONS OF VECTOR BUNDLES
We show that the various operations one can perform on A-vector spaces (quotients, direct sums, tensor products, exterior powers, homomorphisms) have their analogonfor vector bundles. Even the constructions of new linear maps out of old ones have their counter part in vector bundle morphisms. As a byproduct ofthese constructions, we explain why we took the full A-vector space as typical fiber for vector bundles instead ofonly its even pan.
4.1 Discussion. Let p : B ---+ M and q : C ---+ M be vector bundles over the same base space with typical fibers E and F. For m E M this implies that Bm is isomorphic to E and C m to F. Since these spaces are free graded A-modules, we can perform various operations on these fibers, e.g., taking the direct sum Bm EB Cm, or the tensor product Bm 0 Cm. Whence the obvious question: can we construct in a natural way out of Band C a vector bundle r : D ---+ M whose typical fiber is, say, E EB F such that we have a natural identification Dm ~ Bm EB Cm. The obvious way to do this is to define D = UmEM(Bm EBCm ). The only draw-back is that we have to indicate the structure of an A-manifold on this D. In the next constructions we will do exactly that, although in a roundabout way. We will define vector bundles with
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the appropriate typical fiber, and we will show once (in [4.5] for the direct sum case) how this constructed vector bundle relates to D.
4.2 Construction (quotient bundle). Let P : B -+ M be a vector bundle with typical fiber E, C a subbundle with typical fiber F, and U a trivializing atlas adapted to the subbundle C. Let G be the subgroup of Aut(E) that leaves the graded subspace FeE invariant. Since Gleaves F invariant, it induces an action on the quotient ElF, i.e., there exists a group morphism p : G -+ Aut(E / F). Using a basis as in [3.13] shows that p is smooth (it is given by
(~ ~)
f---4
d). We thus can construct the associated vector bundle
with typical fiber ElF. It is called the quotient bundle of B by C and denoted by BIC.
4.3 Construction (direct sums). Let Pi : Bi -+ M, i = 1, ... ,n be a finite family ofvector bundles over M with typical fibers E i . Let { Ua, 'l/Ji,ba } be an atlas and transition functions defining the bundle B i . Indeed, by taking n-fold intersections we may assume that the charts Ua are the same for all bundles. Since the maps 'l/Ji,ba : Ua n Ub -+ Aut(Ei ) are smooth, it follows that the map 11 'l/Ji,ba : Ua n Ub -+ 11 Aut(Ei ) c Aut( EBiEi) defined as (11 'l/Ji,ba)(m) = 11 'l/Ji,ba(m) [1.7.7] is smooth. From the definition of 11 'l/Ji,ba(m) we deduce the equality
on the triple intersection Ua n Ub n Uc. Hence { Ua, 11 'l/Ji,ba } defines a vector bundle over M with typical fiber EBiEi whose structure group can be reduced to 11 Aut(Ei ). This bundle, denoted by EBiBi, is called the direct sum bundle of the vector bundles B i ; it is also called the Whitney sum of the B i . In case there are only two vector bundles involved, one writes Bl EB B2 for this direct sum bundle.
4.4 Remarks. • The definition of a direct sum of two vector bundles is essentially the same as the definition of a direct product bundle. Strictly speaking we should have indexed the EB in Bl EB B2 with an M to distinguish this sum from the direct sum of A-vector spaces. However, a vector bundle will hardly ever be at the same time an A-vector space, so no confusion is possible, since a direct sum of two A-manifolds is not defined . • Since we can reduce the structure group ofEBiBi to 11 Aut(Ei ), it should be fairly obvious that each separate bundle Bi can be seen as a subbundle of EBiBi.
4.5 Discussion. Let P : B -+ M and q : C -+ M be vector bundles over the same base space with typical fibers E and F. It follows that B EB C is a vector bundle with typical fiber E EB F. In [4.1] we suggested the space D = UmEM(Bm EB Cm), with the obvious projection r : D -+ M, as a candidate for this vector bundle. Let us show that we can
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identify these two spaces, i.e., that we can equip D with the structure of a vector bundle over M with the same transition functions as the bundle B EB C. Let U be a trivializing atlas for both bundles at the same time with diffeomorphisms 'l/Ja : p-1(Ua) -+ Ua x E and Xa : q-l(Ua) -+ Ua x F that induce the transition functions 'l/Jba and Xba. By definition of the structure ofa free graded A-module on each fiber, the maps 'l/Ja(m) : p-l(m) -+ E and Xa(m) : q-l(m) -+ F are isomorphisms. We thus obtain an isomorphism
'l/Ja(m)
X
Xa(m) : Bm EB Cm
-+
E EB F .
Varying mover Ua gives us a bijection between 1'-1 (Ua ) = UmEuJB m EB Cm) c D and Ua x (EEB F). Declaring this to be a diffeomorphism equips 1'-1 (Ua ) with the structure of an A-manifold, compatible with the projection 1'. It remains to show that these structures coincide on overlaps Ua nUb. But on overlaps we have (by definition of the transition functions) the equality ('l/Jb(m) x Xb(m)) 0 ('l/Ja(m) X Xa(m))-1 = 'l/Jba X Xba, which is a smooth function. It thus follows at the same time that the smooth structures coincide on overlaps and that the transition functions of the vector bundle D (yes, we just proved it is one!) are exactly the same as those of B EB C. In exactly the same spirit one should interpret the fibers of BIC (assuming of course that C is a subbundle of B) as being the quotient of the corresponding fibers in Band C :
Similar remarks apply to the forthcoming constructions of tensor product bundles, exterior powers and homomorphism bundles.
4.6 Definition. Let p : D -+ M be a vector bundle with typical fiber E and let B and C be subbundles with typical fibers Fl and F2 respectively. We will say that B and C are supplements if Vm EM: Bm EB Cm = Dm, i.e., if Vm EM: Bm and Cm are supplements to each other in the sense offree graded A-modules [11.2.1], [11.3.4].
4.7 Proposition. Let p : D then
-+
M, B, and C be as in [4.6J. If Band C are supplements,
(i) if(Si)~=1 E ru(D)is a set of local trivializing sectionsforB, if (t j )}=1 E ru(D)
is a set of local trivializing sections for C, then (Si' tj) is a set of local trivializing sectionsfor D, and (ii) B EB C is isomorphic as a vector bundle to D. Proof For (i) it suffices to note that by [3.14] Vm E U (si(m)) is a basis for Bm, and that (tj(m)) is a basis of Cm, and thus (si(m), tj(m)) is a basis of Dm because Bm EB C m = Dm (which means that the map (b, c) I--> b + c, Bm EB C m -+ Dm is an isomorphism). The announced result then follows again by [3.14]. For (ii) we use the interpretation [4.5] of B EB C = UmEM Bm EB C m and define (set theoretically) the isomorphism h : B EB C -+ D by hIB",EBc", : Bm EB C m -+ Dm,
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(b, c) I--> b + c. Since Band C are supplements, h is an even fiber bundle map, an isomorphism when restricted to each fiber, and inducing the identity on M. According to [3.6] it thus suffices to show that h is smooth in order to prove the result. To prove that h is smooth, we choose local trivializing sections (Si) E fu(B) C fu(D) and (tj) E fu(C) C fu(D) around a point mo E M (by shrinking we can always assume that we have the same U). We then denote by 'Ij; the local trivialization of B associated to the (Si) [3.10], by X the local trivialization of C associated to the (tj), and by ¢ the local trivialization of D associated to (Si, tj). Using the notation UmEM'Ij;(m) X x(m) as in [4.5], we have a commutative diagram r-I(U) U ",EM
=
U Bm mEM
1j;(m) xx(m)
U
X
EB C m
1
(FI EBF2 )
(m, (II, h))
h -----.
U Dm= mEM
1
U
",EM
-----.
p-I(U)
(m)=
UxE
-----. (m,lI+h),
just because of our choice of ¢. Since we know that all maps except h are diffeomorphisms, we conclude that h is smooth on r-I(U). Since the U cover M (by varying mo), we conclude. IQEDI
4.8 Remark. If B and C are subbundles of a vector bundle p : D -+ M, one might think that, in analogy with sums of graded subspaces, we could define a sum bundle as UmEM(Bm + Cm). However, this is a very unstable definition because there is no easy way to guarantee that Bm + Cm C Dm has a graded dimension independent of m EM. And if this graded dimension is not constant, we do not get a vector bundle in our sense.
4.9 Construction (tensor products). Let Pi : Bi -+ M, i = 1, ... , n be a finite family of vector bundles over M with typical fibers E i . Let { Ua , 'lj;i,ba } be an atlas and transition functions defining the bundle B i . Since the maps 'lj;i,ba : Ua n Ub -+ Aut(Ei ) are smooth, it follows that the map 0}=n 'lj;i,ba : Ua n Ub -+ 0}=n Aut(Ei ) C Aut( 0f=1 E i ) defined as (0}=n'lj;i,ba)(m) = 0}=n'lj;i,ba(m) [1.7.12] is smooth. (Nota Bene. We only have an inclusion 0;=n Aut(Ei ) C Aut( 0f=1 E i ) because the even part of a tensor product is not the tensor product of the even parts.) From the definition of 0}=n 'lj;i,ba(m) and the fact that the 'lj;i,ba (m) are even, we deduce the equality
on the triple intersection Ua n Ub n Uc. Hence { Ua , 0}=n 'lj;i,ba } defines a vector bundle over M with typical fiber 0f=1 E i . This bundle, denoted by 0f=1 Bi (or by BI 0···0 Bn to make the order even more explicit), is called the tensor product bundle of the vector bundles B i .
§4. Constructions of vector bundles
167
4.10 Construction (exterior powers). Let p : B ---; M be a vector bundle with typical fiber E, let { Ua, 'l/Jba } be an atlas and transition functions defining the bundle B and let k be a positive integer. Since the map 'l/Jba : Ua n Ub ---; Aut( E) is smooth, it follows that the map N'l/Jba : Ua n Ub ---; Aut(N E) defined as (N 'l/Jba)(m) = N'l/Jba(m) [1.7.15] is smooth. From the definition of!\k 'l/Jba(m) we deduce the equality
on the triple intersection Ua n Ub n Uc . It thus follows that {Ua,!\k 'l/Jba} defines a vector bundle over M with typical fiber !\k E. This bundle, denoted by !\k B, is called the k-th exterior power of the vector bundle E. According to [1.7.15] we have N 'l/Jba(m) = 'l/Jba(m) and N 'l/Jba(m) = id(A). Together with N E = E and N E = A this gives N B = Band N B = M x A.
4.11 DefinitionlDiscussion. Let p : B ---; M be a vector bundle with typical fiber E. Since elements of Aut(E) are even, they preserve Ea (a E Z2); we thus obtain a smooth action of Aut(E) on the A-manifold Ea. Since the action of Aut(E) on Ea is pseudo effective [1.5], we can apply the construction of an associated bundle with the identity map p = id(Aut(E)), but with an action on different A-manifolds. The resulting fiber bundle will be denoted by p : Ba ---; M, called the degree-a part ofE. Another way to define Ba is as a submanifold of B by Ba = {b E Bib E p-l(p(b))a}. That it is a submanifold of B follows immediately from the fact that Ea is a submanifold of E ~ Eg. That it is indeed the same as the associated bundle defined previously is left to the reader. In order to study more carefully the above situation, let us denote by 'l/Jba the transition functions of the bundle B, and by 'l/Ja,ba those of the bundle Ba. These maps are all essentially the same, they act on different spaces. Since E = Eo X El in terms of A-manifolds, it is natural to look at the transition functions 'l/JO,ba X 'l/Jl,ba who act on Eo x El ~ E, as do the 'l/Jba. An elementary calculation shows that these two sets of transition functions are the same. We conclude that the product bundle Bo XM Bl is isomorphic to the original bundle E. Although the fiber bundle Ba is not a vector bundle, it resembles it very closely. The abelian group Ea is stable under multiplication by elements of AD, i.e., Ea is an Aomodule. The action of Aut(E) on Ea preserves this structure and thus the fibers of Ba have the structure of an AD-module. Copying the discussion for vector bundles shows that f(Ba) is a CCXl(M)o-module. More precisely, one can show that we have an identification r(Ba) = r(B)a, a result in complete agreement with the corresponding fact for smooth functions CCXl(M; F)a = CCXl(M; Fa), and a result which is rather obvious once we see Ba as a submanifold of E. It then follows that we can apply (adapt) the results of [3.9] to sections of Be in particular that local sections of Ba can be glued together to a global section by means of a partition of unity.
4.12 Discussion. Let us now explain why we defined vector bundles as fiber bundles with as typical fiber a full A-vector space E (seen as the even part of E~), instead of the
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even part of an A-vector space. The reason is that we could not have performed all the constructions of vector bundles as we did, in particular not the tensor product. Let us show the problem. As usual we take A = 1\ X where X is an infinite dimensional vector space over R with basis (ei)iEN. We then consider the A-vector space E of dimension 011, for which we have a natural identification E 0 E ~ A (if bEE is a basis, we can identify b 0 b with 1 E A). If we consider only Eo, it will not be an A-module, but only anAo-module. Hence we cannot take the tensor product over A, but only over AD. The interesting question is whether this tensor product Eo 0Ao Eo is in a natural way the even part of an A-vector space, and more in particular whether it is the even part of E 0A E. As a set Eo is isomorphic to AI, and by definition of A, the ei E 1\1 X C Al generate Al as an AD-module. It follows that the elements ei 0 ej generate the tensor product Eo 0Ao Eo (over AD). As said before, E 0A E ~ A, and in particular (E 0A E)o ~ AD is a free AD-module on one (even) generator. Now suppose that Eo 0Ao Eo is a free Ao-module on one generator gEED 0Ao Eo. Since the ei 0 ej generate Eo 0Ao Eo, there exist Aij E AD (finitely many!) such that 9 = I:ij Aijei 0 ej. If we now take n E AD to be the (exterior) product of all different e/s appearing in this sum (adding one not in that sum if there is an odd number of different ei 's), then n is not zero (X is infinite dimensional), but ng is zero. The conclusion of these computations is that in general the AD-modules (E 0 A F)o and Eo 0Ao Fo are not the same. On the other hand, if Band C are vector bundles with typical fibers Eo and Fo respectively, it is quite natural to require from a tensorproduct of bundles that the typical fiber of B 0 C is (E 0 F)o. Since this cannot be done by means of the tensor product of AD-modules, we use the full A-vector space E as fiber, take the tensor product in the category of A-vector spaces, and restrict later to the even (or odd) part.
4.13 Discussion. Of course all our definitions and constructions of fiber bundles and vector bundles are also valid in the context of R-manifolds, i.e., in the special case A = R. Moreover, the body map B maps fiber/vector bundles in the category of A-manifolds to the corresponding objects in the category of R-manifolds. But in the category of R-manifolds something special happens with vector bundles. If X = Xo EB Xl is an R-vector space of dimension plq, its separate parts Xc< can also be seen as R-vector spaces: Xo can be seen as the R-vector space Xo EB {O} of dimension plO, and Xl can be seen as the R-vector space {O} EB Xl of dimension 0lq. Since automorphisms of X are even, they respect this decomposition, and it is not hard to see that we have an identification Aut(X) ~ Aut(Xo) X Aut(X l ) (regarding Xc< as an R-vector space). From these observations it follows that if p : B --+ M is a vector bundle in the category of R-manifolds, then the Bc< are also vector bundles in this category. We are thus able to write B ~ Bo x M Bl ~ Bo EB B l , i.e., in the category of R-manifolds the homogeneous parts of a vector bundle are vector bundles in their own right and a vector bundle is the direct sum of its even and odd part. We said that R-manifolds are nothing more (nor less) than manifolds in the classical sense. We now see that the same is not true for vector
§4. Constructions of vector bundles
169
bundles: vector bundles in the category of R-manifolds are not vector bundles in the classical sense, they are a direct sum of two vector bundles in the classical sense.
4.14 Discussion. Let p : B ---+ M and q : C ---+ N be vector bundles with typical fibers E and F respectively, and let h : B ---+ C be a right linear vector bundle morphism, inducing the map 9 : M ---+ N on the base spaces. Let furthermore V be a trivializing atlas for C with transition functions Xdc and U a trivializing atlas for B with transition functions 'l/Jba such that for all Ua E U there exists a Vg(a) E V such that g(Ua) C Vg(a). (As in [2.2] we use the same name for the function on the base space as on the indices.) By definition of a fiber bundle morphism there exist smooth functions ha : Ua x E ---+ F determined by (Xg(a) 0 h 0 'l/J;;l )(m, e) = (g(m), ha(m, e)). Since h is a vector bundle morphism, the maps ha (m, e) are right linear in e for fixed m. According to [3.17] we can interpret iliem as smooth functions ha : Ua ---+ HomR(E; F). Since h is globally defined, the ha are related on overlaps Ua n Ub by (4.15) On fue other hand, given a set of smooth functions ha : Ua ---+ HomR( E; F) verifying (4.15), we can construct a right linear vector bundle morphism h : B ---+ C inducing the map 9 : M ---+ N as follows. The restriction of f to p-l(Ua) is defined as the smooth map
where 9 x ha denotes the map Ua x E ---+ Vg(a) x F, (m, e) I--> (g( m), ha(m)( e)). Using the condition (4.15) it is elementary to show that fuese local expressions of f coincide on overlaps. In this way we obtain a bijection between right linear vector bundle morphisms and sets of smooth local functions ha : Ua ---+ HomR(E; F) verifying (4.15). And obviously a similar result is valid in the left linear context.
4.16 Construction (Hom-bundles). Let p : B ---+ M and q : C ---+ M be two vector bundles with typical fibers E and F respectively. Let U be an atlas of trivializing charts for both bundles and denote by 'l/Jba : Ua n Ub ---+ Aut(E) and Xba : Ua n Ub ---+ Aut(F) their transition functions. For a fixed mE Ua n Ub we obtain an automorphism Hba(m) of HomL(E; F) defined by (4.17) Using [1.7.12], the identification HomL(E; F) ~ *E0F, and that'l/Jba(m) and Xba(m) are even (and thus left and right linear), we can identify Hba(m) with Xba(m) 0 *'l/Jba(m)-l. We thus obtain smooth maps Hba(m) : UanUb ---+ Aut(HomL(E;F)) ~ Aut(*E0F).1t is by now elementary to show that HCb(m) 0 Hba(m) = Hca(m) on the triple intersection Ua n Ubn Uc. It thus follows that { Ua, Hba } defines a vector bundle over M with typical fiber HomL(E; F). This bundle is denoted by HomL(B; C) and called the bundle of (left)
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Chapter IV. Bundles
homomorphisms from B to C. In the same way we obtain a vector bundle HomR( B; C) with typical fiber HomR(E; F): the transition functions Hba are given by the same expression (4.17), but in the identification HomR(E; F) ~ F 0 E* they are given as Hba(m) ~ ('l/Jba(m)-l)* 0 Xba(m). If C is the trivial bundle C = M x A, the typical fiber of the bundle HomL(B; C) is HomL(E; A) ~ *E. One usually denotes this bundle HomL(B; M x A) by *B and calls it the (left) dual bundle of B. Since for the trivial bundle M x A all transition functions Xba are the identity on A, it follows that the transition functions of the dual bundle are given by *'l/Jba,l 0 id(A) ~ *'l/J/;a1 . In a similar way one defines the right dual bundle B*.
4.18 Discussion. Let p : B -+ M and q : C -+ M be two vector bundles with typical fibers E and F respectively. The name bundle of homomorphisms for the vector bundle HomL(B; C) suggests that there might be a relation with vector bundle homomorphisms h : B -+ C (which by convention induce the identity on M). We will show that there is indeed such a relation: global smooth sections S ofHomL(B; C) are in bijection with left linear vector bundle morphisms h : B -+ C. Given a trivializing atlas, we have seen in [1.20] that smooth sections s of HomL (B; C) are in 1-1 correspondence with families of smooth maps Sa : Ua -+ HomL(E; F) satisfying the relations (1.21) with the transition functions Hba' But these conditions are exactly the conditions (4.15) (in the left linear version), once one realizes that 9 is the identity and that we use the same trivializing atlas for both bundles. We thus obtain a left linear vector bundle morphism h : B -+ C. We leave it to the reader to prove that the correspondence S I--> h is a bijection. At the intrinsic level this correspondence is very easy to understand: a section S E f(HomL(B; C)) determines, for every mE M, an element s(m) E HomL(B; C)m ~ HomL(Bm; Cm), and a left linear vector bundle morphism determines a left linear map hlB", : Bm -+ Cm. The given identification S I--> h is nothing more than saying that these two are the same: s( m) = hi B",'
4.19 Proposition. Let p : B -+ M and q : C -+ N be vector bundles and let h : B -+ C be a (left/right linear) vector bundle morphism inducing the map 9 : M -+ N. Then g* q : g* C -+ M is a vector bundle with the same typical fiber as C [2.2], 9 : g* C -+ C is an even vector bundle morphism, and hind: B -+ g*C is a (left/right linear) vector bundle morphism [2.4].
4.20 Proposition. Let 9 : M -+ N be a smooth map between A-manifolds. The operation of pull-back bundle, which transforms a fiber bundle over N into a fiber bundle over M with the same typical fiber, commutes with the constructions of vector bundles given above. E.g., if P : B -+ Nand q : C -+ N are vector bundles over N, then the vector bundles g* (B EB C) and (g* B) EB (g*C) are the same (isomorphic), as are g* (B 0 C) and
(g* B)
~
(g*C).
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§4. Constructions of vector bundles
Proof Let V be a trivializing atlas on N for the bundles over N that are involved, and let U be an atlas of M such that for each Ua E U there exists a Vg(a) E V such that g(Ua ) c Vg(a). In [2.2] it was shown that the 'l/Jba = Xg(b)g(a) 0 9 form the transition functions of the pull-back bundle over M when the Xdc are the transition functions of the bundle over N. Let us take the tensor product of two bundles as typical example, with transition functions Xl,dc and X2,dc. The transition functions of g* (B 0 C) are given by (Xl,g(b)g(a) 0 X2,g(b)g(a)) 0 g, which is the same as (Xl,g(b)g(a) 0 g) 0 (X2,g(b)g(a) 0 g), which are the transition functions of the bundle (g* B) 0 (g*C). IQEDI
4.21 Discussion. This result is not really surprising when we recall the general idea behind the vector bundle constructions. Let us again take the tensor product as typical example. The tensor product bundle B 0 C is the bundle whose fibers are just the tensor products of the originalfibers: (B 0 C)m ~ Bm 0 Cm. On the other hand, the pull-back bundle g* B is a bundle whose fiber(g* B)m is exactly the fiber Bg(m)' We thus see that all these constructions are done fiberwise, making it easy to understand the commutativity stated in [4.20].
4.22 Nota Bene. Restricting a bundle p : B ~ M to a submanifold N c M is a particular instance of a pull-back. It follows that restriction commutes with the vector bundle constructions. This fact will be frequently used in chapters V and VI without further mentioning.
4.23 Proposition. Let p : B ~ M and q : C ~ M be vector bundles over the same base space with typical fibers E and F respectively. Let 'l/Jba and Xba be the transition functions with respect to a joint trivializing atlas U. Finally, let J E BHom( E; F) be a smooth morphism of A-vector spaces. IfJ intertwines 'l/Jba and Xba, i.e., iffor each m E M we have a commutative diagram
E~F 1/Jba(m)
1 E
1
Xba(m)
J
------+
F,
then it induces a mo!yhism of vector bundles J : B ~ C. If J is ofparity a, is an isomorphism, J is an isomorphism of vector bundles.
J is too; if J
Proof For each Ua E U we define the map ha : Ua ~ Homs(E; F) as being the constant J. By commutativity of the given diagram, these functions ha verify (4.15) with 9 = id(M). Hence they define a vector bundle morphism i The remaining assertions are left to the reader.
IQEDI
Chapter IV. Bundles
172
4.24 Nota Bene. A trivializing atlas is never unique, and neither is a joint trivializing atlas. It follows that the transition functions 'l/Jba and Xba in [4.23] are not unique. The condition in [4.23] thus can be read as a search for a joint tri vializing atlas such that J intertwines the transition functions. However, in most applications a natural candidate for ajoint trivializing atlas is available whi~h satisfies the conditions. But the reader should be aware that the induced identification J depends upon the chosenjoint trivializing atlas.
4.25 Discussion. Using [4.23] we can construct many (iso)morphisms of vector bundles. Since all morphisms between various constructions of A-vector spaces as given in §1.7 (see [1.8.18]) intertwine the relevant transition functions, we obtain, among others, the following isomorphisms (finite dimensional A-vector spaces are f.g.p!) of vector bundles: HomR(B; 0) ~ 00 B*, *(B EB 0) ~ *B EB *0, etcetera. We also obtain an even morphism HOillR(B; 0) 0 B --+ 0, which can be interpreted as fiberwise evaluation. Even the interchanging map ~ satisfies the requirements, so we obtain a vector bundle interchanging map 9l : B 00--+ 00 B (abuse of notation).
4.26 Construction. In §1.7 we have given a number of constructions of new linear maps out of a family oflinear maps: the sum [1.7.2], the direct sum [1.7.6], the product [1.7.7], the tensor product [1.7.12], and the exterior power [1.7.15]. We will show that we can obtain, under suitable hypotheses, similar results for vector bundle morphisms. The basic idea is easily explained. Let the Bi be vector bundles over M, let the 0i be vector bundles over N, and let hi be a vector bundle morphism between the bundles Bi and Oi. If all vector bundle morphisms involved induce the same map 9 between the base spaces, they all map fibers above a point m E M to fibers above g( m) E N. Since the restrictions are linear, we can perform our constructions of new linear maps on these restrictions. This will certainly define an abstract map. Using [4.14] one can show that the new maps are vector bundle morphisms inducing g. We will state the precise result only for exterior powers; the other cases are similar and are left to the reader (cf. [1.7.24]).
4.27 Proposition. Let p : B --+ M be a vector bundle over M, let q : 0 --+ N be a vector bundle over N, and let h : B --+ 0 be an even (left) linear vector bundle morphism inducing 9 : M --+ N. Then there exists a canonically defined even linear vector bundle h: B --+ 0 inducing g. morphism
N
N
N
Proof Let V be a trivializing atlas for C with transition functions Xdc, and let U be a trivializing atlas for B with transition functions 'l/Jba such that the conditions of [4.14] are verified, i.e., we have local smooth maps ha : Ua --+ HomdE; F)o satisfying (4.15), where E is the typical fiber of Band F the typical fiber of C. Now recall that the transition functi ons of !\k B are given by !\k 'l/Jba, and those of !\k 0 by !\k Xdc. One then can prove easily the equality N hb(m) = (N Xg(b)g(a))(g(m)) 0 (N ha(m)) 0 (N 'l/Jba) (m)-l. As in [4.10] the maps m I--> !\k ha(m) are smooth. This proves that the local maps !\k ha : Ua --+ Homd!\k E; !\k F)o define a left linear even vector bundle morphism,
§S. Operations on sections and on vector bundles
173
which we denote by !\k h. We leave it to the reader to verify that it coincides with the map IQEDI described intuitively in [4.26].
N
N
N
4.28 Remark. For k = 0 the map h : B -+ C reduces to the "trivial" map M x A -+ N x A, (m, a) f-7 (g(m), a). See also [1.7.15].
5.
OPERATIONS ON SECTIONS AND ON VECTOR BUNDLES
In §4 we discussed various constructions of vector bundles. But in §3 we showed that the set of all sections ofa vector bundle is a graded C=(M)-module, and thus we can perform analogous constructions on the modules ofsections. In this section we show that there is a correspondence between these two sets ofconstructions: they commute with the operation of taking sections. For our purpose the most important result of this section is [5.14J, which says that sections of the k-th exterior power of the left dual bundle of a vector bundle B -+ M can be interpreted as graded skew-symmetric k-linear (over C=(M)) maps from sections of B to C=(M).
5.1 Discussion. If Pi : Bi -+ M are vector bundles over M with typical fibers E i , we know that the sets r(Bi) are graded C=(M)-modules. Since C=(M) is a graded Ralgebra, we can perform our constructions of direct sums, tensor products, exterior powers, etcetera on these modules. It is thus natural to ask whether these operations "commute" with the corresponding operations on the vector bundles, e.g., whether r(Bl EB B 2 ) is isomorphic to r(B 1 ) EB r(B2)' where the second direct sum is in the category of graded C=(M)-modules. The answer to this general question is positive, although the proofs have an increasing complexity when we go from direct sums via morphisms to tensor products. To be more precise, we will construct new sections out of old ones, and these constructions will all be pointwise, i.e., the new section at m E M will depend only on the values of the old sections at m. The first problem we then have to solve is whether the new section if smooth when all the old ones are, and whether the correspondence so obtained is an even morphism of graded C=(M)-modules. But the "real" problems come when we want to prove that these identifications are isomorphisms.
5.2 Construction. Let Pi : Bi -+ M, i = 1, ... ,n be a finite number of vector bundles with typical fibers E i . If sds a section of B i , we define a section J(EBiSi) of EBi Bi by
where we have used that the fibers of EBiBi are canonically isomorphic to the direct sum of the corresponding fibers in B i . To see that this is indeed a smooth section, it suffices to
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Chapter IV. Bundles
verify it on trivializing charts. If U is a trivializing atlas, the sections Si are represented by functions Si,a : Ua -+ E i , and the section J(EBiSi) by the function Ua -+ EBi B i , m f---4 EBisi(m). One could take this as a definition ofJ(EBisi), of course after having verified that these local functions glue together to form a global section [1.20] (which they do by definition of the transition functions of EBi Bi). Since the Si,a are smooth, their direct sum is smooth as well, proving that J( EBiSi) is a smooth section of EBi B i . Moreover, it is an elementary exercise (left to the reader) to show that J thus defined is an even morphism of graded C=(M)-modules.
5.3 Proposition. J : EBJ(Bi )
-+
f( EBi B i ) is an isomorphism of graded C=(M)-
modules. Proof IfJ(EBisi) = 0, then in all charts Ua we have EBisi,a(m) = 0, implying that the Si,a are identically zero on Ua, i.e., the Si are zero. We conclude that J is injective. If a section U of EBi Bi is represented by local functions U a : Ua -+ EBi Ei = I1 E i , we can project on the separate factors to obtain smooth functions Si,a : Ua -+ Ei such that ua(m) = (si,a(m))i=l = EBisi,a(m). By definition of the transition functions of EBi B i , these functions glue together to form global sections Si E f( Bi). Obviously J( EBiSi) = u,
IQEDI
proving surjectivity.
5.4 Construction. Let p : B -+ M be a vector bundle with typical fiber E and let C be a subbundle with typical fiber F. If S is a section of E, we define a section J(s) of E/C by
J(s)(m)
= 7r(s(m))
E
Bm/Cm
~
(B/C)m ,
where 7r denotes the canonical projection of Bm on Bm/Cm , and where we have used that the fibers of E/C are canonically isomorphic to the quotient of the corresponding fibers in E and C. To see that this is indeed a smooth section, it suffices (as for direct sums) to verify it on trivializing charts. If U is a trivializing atlas for E adapted to the sub bundle C, the section S is represented by the function Sa : Ua -+ E and the section J (s) by the function J(s)a : Ua -+ E/F, m f---4 7r(sa(m)) (abuse of notation: this 7r denotes the canonical projection E -+ E / F). (As for direct sums, we could take this as a definition of J( S ).) In a suitable basis for E[3.13] the last map is given as ignoring the last coordinates of Sa(m). Since Sa is smooth, J(S)a is smooth as well, proving that J(s) is a smooth section of E/C. Again it is straightforward to prove that J is an even morphism of graded C= (M)-modules.
f(B /C) induces an isomorphism of graded C=(M)-modules between f(B)/f(C) and r(B /C). 5.5 Proposition. The map J : r(B)
-+
Proof First note that, since C is a subbundle of E, each section of C automatically is a section of E, i.e., that r(C) is a graded submodule ofr(B). To prove injectivity, it thus
§S. Operations on sections and on vector bundles
175
suffices to show that ker(J) = r( C). But J (s) = 0 if and only iffor all m E Ua we have This is equivalent to saying that Sa takes its values in F, i.e., that S is a section of the subbundle C. To prove surjectivity, let u : M -+ B /C be a global smooth section, represented by local functions U a : Ua -+ E / F. Since F is a graded subspace of E, it follows that there exist smooth functions Ta : Ua -+ E such that U a = 7r 0 Ta. These local functions Ta represent local sections ta E ruJB), but there is no reason to assume that they satisfy condition (1.21). We thus choose a partition of unity Pa subordinated to the open cover U and define the global section s = Ea Pata [3.9]. We claim that J(s) = u, finishing the proof. For m E M we compute: J(s)(m) = 7r(s(m)) = Ea Pa(m)7r(ta(m)) = Ea Pa(m)u(m) = u(m). The last equality follows because Ea Pa(m) = 1 and the next to last equality follows because by construction all local sections ta (whenever defined) induce u when taken modulo Cm. IQEDI
o = J (s)a (m) = 7r( Sa (m)).
5.6 DiscussionINotation. As said in [5.1], the constructed identifications are all pointwise, and they work also if the initial sections are not smooth. To be more precise, let us denote by r- 1 (B) the set of all sections s : M -+ B of a vector bundle p : B -+ M, and let us denote by C-l(M) = r-l(m x A) the set of all A-valued functions on M. It follows immediately that r- 1 (B) is a graded C-l (M)-module. If we now look at [5.2], it is 1 obvious that the identification J : EBi (B i ) -+ r- 1 (EBi B i ) is an even morphism of graded C- 1 (M)-modules. It is also not hard to show that it is an isomorphism (just copy the proof of [5.3]). Since C=(M) is a subset of C-l(M), we can see this identification as a morphism of graded C=(M)-modules. [5.3] then tells us that when we restrict to smooth sections r(Bi) C r- 1 (B), we get an isomorphism onto the smooth sections r(EBi B i ) C r- 1 (EBi Bi). Obviously, a similar remark holds for quotients.
r-
5.7 Construction. Morphisms can be left or right k-linear and may eventually be graded skew-symmetric. We will take the right k-linear graded skew-symmetric maps as a typical example, leaving the other cases (such as the isomorphism r(Homs(Bl, ... , B k ; C)) ~ Homs(r(Bl)"'" r(Bk); r(C)) ) to the reader. We thus suppose that p : B -+ M and q : C -+ M are vector bundles with typical fibers E and F respectively, and we want to construct an isomorphism between r(Hom}f(Bk; C)) and Hom}f(r(B)k; r(C)). For a section ¢ ofHom}f(B k ; C) we define J(¢) E Hom}f(r(B)k; r(C)) as follows. For Si E r(B) the section J( ¢) (SI, ... , Sk) of C is defined by
Since ¢ and the Si are smooth sections, J (¢ ) (s 1, ... , Sk) is also smooth. It is straightforward to show that J(¢) is k-linear over C=(M) and graded skew-symmetric. At a higherlevel, J is left-linear over C=(M) in ¢ and even. Before we can prove that J is an isomorphism, we need some preparations. As in [5.6] we denote by r- 1 (B) the set of all sections of B and by C-l(M) the set of all A-valued functions on M.
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Chapter IV. Bundles
5.8 Lemma. Let p : B
-+
M be a vector bundle with typical fiber E, and let q : C
-+
M
be a vector bundle.
r-l(C) is a right k-linear graded skew-symmetric morphism of graded COO (M)-modules, thenforsectionssi E r(B) andm E M arbitrary, the value i!!(Sl, . .. , sk)(m) E Cm depends only upon the values si(m). (ii) !fin (i) we replace r(B) by r-l(B) andCOO(M) by C-l (M), then the conclusion still holds. (iii) Ifi!! : r(B)k -+ r-l(C) is a right k-linear graded skew-symmetric morphism of graded CXJ(M)-modules, then there exists a unique right k-linear morphism of graded C-l(M)-modules W : r-l(B)k -+ r-l(C) such that WIr(B)k = i!!. (i) Ifi!! : r(B)k -+
Proof • The proof of (i) breaks into two steps. Let Si and s~ be two sets of sections that coincide on an open set U 3 m. If p is a plateau function around m in U [111.5.21], we have the global equalities p' Si = P . s~ (because supp(p) C U). Since p is even, p(m) = 1, and i!! is k-linear over COO(M), we compute:
i!!( Sl, ... ,Sk) (m) = p( m)k . i!!( Sl, ... , Sk)( m) = i!!(pSl, ... , pSk) (m)
= i!!(ps~, .. . ,ps~)(m) = i!!(s~, ... , sU(m)
.
This proves that the value of i!!(Sl,"" sk)(m) does not depend upon the particular Si, provided they coincide on an open neighborhood of m. To prove that it only depends upon the values si(m) we assume that Si and s~ are two sets of sections such that Si (m) = s~ (m). Let (U, 'l/J) be a tri vializing chart for B, let p be a plateau function around min U, and let V cUbe an open set such that plv == 1. If (ej)';=l is a basis of E, we define local sections fj E ru(B) by fj(m') = 'l/J-l(m',ej)'
E j fj . S{. Since the support of p is contained in U, the support of the s{ (which is by definition contained in U) is contained in the support of p, and thus supp (s~) is closed in M, hence the s{ can be regarded as smooth functions on M. By the same argument Pfj can be seen as a global smooth section of B [3.9]. We thus obtain the equality of global sections Hence there exist smooth functions s~ E COO(U) such that (psi)lu =
U sing the k-linearity of i!! we compute by the first step
where
£il ... i k (Xl,
(from PSj =
... ,Xk) is a function on Ak obtained by shifting the coefficients s~j
Ei e i J -
1
sJij
)
out of i!! by k-linearity. These functions can be inductively
defined by £i(X) = X and £i1, ... ,ik+l (Xl"'" Xk+l) = 1t"'(Xl) . £i2, ... ,ik+l (X2,"" Xk+l) with a = E( eiJ + ... + E( eik+l)' Obviously the value of £il ... ik (si\ ... , s1k ) at m depends only upon the values of the s;j at m. Since the same is true for the sections s~, it
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§S. Operations on sections and on vector bundles
follows from the equality Si (m) = s~ (m) that ( S1, ... , Sk)( m) = ( si, ... , sU( m) as claimed . • To prove (ii), we can copy the proof of (i), except for the fact that the functions s{ need not be smooth. However, there is a much shorter proof. Suppose that Si and s~ are sections that coincide at m. Then we can define the (non-continuous) function p by p( m) = 1 and p( m') = 0 for all m' =I m. Then PSi = ps~ and hence linearity over C-1(M) proves (ii) . • To prove (iii) we start with uniqueness. So we suppose that W exists, and that Si are arbitrary sections. If (U, 'lj;), V, p, and sj i are as above (and note, the sj i need not be smooth), we know that
L W(P~i,,···, p~ik)(m) . £i, ... ik (si',···, Skk) = L (P~i,,··· ,p~iJ(m) . £i, ... ik (si',···, Skk) ,
W(S1,"" sk)(m) =
where the last equality is by the assumption WIr(B)k
=
. This proves that the element
W(S1,"" sk)(m) is completely determined by , hence it must be unique. On the other hand, we can use this formula to define W. More precisely, for sections Si : M -+ B and a point mE M we define W(S1,"" sk)(m) by
It remains to show that this is well defined, right k-linear over C-1(M), and graded skew-symmetric. To that end we note that by (i) the right hand side does not depend upon the specific choice for p, and hence neither does it depend upon U or V. An elementary but tedious computation using the explicit expression for £i, ... ik shows that it is also independent of the choice of the trivializing sections ~i (two different choices are related by a matrix of smooth functions, which can be brought inside itself is k-linear and graded skew-symmetric. To prove that this ¢ : M ---+ Homlf(B k ; C), m f---+ ¢(m) E Homlf(B~; Cm) is smooth, it suffices to show that in a trivializing chart (U, 'IjJ) the associated function ¢-.p : U ---+ Homlf(E k ; F)) is smooth. Now this function is smooth if and only if its coefficients with respect to a basis are smooth. If we use (U, 'IjJ), V, and p as in he proof of [5.S], these coefficients are given by ¢-.p (m) (eil , ... , eik) (or more precisely, by the coefficients of this vector with respect to a basis of F), and these are given on the neighborhood V (see above) by the formula
which is manifestly smooth.
5.11 Discussion. Let us introduce some names for spaces of right k-linear graded skewsymmetric morphisms: Homlf(r- 1(B)k; r-1(C)) will contain those that are linear over C-l(M), Homlf(r(B)k; r-l(C)) and Homlf(r(B)k; r(C)) those that are linear over C=(M). For any 1lf E Homlf(r-1(B)k; r-l(C)) we can take its restriction to r(B)k : Ii> = 1lflrCB)k E HomRk(r(B)k; r-l(C)). [5.S-iii] shows that the map 1lf f---+ Ii> is a bijection. On the other hand, Homlf(r(B)k; r(C)) is obviously a graded submodule of HomRk(r(B)k; r-1(C)). We conclude that we can interpret Homlf(r(B)k; r(C)) as a graded submodule of Homlf(r-1(B)k; r-1(C)). But the reader should be aware that the first space concerns morphisms linear over C= (M) and that the second one concerns morphisms linear over C-l(M). Looking at the definition of 'J in [5.7], it is obvious that it defines an identification r-1(Homlf(B k ; C)) ---+ Homlf(r-1(B)k; r-l(C)). Looking at the proof of [5.9], it is not hard to show that this is an isomorphism of graded C-l(M)-modules. The result of [5.9] can now be interpreted as saying that if we restrict attention to smooth sections in the source space, then we get an isomorphism onto the graded submodule Homlf(r(B)k;r(C)) C Homlf(r-1(B)k;r-1(C)). An interesting consequence is the following. Suppose we have an arbitrary section ¢ of Hom (Bk; C) and suppose that under the identification 'J it maps r(B)k to r( C), i.e., it maps smooth sections of B to smooth sections of C. Then ¢ must be smooth.
If
5.12 Definition. If h : B ---+ C is a (left or right linear) vector bundle morphism of bundles over the same base space, we define an induced map h* : r( B) ---+ r( C) by h* (s) = h 0 s. For a section S E reB), the section h*(s) E r(c) is usually called the pushforward of
§5. Operations on sections and on vector bundles
179
S by h. If h is left/right linear of parity a, the same is true for h* (which "obliges" us to write (( S II h*)) = soh in the left linear case). Also, if h is an isomorphism, so is h*. This applies especially to the isomorphisms given in [4.2S], which gives, for example, an isomorphism r(HomdB; C)) ~ r(*B 0 C).
5.13 Discussion. Let p : B ----+ M and q : C ----+ M be vector bundles over M. In [4.18] we argued that sections of HomdB; C) are in bijection with left linear vector bundle morphisms B ----+ C. With theidentification r(HomdB; C)) ~ Homdr(B); r(C)) and the push forward we can describe this bijection in a roundabout way. For a left linear vector bundle morphism h : B ----+ C we obtain a left linear morphism of graded C= (M)-modules h* E Homdr( B); r( C)). Under the identification of this space with r(HomdB; C)), we obtain an element J-1(h*) E r(HomL(B; C)). We leave it to the reader to verify that this is exactly the section of HomdB; C) we have defined in [4.18]. An interesting consequence of all these isomorphisms is that any morphism r( B) ----+ r( C) of graded C= (M)-modules is necessarily of the form h* for some vector bundle morphism h : B ----+ C.
5.14 Discussion. Using the bundle isomorphisms Homtk(Bk; C) ~ HomdN B; C) ~ *(N B) 0 C ~ N (*B) 0 C, the induced isomorphisms (by push forwards) on the modules of sections, and [S.9], we obtain an isomorphism between r(l\k (*B) 0 C) and Homtk(r(B)k; r(c)). If we take for C the trivial bundle C = M x A, we thus obtain the isomorphism
(5.15) and in particular for k = 1 :
r(*B)
~
*r(B)
and similarly
r(B*)
~
r(B)* .
One of the consequences of [S.9] for these particular examples is the following (see also [S.l1]). Let w : M ----+ I\k *B be any section and let Si : M ----+ B be smooth sections. If the function m f---+ L(sl(m), ... , sk(m))w(m) is smooth for all possible choices of the
Si,
then w must be smooth.
5.16 Construction. Let P : B ----+ M and q : C ----+ M be vector bundles with typical fibers E and F respectively. If S is a smooth section of Band t a smooth section of C, we define a section J( S 0 t) of B 0 C by
J(s 0 t)(m) = sCm) 0 t(m) E Brn 0 C m
~
(B 0 C)rn ,
where we have used that the fibers of B 0 C are canonically isomorphic to the tensor product of the corresponding fibers in Band C. Using arguments similar to those used for direct sums and a quotient, one can easily show that this defines indeed a smooth section and that J thus defined is an even morphism of graded C=(M)-modules.
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Chapter IV. Bundles
5.17 Proposition. J: reB) 0r(C) modules.
----+
r(B0C) is an isomorphism of graded C=(M)-
Proof An elementary but tedious verification shows that we have the following commutative diagram:
reB) 0r(C) [1.8.5]1
reB 0 C)
~
~
r(*(B*)) 0 r(C)
[1.8.5]
r(*(B*) 0 C)
[5.14]1 ~
*r(B*) 0 r(c)
r r
~ ------t
[L7.11]
Homdr(B*); r(C))
~ ~
[1.8.8]
f(HomdB*; C)) ,
[5.9]
in which two arrows are not yet known to be isomorphisms. In [7.25] we will prove that for any vector bundle P : B ----+ M the graded C=(M)-module r(B) is f.g.p, and thus by [1.8.8] the identification *r(B*) 0 r(c) ----+ Homdr(B*); r(c)) is an isomorphism. Hence the last arrow must also be an isomorphism. IQEDI
5.18 Discussion. If we have a finite family Pi : Bi ----+ ]'\11 of vector bundles with typical fibers E i , it is by now obvious how to construct an identification J : 0ir(Bi) ----+ r( 0iBi) : if the Si are sections of B i , J (S1 0 ... 0 S k) is defined by
Since we know that this identification is an isomorphism of graded C=(M)-modules when k = 2, we can use induction and associativity of the tensor product to prove that it is an isomorphism for all kEN.
5.19 Construction. Let P : B ----+ M be a vector bundle with typical fiber E. If S1, are smooth sections of E, we construct a section J(S1 1\ ... 1\ Sk) of I\k E by
•.. , Sk
By now standard arguments show that this defines a smooth section and that the map N reB) ----+ r(N B) is an even morphism of graded C=(M)-modules.
J:
5.20 Proposition. J modules.
N reB)
----+
r(N B)
is an isomorphism of graded C=(M)-
181
§6. The pull-back of a section
Proof A rather tedious verification shows that the map 'J can also be written as the following series of isomorphisms/identifications: [1.8.5]
Nr(B)
~
[5.14]
N r(*(B*))
[1.7.16]
N *r(B*)
-----'>
*(N r(B*))
[5.9]
[1.5.5]
~ [1.5.5]
r(Homtk((B*)k; M x A))
[1.8.16]
[1.8.5]
r(N B).
In [7.25] we will show that r(B) is f.g.p [7.25], and thus by [1.8.16] the identification ----+ *(N r(B*)) is an isomorphism. IQEDI
N *r(B*)
6. THE
PULL-BACK OF A SECTION
In this section we introduce operations which are crucialfor differential geometry: the exterior or wedge product ofsections and the contraction ofa section ofa bundle with a section of the k-th exterior power of its dual bundle. We also introduce the notion ofthe pull-back ofa section and we show how this relates to the various operations discussed before.
6.1 Discussion. Let p : B ----+ M be a vector bundle with typical fiber E. In [1.5.8] we defined the wedge product as a bilinear map I\k E x 1\£ E ----+ I\kH E, factoring in a linear E ----+ I\kH E. It is straightforward to show that this map intertwines map I\k E 0 the corresponding transition functions [4.23], giving us an even bundle morphism I\k B 0 1\£ B ----+ I\kH B. Now taking the push forward of this morphism acting on sections, we can form the even bilinear morphism of graded C=(M)-moduJes
N
Tracing the various morphisms immediately shows that this map is given by the pointwise formula 'J t\ (a,;3) (m) = (a( m)) 1\ (;3( m)). In view of this pointwise identification we will denote this bilinear map 'J t\ : B) x B) ----+ B) also by just a wedge product: 'Jt\(a,;3) == a 1\;3. Since the original map for A-vector spaces (or better, family of maps, since they depend upon the powers k and £) satisfies the relation A 1\ B = (_l)(€(A),€(B))+kf.B 1\ A, it follows immediately from the fact that this identification is pointwise that the same is true for the induced map on sections as described above. It follows that the graded C=(M)-moduJe EBkEZ B) can be given the structure of a (Z x Z2-graded) C=(M)-algebra. Combined with the identifications N r(B) ----+ B), which is also pointwise, we thus obtain a morphism of graded
r(N
r(N
r(N+£
r(N
r(N
182
Chapter IV. Bundles
commutative Coo (M)-algebras 00
00
J : /\r(B) == E9N r(B) ---., E9r(N B) , k=O
where [5.20] shows that it is in fact an isomorphism. Note that we carefully did not write r(EBk /\k B), as EBk /\k B is not (because of the infinite sum) a vector bundle in our sense. For2t-graded A-modules E we have defined in [1.6.16] (see also [1.6.21]) a contraction operator [ : E 0/\k *E ---., /\k-1 *E such that, for a fixed e E E, the operator [(e) is a left derivation of /\ *E. It is elementary to verify that this map [ satisfies the conditions of [4.23] and thus induces a morphism of the corresponding bundles. Taking sections, we can construct the morphism
[ : r(B) x r(N *B) ---., r(B) 0 r(N *B) ---., r(B 0
N *B)
---., r(N-1 *B) .
There are two justifications for using the same symbol [ for this morphism on sections. In the first place, as the reader can check, this operation is given for s E r(B) and IP E r(N *B) by the pointwise formula
[(s)1P == [(s 01P) : m
f-*
[(s(m))IP(m) E N-1 *Bm .
It follows easily that we obtain an even morphism of graded Coo (M)-modules 00
[ : r(B) ---., DerdE9r(N *B)) . k=O
On the other hand, we could have applied [1.6.16] directly to the graded Coo (M)-module
*r(B) itself and have obtained a contraction operator [ : r(B) ---., Derd/\ *r(B)). We leave it (again) to the reader to show that these two contraction operators coincide under the identification /\ *r(B) ~ /\ r(*B) ---., EB~o r(N *B), i.e., that we have a commutative diagram
/\ *r(B)
1 Now let
S1, ... , Sk
.( s) -----+
.(s) -----+
/\ *r(B)
1
EB~o r(N *B) . EB~=o r(N *B) E r(B) be arbitrary, then we obtain, by composition, a map
On the other hand we have the evaluation map
We leave it to the reader to check (as always using that all identifications are pointwise) that these two maps coincide under the identification r(N *B) ~ HOffiLk(r(B); Coo(M)) (5.15) (see also [1.7.21]).
§6. The pull-back of a section
183
6.2 Discussion. Since all our identifications are pointwise, they commute with restrictions to submanifolds (and thus in particular with restrictions to open subsets). Let us elucidate this by a few examples, in which N c M is a submanifold. Let Band C be vector bundles over M and let s E reB) and t E r(C) be smooth sections. Then the identification in [5.2] satisfies J(sIN EB tiN) = (J(s EB t))IN ,
obviously because for mEN we have sIN(m) = sCm) and tIN(m) = t(m). In a similar way, if ¢ E r(HomFf(B k ; C)) ~ r(N B* 0 C) and Si E reB) are smooth sections, then, using the identification J from [5.7], we have
As a final example, consider ¢ E
r(N B*)
and
S
E reB) smooth, then
6.3 Definition. So far we have always considered bundles over the same base space, but now we will look at bundles over different base spaces. Let p : B ---+ M and q : C ---+ N be vector bundles and let h : B ---+ C be a right linear vector bundle morphism inducing the map 9 : M ---+ N on the base spaces. For each m E M we have a right linear map hm = hlB~ : Bm ---+ Cg(m)' and thus we can form its right dual h;" : C;(m) ---+ B;". However, the collection of all h;" does not define a vector bundle map C* ---+ B* unless 9 is bijective: it is only defined on fibers of the form Cg(m), missing fibers of C* if 9 is not surjective, it could be multiply defined if 9 is not injective, and it does not induce a well defined map N ---+ M (unless g-1 exists and is smooth). On the other hand, if we take a section r : N ---+ C* of the right dual bundle of C, we can form a section (( r I h * > : M ---+ B* of the right dual of B by the formula
(rllh*))(m) = (r(g(m)) Ilh;")) = r(g(m)) ohm: Bm
(6.4)
---+
Cg(m)
---+
A,
or equivalently by the commutative diagram h*
B;" 3 ((r(g(m)) I h;,,> ~ r(g(m)) E C;(m)
((TW»I m
IT ~
gem).
9
The section (( r I h*)) is called the pull-back of r by h. By varying r we obtain a map 1 1 h* : (C*) ---+ (B*) which is called the pull-back map. It follows easily from the definition that h* is additive and verifies for any ¢ : N ---+ A :
r-
r-
(¢. rllh*)) = (¢og). (rllh*> .
Chapter IV. Bundles
184
This looks like a left linear map (the reason why we wrote it as (( r II h*))), but not quite: r-1(c*) is a graded module over C-l(N), while r-1(B*) is a graded module over C-1(M). This is also apparent from the factthat on the right we have to multiply by ¢ 0 9 instead of just by g. We now show that if the section r is smooth, i.e., r E r( C*), then the pull-back (( r II h *)) is also smooth. For that we evaluate on a smooth section s E r (B) according to [5.14], which gives
«rllh*))(s): m f-* (k 'Pba,2i(X a )' (~a)2i is nilpotent. We deduce the existence of smooth functions Xba,2i, i ?:.-k defined on O::'bo. d such that
The term of order 2k interests us most and can be computed as
°
where 'P~~ is (the equivalent of) the function jCl) in (111.2.28). We next compute a relation between various xs, needed to prove that we indeed will increment k by 1. For a point (x a , ~a) E Oab n Oac one has
Xc
= 'Pca,O(Xa + LXca,2i(Xa)' (~a)2i) = 'Pcb,O('Pba,O(Xa + LXca,2i(Xa)' (~a)2i)) i~k
Xc
i~k
= 'Pcb,O(Xb(Xa,~a) + LXcb,2i(Xb(Xa,~a))' (~b(Xa,~a))2i)
.
i~k
Applying 'Pbc,O, substituting (8.3) for Xb (xa, ~a ) ,expanding into powers of ~a and retaining only (~a)2k, we obtain the equality
(8.4)
Xcb,2k('Pba,O(X a ))' ('Pba,l(X a )' ~a)2k
+ 'Pba,2k(X a )' (~a)2k
= 'P~~~o(Xa) . Xca,2k(X a ) . (~a)2k .
With these preparations, we can introduce a coordinate change that eliminates the (~)2k in all transition functions at once. To do so, let Pa be a partition of unity subordinated to the cover U, and define the functions Pa : Oa ----+ Ao as Pa = Pa 0 'P;;-l. We then note
198
Chapter IV. Bundles
that if (J" is a smooth function defined on Oab, then Pb . (J" is a smooth function defined on Oa (supp(Pbloab) C Oab is closed in Oa). We thus can define the diffeomorphisms 'l/Ja : Oa ----+ Oa, (Xa, ~a) f-* (Ya, ~a) by (8.5) c
That 'l/Ja is indeed a diffeomorphism follows from [III.3.l8]. With these diffeomorphisms we create new charts <Pa : Ua ----+ Oa for M by <Pa = 'l/Ja 0 'Pa; they also form an atlas for M. To see how the transition functions auaj If(m) ) , y
whereas (X I)(m) is given as 'L,j (X Ji)(m)fj. Obviously these two objects are not the same. Still, there is a link. In order to investigate this link in detail, we introduce, in analogy with the projection 7r : F~ ----+ F defined in [III. 1.26], the left linear map 7rf) : T F = F x F~ ----+ F defined as
7rf)(v
. a = L' . (w +TI:P)f" L (w .ayla +vp-)) &yl. 1
'.
1 _.
1
1
1
It then follows immediately that 7rf)((X(m) liT!)) = (XI)(m). Although 7rf) is not injective, there are several situations in which knowledge of X I is sufficient to reconstruct (( X ( m) I T I))· In the first place if I is an even function, in which case all (( 11 I jf)) are zero, and thus <X (m) I T I)) = 'L,j (( (X I) (m) I jf > yj . It also happens if I is an odd function
a
(with all ((fo I jf)) zero), in which case we have <X(m) I T!)) = 'L,j ayj
+ = O. Then we define the map X f : M ---+ B by
Xf = X
0
Tf
0 7rf) •
This makes sense because X oT f takes values in ker(Tp) which is the domain ofdefinition of 7rf). According to [3.5] this definition coincides with the old one in case N reduces to a single point, i.e., B = F an A-vector space.
3.11 Discussion. In order to see that [3.10] fulfills our idea of [3.6], we consider a local trivialization U x F of the bundle p : B ---+ N. In this trivialization the map f restricted to a suitably chosen V c M is given as f( m) = (p(J (m)), h( m)) for some smooth map h : V ---+ F. In other words, in the trivialization U x F the map f is given as the couple (po f, h). According to [2.22] the tangent map Tf is given as (T(J 0 p), Th). Since X 0 T f takes values in ker Tp, we have X 0 T f = (Q, X 0 T h). And then (3.9) and [3.5] tell us that on the trivialization U x F we have
(3.12)
Xf == X oTf 0
7rf)
= (J 0 p, X 0 Th 07rf)) == (J 0 p, Xh) .
We conclude that the formal definition of X f in the bundle setting indeed gives us what we expected in [3.6].
3.13 Lemma. Let f : M ---+ B be a smooth map, p : B ---+ N a vector bundle, and X a vector field on M such that X 0 T f takes values in ker Tp. If either X or f is homogeneous (f taking values in Ber.), then the map X f = X 0 T f 0 7rf) completely determines the map XoTf. This means in particular that for all m EM: (X!)(m) = 0 ifandonly if (X oTf)(m) == ((XmIITJ)) = O.
Proof This is an immediate consequence of [3.5] and the local expression (3.12) for Xf. IQEDI
3.14 Discussion. We now have sufficient material to attack the question of triviality of vector bundles. Let p : B ---+ M be a vector bundle with typical fiber the A-vector space F. Ifwe suppose that B is trivial as a vector bundle, we have a vector bundle isomorphism h : B ---+ M x F. Since the restriction of h to a fiber Bm is an even linear map, it preserves the parity, and thus the restriction of h to the subbundle Ber. [IVA.11] gives us a diffeomorphism h : Ber. ---+ M x Fer.. We conclude that the subbundles Ber. are trivial as fiber bundles.
224
Chapter V. The tangent space
3.15 Proposition. Let p : B ----+ M be a vector bundle with typical fiber F. Then B is trivial as a vector bundle ifand only ifeither Bo or Bl is trivial as a fiber bundle. Proof In view of [3.14] we only have to prove that if Ber. is trivial, then B is trivial as a
vector bundle. For simplicity we only treat the case a = 0, the case a = 1 being similar. So let us suppose that 9 : M x Fo ----+ Bo is a fiber bundle isomorphism. This implies that the restriction of 9 to a fiber {m} x Fo is a homeomorphism onto the fiber (Bm)o, but there is nothing that says that this should be linear (in whatever sense). Let (Xi) be coordinates on Fo with respect to a basis (fi) for F. Then we define vector fields Vi on M x Fo by Vi(m, x) = axi. In terms of a local trivialization U x F of B, the map 9 is described by g( m, x) = (m, g( m, x)) with 9 : U x Fo ----+ Fo a smooth map. It follows that Vi 0 Tg is given in this trivialization as (3.16)
((ViI(m,xdTg~ =
L
agJ
a
axi (m,x) ax j
•
j
Since Bo is a submanifold of B, its tangent bundle T Bo is a subbundle of T BIBa [2.16]. From (3.16) it is obvious that ((Vil(m,x) I Tg)) lies in kerTp [3.7]. We thus can use the map 7r{) to define the smooth sections Si : M ----+ B by
i.e., si(m) = (Vig)(m,O) [3.10]. We claim that these sections satisfy the condition of [IV.3.10-ii], and thus that B is trivial as a vector bundle by [IV.3.10-i]. First we note that the vector fields Vi are homogeneous and everywhere independent, and thus, because T 9 is a diffeomorphism, the tangent vectors (Tg 0 Vi)( m, 0) are homogeneous (of the same parity as fi) and independent vectors in Tg(m,O)BO C Tg(m,O)B. Since 9 takes values in Bo, the vectors (Tg 0 Vi)( m, 0) do not have components in the directions associated to the overlined vectors in F~. But the projection 7r : F~ ----+ F is an even bijection when restricted to the graded subspace generated by the non-overlined basis vectors (see [111.1.26]). It follows that the vectors (7r() 0 T 9 0 Vi) (m, 0) are independent and homogeneous of parity c(fi)' Since Bm is isomorphic to the A-vector space F, an independent set of the correct number of vectors is also generating, i.e., a basis. We conclude that the sections Si satisfy [IV.3.1O-ii], and thus that B is trivial as a vector bundle by [IV.3.10-i]. In the odd case (a = 1), the only difference worth mentioning is that there the vectors (Tg 0 Vi)( m, 0) lie completely in the graded subspace generated by the overlined vectors. And then the restriction of 7r : F~ ----+ F to that graded subspace reverses the parity of these vectors. IQEDI
axi
3.17 Counter example. Let p : B ----+ M be a vector bundle with typical fiber F. If B is trivial as a vector bundle, we have seen that then Ber. is trivial as a fiber bundle. Applying the body map, we deduce that BB ----+ BM is trivial as an R-vector bundle and that BBer. ----+ BM are also trivial as a fiber bundle. Now let F be of dimension 111 with basis fo, fl (fer. of parity a), and let M = GS 1 be the A-manifold without odd dimensions whose underlying R-manifold is the circle.
225
§3. Advanced properties of the tangent map
We cover M by two charts U± given by U+ ~ {() E AD I 0 < B() < 27r} and U_ ~ {() E AD I -7r < B() < 7r}. The change of charts ip_+ : U_ ---+ U+ is given by () 1-* () if B() > 0 and () 1-* () + 27r is B() < O. Over M we define the vector bundle p : B ---+ M with typical fiber F by the transition functions 'I/J-+ given by () 1-* (~ ~ ) if B()
> 0 and() 1-* (~1 ~1) ifB() < O.
This vector bundle is essentially twice (in the 10 direction and in the h direction) the Mobius bundle over the circle. We claim that this bundle is trivial as a fiber bundle, but not trivial as a vector bundle. To prove that it is not trivial as a vector bundle, we consider the fiber bundle BBo ---+ BM. This is a fiber bundle over the circle S1 with typical fiber R ~ BFo given by the transition functions (f;-+ (()) = sign () E Aut(R), i.e., the Mobius bundle, which is not trivial. Hence B can not be trivial as a vector bundle. On the other hand, let us denote by (x,~, y, 7]) the even and odd coordinates in F ~ Fg according to (x + ~)10 + (y + 7])h = xlo + 7]h + {fo + yf1' Then we can define a fiber bundle isomorphism h : M x F ---+ B in the trivialization U+ by
h((), x,~, y, 7]) = ((), cos(()/2)x + sin(()/2)y, cos(()/2)~ + sin(()/2)7], - sin( () /2)x
+ cos( () /2)y, -
sin( () /2)~
+ cos( () /2)7])
.
We leave it to the reader to show that this can indeed be extended to a global smooth diffeomorphism. Since it obviously respects the fibers, it is a trivialization of B as a fiber bundle. This example shows that a vector bundle which is trivial as fiber bundle need not be trivial as a vector bundle. The obstruction lies in the fact that the group Aut(F) is more or less incapable of mixing even and odd basis vectors. This is seen most clearly when we look at a matrix representation of B Aut(F), which decomposes as Gl(p, R) x Gl( q, R) if the dimension of F is plq.
3.18 Remark. In the special case A = R, a vector bundle p : B ---+ M is trivial as a vector bundle if and only if both Bo and B1 are trivial as fiber bundles. The reason is that for generic A (satisfying (C [=l)) the subbundle Bo still contains information about B1 in the form of the odd coordinates with respect to the odd directions, while for A = R one loses this information. This is confirmed by the fact that for A = R the group Gl(plq, R) decomposes as the direct product Gl(p, R) x Gl( q, R), whereas for generic A there is a cross-over between the even and odd directions in the form of odd matrix elements.
3.19 Discussion. Let M, N, and Q be three A-manifolds and 1 : M x N ---+ Q a smooth map. For a fixed m E M we define the map 1m : N ---+ Q by 1m (n) = 1(m, n). According to [III.1.23-g], if m has real coordinates, then 1m is smooth. On the other hand, if m does not have real coordinates, there is no reason to assume that 1m will be smooth (in general it will not be). Thus we can not speak about the tangent map T 1m for an arbitrary fixed m E M. Since it is often desirable to do so anyway, we will circumvent this problem as follows. We define the generalized tangent map T 1m : TN ---+ TQ by
226
Chapter V. The tangent space
Since T f is even and left linear on any fiber T( m, n) M x N, T f m is even and left linear on any fiber TnN. But for general m it will not be a smooth map. However, if we do not look at a fixed value of m, but see it as a variable, we get a map M x TN ---+ TQ. Using the identification T(M x N) ~ T M x TN [2.21], we can see this map as the composition of the smooth maps Q. x id(T N) : M x TN ---+ T M x TN and T f. It follows that the family of maps Tfm with varying m E M is smooth. ---+ Q by fn( m) = f(m, n). In a similar way wecanfixapointn E N and define And then we define the generalized tangent tangent map T f n by
in : Ai.
the maps f m, this is an even (left) linear map when restricted to a fiber T mM, but T f n need not be smooth. However, when we see n as a variable, we get a smooth map T M x N ---+ TQ (the restriction ofT f to TM times the zero section), which is smooth. More generally, let 0 C M x N be an open set and f : 0 ---+ Q a smooth map. For a fixed m E M we define the set Om C N by 0 n {m} x N = {m} X Om. It is an open subset of N on which we have a map fm : Om ---+ Q defined by fm(n) = f(m, n). As before, if m does not have real coordinates there is no reason to assume that f m is smooth, but we can define a generalized tangent map T f m : TOm ---+ TQ by the formula (( Yn I T f m> = (( (Q.m, Yn ) I T f)). To show that the family of tangent maps T f m is smooth, we need a more careful discussion about the domain of definition of this family. Since we will not need it, we will not go into these details. A~...f0r
3.20 Lemma. Let M, N, and Q be A-manifolds of which N is connected, and let ---+ Q be smooth. Then the following three assertions are equivalent.
f :M x N
(i) There exists a smooth map 9 : M ---+ Q such that f = go 7rM. (ii) There exists a collection of vector fields X on N such that at each n E N the Xn generate TnN and such that (Q. x X) 0 T f = O. (iii) \f(m, n) EM x N: Tnfm = O.
In the case Q is an A-vector space, the condition (Q. x X) 0 T f = 0 in (iO can be replaced by (Q. x X) f = O. Proof. (ii) , hence if Tnfm = 0 then (Q. x X) 0 Tf = O. On the other hand, if the Xn generate TnN, it follows that (ii) implies that Tnfm is zero on a generating set, i.e., Tnfm = O. • (i) ': : Al X Al X M ---+ Al X Al X M with the property (>.:)3 = id.
am.
236
5.
Chapter V. The tangent space
COMMUTING FLOWS
In this section we show that two integrable homogeneous vector fields commute if and only if their flows commute. As a byproduct we show that if X is an integrable homogeneous vector field with flow 1>t and Yan arbitrary vector field, then the commutator [X,y] equals atT1>-tY at t = O.
5.1 Definitions. Let X and Y be two homogeneous integrable vector fields on an Amanifold M with flows 1> X and 1>y respectively. We say that the flows commute if we have the equality 1>x (t, 1>y(s, m)) = 1>y(s, 1>x(t, m)) for all points (t, s, m) for which both sides make sense. It is the purpose of this section to show that the flows commute if and only if the vector fields commute, i.e., [X, Y] = O. The proof of this statement requires some preparatory lemmas, some of which have an interest on their own.
5.2 Lemma. Let M and N be two A-manifolds. Let I be a homogeneous open interval, W a wave in I x M around to for some real value to E f. Let Y be a homogeneous vector field on N of the same parity as 1. Finally let fi : W ---'>- N be two smooth maps that coincide on {to} x M such that T fi Oat = Yo J;. Then II = h Proof Define gi : W x N ---'>- M x N by gi(t, m, n) = (m, fi(t, m)). It follows that T gi Oat = (Q. x Y) ° gi and that the gi coincide on {to} x M x N. The result now follows IQEDI from uniqueness of (local) flows with initial condition [4.8] and [4.17].
5.3 Corollary. Let mo E BM be a point with real coordinates and X a homogeneous vector field on M with flow 1> X such that X (mo) = O. Then for all t : 1> X (t, mo) = mo· Proof Let Wx be the domain of definition of 1>x and fmo = Wx n Aa x {m o }. If II : fmo ---'>- M is defined as lI(t) = 1>x(t, m o), then by definition of the flow we have Tho at = X ° II. But for h : fmo ---'>- M defined by h(t) = mo we also have Tho at = 0 = X (m o ) = X ° f2. We conclude by [5.2]. IQEDI
5.4 Corollary. Let f : M ---'>- N be a smooth map and let X be an integrable vector field of parity a on M with flow 1> x· Then T foX = 0 if and only iff ° 1> X = f ° 7r2 on W x, where 7r2 : Aa x l\!f ---'>- M denotes the canonical projection. Proof If f ° 1> X = f ° 7r2 then, by applying the tangent map, we obtain the equation 0= TfoT7r2 Oat = TfoT1>xoat = TfoXo1>x. Restriction of this map to {O} xM gives T foX = O. If on the other hand T foX = 0, we deduce that T(f ° 1>x) Oat = Q. and T(f ° 7r2) Oat = Q.. Since the two maps f ° 1> X and f ° 7r2 coincide on {O} x M, the result follows from [5.2]. IQEDI
§5. Commuting flows
237
5.5 Lemma. Let 1 : M ----+ N be smooth, let X be an integrable vector field of parity a on M, and let Y be an integrable vector field of the same parity a on N. Ifwe have ¢y 0 (id x 1) = 1 0 ¢x on an open set containing {O} x M then X and Yare related by f. On the other hand, if X and Yare related by 1, then ¢y 0 (id x 1) = 1 0 ¢ X on W x· Proof Suppose ¢y 0 (id x 1) = 1 0 ¢ x. Applying the tangent map of this relation to the on Ao x M (more precisely, on the open set containing {O} x M) gives us vector field
at
T 1 0 T ¢X
T10 T10
~ ~
x x
0 0
0
at = T ¢y
0
T( id x 1) 0
at
(at
x Q) 0 (id x 1) ¢x = Yo ¢y 0 (id x 1) = Yo 1 0 ¢ X
¢ X = T ¢y 0
Restricting this identity to {O} x M proves that Y 0 1 = T 1 0 X. To prove the second part we start with a local statement. Since 1 is smooth, for any point m E M there exist open sets m E U c M and l(m) EVe N, and an open interval I containing 0 such that 1(u) c V, I x U c Wx, and I x V c W y . We compute T(J 0 ¢ x) 0 = Yo (J 0 ¢ x) and T( ¢y 0 (id x 1)) 0 = Yo (¢y 0 (id x 1)). Since the maps 1 0 ¢ X and ¢y 0 (id x 1) coincide on {O} x U, we conclude from [5.2] that the maps coincide on I x U. By gluing these local subsets I x U together, we conclude that 1 0 ¢x and ¢y 0 (id x 1) are defined and coincide on a wave W in Au x M around O. Let Wo be the biggest wave with these properties. We claim that Wo = W x , a fact which is obvious in the odd case. In case X and Y are even, suppose Wo is strictly included in W x. There thus exists a border point (tl' ml) E Wx \ Wo for WOo Since Wo is open and ¢x continuous, there exist open tIE I and ml E U such that (I - I) x ¢ X (I X U) c W o , where I - I = {a - b I a, bEl} is an open interval containing O. Moreover, since W X is open, we may assume I x U c W x. By definition of (it, mI) there exists a t2 E I such that (t2, mI) E WOo Since Wo is open, we may assume {t2} x U C Wo (shrinking U if necessary). For an arbitrary tEl we compute
at
1(¢x(t, m))
at
= 1(¢x(t, ¢x( -t2, ¢X(t2, m)))) = 1(¢x (t - t2, ¢X(t2, m))) = ¢y(t - t2, 1(¢x(t2, m)))
by [4.12] and [4.13]
= ¢y(t - t 2, ¢y(t2' l(m)))
because {t2} xU
= ¢y(t, l(m))
by [4.12] and [4.13].
We conclude that I x U proves that W X = WOo
because (I - I) x ¢x(I x U)
c Wo bymaximality, contradicting (tl, mI)
c Wo
c Wo
E I x U\ Woo This
IQEDI
5.6 Corollary. If 1 : M ----+ N is a diffeomorphism and X an integrable homogeneous vector field on X with flow ¢ x, then Y = T 1 0 X 0 1-1 is an integrable vector field on N with flow ¢y = 1 0 ¢x 0 (id xI-I).
238
Chapter V. The tangent space
5.7 Lemma. Let X be a homogeneous integrable vector field on M and let P be any open subset ofN x M. If ¢x : Wx ----+ M is the flow of X on M, then the flow of Q x X on P is given by
(5.8)
¢QxX : (t, n, m)
f-*
(n, ¢x(t, m)) .
Its domain of definition WQxx is Al x P for odd X; if X is even it is given by WQxx
=
{(t, n, m) E AD x P
I (t, m)
E Wx & :JIo,t :
{n} x ¢x(Io,t, m)
c P} ,
where Io,t denotes an open interval containing both 0 and t. Proof We apply [5.5] to the two canonical projections 7rN : P ----+ Nand 7rM : P ----+ M, which satisfy T7rM 0 (Q X X) = X 07rM and T7rN 0 (Q X X) = Q0 7rN. This gives us as result that 7rM 0 ¢Qxx = ¢x 0 (id X 7rM) and 7rN 0 ¢Qxx = ¢Q 0 (id x 7rN)' This proves that (5.8) is valid on the domain of definition of ¢Qx x. This finishes the proof for odd X. For even X it remains to show that the given W Qxx coincides with the domain of definition of ¢Qx x, which we temporarily denote by WOo Since ¢ X is continuous, P open and WX a wave, it follows that WQx X is a wave in AD x P around 0 on which the map 'IjJ : (t, n, m) f-* (n, ¢x(t, m)) makes sense. An elementary calculation shows that T'IjJ 0 at = Q x X 0 'IjJ. By maximality ofWo this implies that WQxx CWo. On the other hand, if I, U C NI, and V C N are open such that I x V x U cWo, it follows from our first observation that I x U c W x. Moreover, since W X is a wave, we may assume that o E I. It follows that I x V x U C WQxx, proving the other inclusion. IQEDI
5.9 Corollary. Let X and Y be integrable homogeneous vector fields on M. Their flows commute if and only if Y and Q x Yare related by ¢x (ifand only if X and Q x X are related by ¢y). Proof Consider the vector field Q x Y on W X . It follows from the explicit expression (5.8) that the flows of X and Y commute if and only if ¢y 0 (id x ¢x) = ¢x 0 ¢QxY. According to [5.5] this is true if and only if T ¢ X 0 (Q x Y) = Yo ¢ x. Interchanging the IQEDI roles of X and Y proves the second part.
5.10 Discussion. We continue with the preparations for our characterization of commuting flows and we take a closer look at the tangent map of the flow ¢ X of a homogeneous vector field X on an A-manifold M. We will use the symbol t for the time parameter, even if the vector field X is odd. In order to simplify this discussion, we will drop (in this discussion) the subscript X in ¢ X . We start by choosing a point (s, t, m) E W X (i.e., such that ¢(s, ¢(t, m)) makes sense) and we choose local coordinate systems x~ around m, x~ around ¢(t, m), and x~ around ¢(s, ¢(t, m)) == ¢(s + t, m). We then note that a T¢: -a i ICt,m) Xa . a T¢. atICt,m)
~ a¢b
f-*
f-*
L j
a~ a -ai (t,m). a j I¢Ct,m) Xa Xb a
_
i
a
~ at (t,m)· a jIW,m) -Xb(¢(t,m))a jl¢Ct,m.), j
Xb
Xb
§5. Commuting flows
239
where the last equality follows by definition of a flow. Since IP3e = id, the same is true for the tangent map: (TIP X)2 = id; applied to the vectors ax~ I(t,m) and at! (t,m)' this identity gives the relations .
k
~ a¢~ a¢b ~ - . (-t,¢(t,m))· -a. (t,m) i
aXb
x~
k
= OJ ,
LX~(m). ~!~(t,m) = Xi(¢(t,m)). i
a
Applying likewise the identity (TIP X)3 = id to the vectors ax~ I(s,t,m) gives the additional relation (5.11)
a¢b a¢~ ~ -a. (t,m). - . (s,¢(t,m)) ~ x' ax) j a b
=
a¢~
-a. (s+t,m). x'a
Now if sand t are close to zero, we may assume that the coordinate systems are the same, i.e., a = b = c. We then obtain in particular the relation
a¢~ (0 m) = Oi
(5.12)
ax~'
)
.
Applying [111.3.13] then gives the useful identity a2¢~
-~. (O,m) =
(5.13)
ax~ ax~
o.
5.14 Discussion. Let us now return to the actual characterization of commuting flows. According to [5.9], the flows of X and Y commute if and only if Y and Q x Y are related by ¢x, i.e., Yo ¢x = T¢x 0 (Q x Y). Composing on the right with the diffeomorphism IP X shows that this is the case if and only if Yo 7r2 = T¢ X 0 (Q x Y) 0 IP x, where 7r2 : W X ---+ M denotes the canonical projection (t, m) f---+ m. Since the left hand side is rather easy to understand, we have to study the right hand side in more detail. We thus define the function 'IjJ : W X ---+ T M by
'IjJ
= IPx 0 (Q x Y) oT¢x .
Now IPx maps (t, m) to (-t, ¢x(t, m)), then Q x Y maps this point to a tangent vector at this point, and finally T¢x, which maps the base point (-t, ¢x (t, m)) back to m, sends this tangent vector to a tangent vector at m. In short, 'IjJ(t, m) E TmM. If'IjJ equals 7r2 0 Y, this implies that 'IjJ(t, m) should be equal to Ym E T mM, independent oft. It now follows easily from the relation (5.12) that 'IjJ(0, m) = Ym , so it remains to show that 'IjJ(t, m) is independent of t. Applying [5.4] to the vector field at on W X shows that this is the case if and only if at 0 T'IjJ = O. Since the vector field at and the map 'IjJ satisfy the conditions of [3.13] and since at is homogeneous, we conclude that at 0 T'IjJ = 0 if and only if at'IjJ = O. To summarize this discussion, we have shown that the flows of X and Y commute if and only if at'IjJ = O. Hence our interest in the quantity at'IjJ, where we recall that t denotes the time parameter whose parity equals the parity of X.
240
Chapter V. The tangent space
5.15 Proposition. Let X be a homogeneous integrable vector field on M and let Y be an arbitrary vector field on M. If 'IjJ = X 0 (Q x Y) 0 T¢ X : W X ---+ T M, then 8t 'IjJ : Wx ---+ TM is given by 8t 'IjJ = x 0 (Q x [X, Y]) oT¢x. Using the notation of the generalized tangent map [3.19J and abbreviating ¢x to ¢, this can be written as (5.16) Proof To simplify the notation, we will, as in [5.10], write ¢ for ¢ X throughout this proof. Let us first consider the last statement. Writing the definition of'IjJ(t, m) explicitly gives
'IjJ(t,m) = 1, but it is as easy. Since the vector fields Xi commute, their flows commute. For i = 2 we thus may write
ax,
'ljJ(XI, ... ,xn) = ¢x 2 (x 2, ¢x, (Xl, ¢x 3 (x 3, ... ¢X k (xk, 0, 0, ... ,0, xk+l, ... ,xn) ... ) . We thus can apply exactly the same argument to prove that Xi is represented by axi in the k. IQEDI coordinates x for 1 < i
s:
6.2 Definitions. Let M be an A-manifold and V a subbundle of TM. V is said to be involutive if for any two smooth vector fields X and Y that take their values in V, their commutator also takes values in V, i.e., X, Y E r(V) ===} [X, Y] E r(V). A subbundle V is said to be integrable iffor each m E M there exists a chart m E U with coordinates x such that Vlu is generated by ax', ... ,axk, where k is the rank of V. An integrable subbundle is also called afoliation.
6.3 Theorem (Frobenius). A subbundle VeT M is involutive if and only if it is integrable. Proof If V is generated on U by ax" ... , axk, it follows immediately from [1.21] that X, Y E ru(V) implies [X, Y] E ru(V). Since this is true all over M, V is involutive by [1.20]. To prove the implication in the other direction, suppose V is involutive and let m E M be arbitrary. Since V is a subbundle of rank k, there exists a chart m E U with coordinates (yl, ... , yn) and homogeneous Xi E ru(TM), 1 i n generating ru(TM) such that the Xi, 1 i k generate VI u [Iv'3.14]. If we define ma = Bm, the coefficients of Xi (ma) with respect to the basis ayj are real; by a constant linear transformation with k. Using these coordinates, real coefficients we may assume that Xi (ma) = ayi, 1 i we define a projection 7rk : U --4 V, (yl, ... ,yn) 1----* (yl, ... ,yk), from U to an open set V in an A-vector space of total dimension k. By construction the map
s: s:
s: s:
s: s:
Tmo7rklv",o : Vmo is bijective, simply because Xi(m o ) =
xl
ayi.
--4
T7rd mo)V
Writing Xi(m) =
s: s:
Lj Xl(m) . ayj,
our
s: s:
assumption implies that (mo) = 5{, 1 i k, 1 j n. It follows that the (m), 1 i, j k is invertible in a neighborhood UI of mo. And square matrix thus Tm 7rk Iv", : Vm --4 T7rdm) V is bijective for all m E UI . Hence there exist vector fields Yi E r u, (V) generating Vlu, such that T7rk 0 Yi = ayi 07rk. (The Yi( m) are given i,j k.) explicitly by multiplying the Xi(m) by the inverse of the matrix X{ (m), 1 We deduce that
xl
s:
s:
s:
s:
a a
T7rkO[Yi,Yj] = ["il',~]07rk =Q. uy' uyJ
Since V is involutive and T7rk bijective on Vlu" it follows that [Yi, Yj] are independent, we can apply [6.1] to conclude.
= 0.
Since the Yi IQEDI
244
Chapter V. The tangent space
6.4 Definition. Let VeT M be a sub bundle of rank k. A smooth map f : N ----t M is said to be tangent to V if for all n E N we have Tf(TnN) C Vlfen). An integral m11nifold of V is a pair (i, N) such that (i) dim( N) = k and (ii) i : N ----t M is an injective immersion tangent to V. If V is a foliation, one defines a leaf ofV to be a connected integral manifold i : L ----t M such that i( L) is maximal with respect to inclusion.
6.5 Corollary. Let V C TM be afoliation of rank k and let ma E BM be a point with real coordinates. Then there exists an integral manifold (i, N) of V such that i : N ----t M is an embedding and such that ma E i(N). Proof Let U be a chart around ma as in the definition of an integrable subbundle and let N CUbe the subset N = {m E U I 'Vi > k : xi(m) = xi(ma)}. Then N is a submanifold of M of dimension k (the xi(ma) are real!) and thus the canonical injection i : N ----t M is an embedding. Moreover, since V is spanned by the axi, i :::; k, (i, N) is an integral manifold for V. IQEDI
6.6 (Counter) Examples. One usually says that a subbundle V C TM is integrable if through every point passes an integral manifold. Defined that way, Frobenius' theorem states that V is involuti ve if and only if through every point passes an integral manifold. However, in the context of A-manifolds problems arise due to the fact that the image of an immersion has to pass through points with real coordinates. The following examples show what can happen, justifying our definition of integrability . • Consider first the A-manifold M = of dimension 210 with coordinates (Xl, x 2 ), on which we define V as the subbundle of rank: 1 generated by the vector field X I = axl. This subbundle is involutive: (Xl, x 2 ) is a coordinate system satisfying [6.2]. If i : N ----t M is an integral manifold, it is fairly obvious that i(N) should be contained in a slice x 2 constant (we will show it explicitly in the proof of [6.9]). But a point with real coordinates in N is mapped to a point with real coordinates in M, implying that x 2 should be real. It follows that no integral manifold passes through points (Xl, x 2 ) with x 2 non-real. • With this example in mind, one might think that it should be sufficient to demand that through every point with real coordinates passes an integral manifold. As the next example will show, this condition is too weak to ensure that a subbundle V is involutive. Consider the A-vector space E of dimension 112 and define M = Eo with coordinates (x, e). The subbundle V of rank 2 is generated by the global vector fields Xl = ax and X 2 (x, = aEl + aE2. Now consider the A-vector space F of dimension 111 and define N = Fo with coordinates (Y,7]). It is elementary to show that fue smooth map i : N ----t M, (y, 7]) ~ (x, = (y, 7], 0) is an integral manifold ofV. Moreover, it passes through every point with real coordinates of M. However, V is not involutive because [X2' X 2] = eae rf- V.
A6
e,
e, e)
ee e, e)
6.7 Lemma. Let V be an involutive subbundle of T M of rank k. Let furthermore U be a chart with coordinates Xi such that VI u is generated by axi, 1 :::; i :::; k. Finally define
§6. Frobenius' theorem
245
the slices Sm C U as Sm = {m' E U I 'Vi > k: xi(m') = xi(m)}. If f: N --4 M is tangent to D, then each connected component off (N) n U is contained in a slice Sm withm E BM. Proof The main problem of this proof is to show that f(N) cannot "fill up" parts of U; the crucial ingredient is that N is second countable. Let F be the A-vector space of the appropriate dimension such that X k +1 ) ... ) xn (n = dim M) are coordinates on F o , and define s : U --4 Fo by s(xl) ... ; xn) = (xk+l) ... ) xn). By definition s is constant on slices and Sm = s-1 (s( m)). Since f is tangent to D, it follows that T( so f) = O. By [3.21] we conclude that so f is constant on connected components of f- 1 (U). In particular if B is a connected component of f- 1(U) and b E B, then f(B) is contained in the slice Sf(b) (because (sof)(B) is constant equal to s(f(b»). Since N is locally homeomorphic to the even part of an A-vector space which is locally connected [III. 1.3], a connected component B of the open set f- 1(U) is open. Hence BB C B and B contains a point b EBB c B with real coordinates. And thus s(f(B)) E BF and f(B) is contained in a slice Sm with mE BM, namely m = f(b). Now f(N) n U = f(f-l(U)) and thus the image f(B) of a connected component B of f-l(U) is contained in a connected component C of f(N) n U. Since each f(B) is contained in a slice, C is the union of (parts of) slices. It is true that the union of two slices is no longer connected, but an arbitrary union of slices could be connected. For instance, U itself (if it were connected) is the union of all its slices. In order to prove that C is contained in a single slice, we invoke the fact that N is second countable and thus that there are (at most) countably many connected components (open!) B of f-l(U). Since for each B the image s(f(B)) is a single point in BF, s(C) is a countable subset of BF. Now C is connected, BF is homeomorphic to some Rd and the only countable connected subsets ofRd are points (the only connected subsets ofR are intervals). Hence s(C) is a IQEDI single point and thus C is contained in a single slice.
6.8 Proposition. Let D be an involutive subbundle of TM, let i : L --4 M be an integral manifold of D, and let f : N --4 M be a smooth map. If f(N) c i(L), then there exists a unique smooth map 9 : N --4 L such that f = i 0 g. Proof Since i is injective, existence and uniqueness of a set theoretic map 9 is guaranteed. The only difficulty is in proving that this 9 is smooth. So let no E BN be arbitrary, mo = f(no) E M and Po = g(no) E L. Let furthermore U, No, and io be as in [6.5]. Finally let U L be the connected component of i-I (U) containing Po and let UN be the connected component of f-l(U) containing no. These sets are open because i and fare smooth. Since i is tangent to D, it follows from [6.7] that i(UL) C io(No). Since io is an embedding, there exists a unique smooth j : UL --4 No such that i = io 0 j [2.18]. Since Ti is injective, Tj is injective; since No and L have the same dimension, j is a diffeomorphism onto its image [2.14]. And now: f(U N ) is connected and contained in i(L) n U, and the connected components ofi(L) n U are contained in the slices Sm [6.7]. Hence f (UN) is contained in the slice Sm o' because f (no) = mo E Sm o' Once again because io is an embedding, there exists a unique smooth map fa : UN --4 No such that
246
I =
Chapter V. The tangent space
io 0 10' It follows that the set theoretic map 9 is given on UN by glUN which is smooth. Since no is arbitrary, we conclude that 9 is smooth.
=
j-I 010'
IQEDI
6.9 Proposition. Let'D be an involutive subbundle of TM of rank k. Through every point mo E M with real coordinates passes a unique (up to diffeomorphism) leaf(i, L). Moreover, ifI : N ----t M is tangent to 'D, if N is connected, and if I (N) n i (L) =I- ¢, then there exists a unique smooth map 9 : N ----t L such that I = i 0 g, and thus in particular I(N) C i(L). Proof Let m E M be arbitrary and choose a chart U as in [6.2]. Shrinking U if necessary, we may assume that all slices Sm' with m' E U are connected. Since M is second countable, there exists a countable set of such charts U = { Ui liE N } covering M. Let Uo E U be such that mo E Uo, and let So = Sma be the slice in Uo containing mo [6.7].
If S is a slice in U E U and S' a slice in U' E U, we will say that Sand S' are related if there exists a sequence Ui E U, 1 :::; i :::; C, and slices Si in Ui such that S = S I (and thus U = UI ), Si n Si+ I =I- ¢, and S e = S'. We now define S as the set of all slices in any U E U that are related to So = Sma' Then we define the topological space X = ilSES S and the continuous map j : X ----t M such that j IS is just the canonical injection of the slice S in M. We finally define an equivalence relation", on X by x '" y {=:} j(x) = j(y), and the topological space L = XI'" with the canonically induced injective continuous map i : L ----t M. We claim that this (i, L) is the sought for leaf passing through mo. The proof of this claim breaks down into several steps . • The first step is to prove that j is an immersion. We will say that a slice S in U E U is a real slice if it is of the form S = Sm with m E BM. It follows that if S is a real slice, the canonical injection i : S ----t M is an integral manifold and an embedding. Now let Sa be a real slice in Ua E U and Ub E U arbitrary. It follows that the connected components of Sa n Ub are contained in the slices of Ub. Hence, if Sb is a slice in Ub, the intersection Sa n Sb is a union of connected components, and thus open in Sa. In particular, Sa n Sb being open in Sa, there is a point with real coordinates in this intersection, i.e., Sb is a real slice. Since So is a real slice, we conclude that all S E S are real slices, and that j : X ----t M is an immersion . • The next step is to prove that L is a proto A-manifold and that i is an injective immersion. If Sa, Sb E S intersect, we have seen that Sab = Sa n Sb is open in Sa. We thus can define 'Pba : Sab ----t Sba by 'Pba = (jISb)-I 0 (jlsJ (use [2.18] with the embedding j ISb : Sb ----t M). It follows immediately that x '" y if and only if x E Sab, y E Sba and y = 'Pba(X) for some indices a, b. We thus have the complete set of ingredients to form a proto A-manifold [111.4.9] (recall that all slices are essentially open sets in a k dimensional A-vector space). We conclude that L = XI'" is a proto Amanifold. The induced map i verifies j = i 0 7r, where 7r denotes the canonical projection 7r : X ----t L. Since 7rISa is a diffeomorphism, we have il7r(Sa) = (jlsJ 0 (7rlsJ. It follows that i : L ----t M is smooth, injective, an immersion, and tangent to 'D, i.e., an integral manifold of'D, except that we do not know that L is an A-manifold.
§7. The exterior derivative
247
• Since i : L ----t M is injective and smooth, BL is Hausdorffbecause BM is. To prove that L is second countable is harder. We will show that S is countable, which implies that X, and thus L is second countable. First fix a sequence Ui E U, 1 ::; i ::; f!.. If Si is a slice in Ui , the connected components of the intersection Si n Ui+l are contained in slices of Ui+l. Since Si is second countable, there are only countably many slices Si+l in Ui+l that intersect Si. It follows that there are only countably many slices related to So = Smo by a sequence of slices contained in the given sequence of Ui E U. Since U is countable, there are only countably many such sequences, proving that S is countable. We conclude that L is a genuine A-manifold, and that i : L ----t M is an integral manifold passing through mo. • Two items remain to be proved: that L is connected and that it is maximal. The connectedness follows from the fact that the slices S in the charts U E U are all connected. The actual argument is a bit tedious and left to the reader. Maximality will be proved at the end. • For the second part, let f : N ----t M be tangent to 'D. For any Ui E U we define the open sets Vij C N as the connected components of f-1(Ui ). It follows from [6.7] that f(Vij) is contained in a slice Si of Ui . By construction of L, if a slice Si in Ui intersects i (L ), it must be contained in i (L ). Thus iff (Vij) n i (L) =I- ¢, it must be that f(Vij) C i(L). If we define NL = {n E N [ f(n) E i(L)}, this implies that Vij is contained either in N L or in its complement. Hence N L is open and closed. By hypothesis N is connected and NL is not empty, so NL = N and f(N) C i(L). The last conclusion follows from [6.8]. • To finish the proof, suppose i' : L' ----t M is an integral manifold passing through mo. By the previous result, i'(L') C i(L), proving that L is maximal. If we have equality i'(L') = i(L), we have induced smooth maps L ----t L' and L' ----t L by [6.8]. Standard arguments using uniqueness of these factorizations then proves that Land L' are diffeomorphic, proving that (i, L) is unique up to diffeomorphism. [QED[
6.10 Remark. The natural idea of proving this proposition using Zorn's lemma does not work. It is true that one can construct an upper bound to any chain of integral manifolds (chain with respect to inclusion of their images in M). This upper bound has a canonical structure of a proto A-manifold, and even its body is Hausdorff. However, in this approach it is very hard to prove that it is second countable.
7.
THE EXTERIOR DERIVATIVE
In this section we define differential forms and the exterior derivative as well as some of its standard properties.' it is a derivation of square zero and commutes with pull-backs. For the last property we define the notion of the pull-back of a differential form as well as a generalization using the generalized tangent map. Defining the Lie derivative of
248
Chapter V. The tangent space
differential forms by the formula of H. Cartan [H.Ca]: £(X) = do L(X) + L(X) 0 d, we also show that £(X) a equals Ot¢;a at t = 0, where ¢t denotes the flow of the homogeneous integrable vector field X.
7.1 DefinitionIDiscussion. Let M be an A-manifold. A k-form on M is a section of the bundle I\k *TM, and thus in particular a smooth k-form is an element of r(l\k *TM). In accordance with standard notation, we denote the set of all smooth k-forms on M by nk (M), i.e., A k-form at m E M is any point in the fiber of I\k *T M above m. It follows that if a is a k-form on M, a(m) == am is a k-form at m E M. Since 1\0 *TM is the trivial bundle M x A, it follows that a O-form is just a function on M, and thus in particular nO(M) = COO(M). For k = 1 we find 0. 1 (M) = r(*TM), i.e., a I-form is a section of the bundle *TM, the left dual bundle of the tangent bundle TM. This left dual bundle *TM is usually called the cotangent bundle of M. According to [IV.5.14] and [1.5.5] we have the identifications (7.2)
nk(M) ~ Hom~k(r(TM)k; COO(M)) ~ HomL(N r(TM); COO(M))
== *(N r(TM)) ,
which tells us that we may interpret a smooth k-form as a (left) k-linear graded skewsymmetric map of smooth vector fields on M with values in the smooth functions on M. Note that the k-linearity is over COO(M) and not over A (which does not make sense because r(T M) is not an A-module). For k = 1 the identifications (7.2) reduce to r(*TM) ~ *r(TM). Even for k = 0 (7.2) makes sense: nO(M) == r(M x A) and *(Nr(TM)) == *COO(M) because N gives the basic ring [1.5.3], which here is COO(M). Since the trivial bundle M x A comes with its canonical trivialization, we have a canonical identification r(M x A) ~ COO(M). And thus for k = 0 (7.2) reduces to the obvious identification COO(M) ~ *COO(M). For future reference we define n(M) as the direct sum over all k : 00
n(M) =
EB
nk(M) .
k=O
Obviously n(M) is a Z x Z2-graded COO (M)-module, where the Z-grading is given by the k from k-form. Using the wedge product of such sections as defined in [IV.6.l],n(M) becomes a Z x Z2-graded commutative COO (M)-algebra. This becomes even more explicit when we also use the identification *(N r(T M)) ~ N *r(T M) [Iy'5.20], which tells us that n(M) is (isomorphic to) the exterior algebra 1\ *r(TM). Now if f E COO(M) ~ nO(M) is a O-form and a E nk(M) a k-form, we can form the wedge product f 1\ a as well as f . a, which uses the COO ( m)-module structure ofnk(M). Since the wedge product is pointwise, [1.5.8] tells us that these two are equal: f 1\ a = f . a (and similarly a· f = a 1\ f).
§7. The exterior derivative
249
7.3 Definition. Let 0: be a k-form on M. The exterior derivative of 0: is the (k do: on M defined by (7.4)
+ I)-form
(_I)k. L(Xo, ... , X k) do: = =
L
i+ L(E(Xp)iE(X i »
(-1)
p
Xi(L(XO, ... , Xi-I, Xi+I, ... , Xk) 0:)
O~i~k
j+ L
(-1)
+
ii L(XO, ... ,Xi-l,Xi+l, ... ,Xk)'Y·Xd. i=O
Using that L(XO, ... , X k ) equals L(XO) 0 • • • 0 L(Xk) [1.7.16], and that L(Xi ) is a right derivation of degree ( -1, c:(Xi )) [1.6.16], one can show by induction that the right hand side of this formula equals L(XO, ... , X k ) ('Y A df). This proves (ii) because the (Xo , ... , X k ) generate (TU)k+l. • The direct way to prove (iii) is to use induction as above and the derivation property of vector fields. An easier way is to note that the exterior derivative is additive and commutes with restrictions. It follows that it suffices to prove it in a chart with 0: = dX i1 A· .. A dxik-f and(3 = dx j1 A· .. Adxjl.g for some homogeneous functions f and g. Using the definition of d on 0-forms, it is immediate that d(f . g) = (df). 9 + f . dg. The result then follows from (ii) and the graded skew-symmetry of the wedge product. • As before, it suffices to prove (iv) on a chart with 0: = dX i1 A· .. A dX ik ·f. Applying (ii) and (i) twice we obtain n
d(do:)
= (_I)k+(k+l)
L
dX i1 A··· A dX ik A dx i A dx j . aja./ .
i,j=l
From [ai,aj] = 0 we deduceajad = (_I)(E(xi)le(xj»aiajf. On the other hand, graded J skew-symmetry of the wedge product gives us dx i A dx j = -( _1)(E(x')IE(x »dx j A dXi. It follows that d( do:) equals its opposite, and thus is zero. IQEDI
252
Chapter V. The tangent space
7.10 Discussion. If Xi is a coordinate on a local chart U of an A-manifold M, it is in particular an element of Coo (U). As such we can calculate its exterior derivative dU(Xi), which is a I-form on U. On the other hand, we have defined the I-form dXi on U as being an element of the basis dual to the basis OJ ofr(TU). It follows immediately from [7.9-i] that these two I-forms coincide:
This observation justifies the name dXi for these I-forms. In particular it follows that the I-form dx i does not depend upon the choice of the other coordinates on U. This in contrast to the vector fields Oi, where anyone of them depends in general upon the whole set of coordinates.
7.11 Lemma. Let M be a connected A-manifold and Then f is constant if and only if df = O.
f :M
--4
A a smooth function.
Proof This is a direct consequence of [3.21] Gust use all vector fields) and the definition of df [7.3], [7.6]. IQEDI
7.12 Discussion. By definition the exterior derivative is a map d : Dk(M) --4 Dk+l(M), so officially it should be indexed by a k. Taking the direct sum over all k gives us a map (still denoted by d): d: D(M) --4 D(M) .
Property [7.9-iii] shows that the map d is a right derivation of degree (1,0). However, there is a pitfall to be avoided: the exterior derivative is not linear over Coo (M)! Luckily R is a subring ofCOO(M), and d is linear over R. It follows that the exterior derivative is a right derivation of the Z x Z2-graded commutative R-algebra D(M). Since commutators of derivations are again derivations, we can compute the commutator [d, d]. Since the degree ofd is (1,0), itfollows that [d, d] = 2d 0 d == 2d 2 . Property [7.9-iv] then can be stated equivalently as [d, d] = O.
7.13 Summary. The exterior derivative on differential forms d : D(M) --4 D(M) is a right derivation of the Z x Z2-graded R-algebra D(M) of bidegree (1,0) and of auto commutator zero.
7.14 Nota Bene. Some readers might wonder about the global factor (_l)k in the definition of the exterior derivative of a k-form. We will give two explanations. In the first place, this sign can be attributed to our way to identify I\k *E with *(I\k E), which allowed us to write
§7. The exterior derivative
253
without an additional sign (_1)k(k-l)/2 [1.7.22]. And indeed, introducing this extra sign will give us on the left hand side of (7.4) the factor (_1)k(k+l)/2 and on the right hand side the factor (_1)k(k-l)/2, and then our global factor (_l)k disappears. In the second place, the given d is a right derivation of degree (1,0), so applying the inverse transpose 'I-I gives us a left derivation 'r-1d. Since linearity is over R for which left and right do not make a difference, the only difference comes from the degree. Now if 0: is a k-form of parity (Z2-degree) a, we have €( 0:) = (k, a) and thus
It follows that we could have defined the left derivation 'r-1d by (7.4) without the additional sign. Looking at [7.9], the conclusions (i) and (iv) remain unchanged when we replace d by 'r-1d; in (ii) the sign (_l)k disappears, and the conclusion in (iii) has to be replaced by [(0: A (3)'r- 1d = 0: A ([((3)'r- 1d) + (_1)l([(o:)'r-ld) A (3, i.e., by the standard property of a left derivation.
7.15 Definition. If X is a smooth vector field on M, we have defined in [1.6.16] (see also [IV.6.1])a right derivation [(X) of the Z x Z2-graded algebra D(M); if X is homogeneous, [( X) has degree ( -1, €( X)). It is a right derivation of the Coo (M)-algebra structure, and hence a right derivation of the R-algebra structure. Since the exterior derivative d is also a right derivation of the R-algebra structure of D(M), we can define a new right derivation £(X) by taking the commutator:
£(X) = [d, [(X)] == do [(X)
+ [(X)
0
d.
The right derivation £ (X) of D( M) is called the Lie derivative in the direction X; if X is homogeneous, £(X) has bidegree (O,€(X)). For O-forms, i.e., for functions, the Lie derivative reduces to £(X) f = [(X)df = Xf because the contraction operator [(X) acts as the zero operator on O-forms. The definition of the Lie derivative in the direction of a smooth vector field X is extended to include an action on smooth vector fields Y E r(T M) by
£(X) Y = [X, Y] , i.e., the action of the Lie derivative of Y in the direction of X is just the commutator [X,Y].
7.16 Proposition. Let X and Y be vector fields on M, then (i) [d, £(X)]
== do £(X) - £(X) d = 0; = [([X, Y]) == [(£(X)Y) = [[(X), £(Y)]. 0
(ii) [£(X), [(Y)]
Proof. The proof of (i) is immediate when using that dod = o. • To prove the equality [£(X), [(Y)] = [([X, Y]), we first note that any operator [(Z) commutes with restrictions, simply because it is defined pointwise. Since the same is true
254
Chapter V. The tangent space
for d, it follows that it also is true for both sides of the equality [.c (X), L(Y)] = L( [X, Y]). It thus suffices to verify this equality on a local chart U c M. But on a local chart U the R-algebra D(U) is generated by the functions Coo(U) and the (local) I-forms dXi. Since both sides of the equality [.c(X) , L(Y)] = L([X, Y]) are right derivations of this algebra, it suffices to verify it on generators. Now on functions both sides act as the zero operator, hence they are equal. To see what happens on a generator dXi, write Xlu = Ei Xi. Oi and YI u = Ei yi . Oi with Xi, yi E Coo (U). Since the equality is linear in X and Y, we may assume that X and Yare homogeneous. We then compute
[.c(X), L(Y)] dx i
.c(X)(L(Y) dx i ) - (_I)(E(X)IE(Y» L(Y)(.c(X) dx i ) = .c(X) yi _ (-I)(E(X)IE(Y»L(Y)(d(L(X) dx i )) =
= X(yi) _ (_I)(E(X)IE(Y»Y(X i ) = L([X, Y]) dx i .
The last equality of (ii) follows by interchanging X and Y and noting that for homogeneous vector fields X and Y we have not only [X, Y] = -(-1) (E(X)IE(Y» [Y, X] but also
[.c(X), L(Y)]
= -( _1)(E(X)IE(Y» [L(Y), .c(X)].
7.17 Corollary. Let X, Y 1 , have the operator equality
... ,
IQEDI
Y k be smooth homogeneous vector fields on M, then we
[.c(X) , L(Yd ° ... ° L(Yk)] = k E (E(X)IE(Yj» 2)-I)l S jo a vu ' ThIS formula tells us that the k-form a~), which does not contain any 7] nor d7], is given in the (x,O coordinates as a~), which does not contain any ~ nor d~, plus terms involving ~s and/or d~s. We now recall that in the coordinate change (x, 0 ~ (Y,7]) the y contain only even powers of~. This implies that all terms a~V with C > 0 must contain ~s and not only d~s, simply because d(~i~j) = ~id~j - ~j d~i. We conclude that Ba~) = Ba~). This proves that, if a is a _
0
k-form on M, then we can decompose it in a local chart as alu =
Lp a~), and then the
local forms Ba~) coincide on overlaps, i.e., they define a global k-form on BM. But the reader has to be careful: Ba is in general not a k-form on BM (the d~s do not necessarily disappear), and the local k-forms a~) do in general not glue together to form a global k-form on M. The latter can be achieved by making a particular choice for the charts U to be used (see the proof of [8.9]).
262
Chapter V. The tangent space
8.4 Definition. Let M be an A-manifold, and a a k-form on M. We define the k-form BMa on BM as being the k-form B(a(O)) constructed in [8.2]. If a is a O-form (a function), this definition coincides with the standard body map: BM f = Bf. If a is a I-form, this definition coincides with the standard body map applied to the even part of a : B M a = B ( ao). In the same spirit we define the map B M on vector fields by
BMX = B(Xo).
8.5 Discussion. Using the Lie derivative£(XE) on vector fields (£(XE)(X) = [XE, Xl) we could have defined an analogous (local) decomposition of a vector field X = X(f) according to the number of times an odd coordinate ~ appears in a (local) description. As for k-forms, one then could associate to any vector field X on M a vector field B(X(O)) on BM. However, as for I-forms, this coincides with the given definition of BM : B(X(O)) = B(Xo) = BMX. On the other hand, this approach to BM on vector fields can be easily extended to other kinds of tensor fields on M (if the need arises).
Le
8.6 Remark. Our construction of B M on k-forms can be stated in a more fancy language. The fact that the number of odd coordinates appearing in a local expression never decreases, implies that we can define the subsets F;(M) c [lk(M) as consisting of those k-forms that do contain at least C odd coordinates in any local coordinate system. We thus obtain a filtration [lk(M) = F~(M) ::l Ff(M) ::l F~(M) ::l .... The existence of our map BM : [lk(M) ----t [lk(BM) then can be rephrased as the existence of a (canonical) map F~(M)/Ff(M)
----t
[lk(BM).
8.7 Lemma. Let M be an A-manifold, a a k-form, (3 an C-form, and X a vector field on M. Then (i) BM : [lk(M) ----t [lk(BM) is R-linear and surjective; (ii) BM(a A (3) = (BMa) A (BM{3); (iii) BM(L(X) a) = L(BMX) BMa; (iv) BM(da) = d(BMa).
Proof The properties (ii), (iii), and (iv) are rather obvious in a local chart, and thus globally when one realizes that the given constructions all commute with restrictions. The IQEDI only non-trivial part is the surjectivity, which will be proven in [8.9].
8.8 Definition. A k-form a on M is said to be closed if da = 0; it is said to be exact if there exists a (k - I)-form (3 on M such that a = d{3. Since d2 = 0 it follows that any exact k-form is closed. An equivalent way to state these definitions is that a is closed if a E ker(d : [lk(M) ----t [lk+l(M)), and it is exact if a E im(d : [lk-l(M) ----t [lk(M)). The implication exact ===} closed gets translated into the inclusion
263
§S. de Rham cohomology
Since d is R-linear and homogeneous, it follows from [1.3.9] and the fact that R is a field that these subsets of n,k (M) are graded R-vector subspaces of the R-vector space n,k(M). (Of course n,k(M) is also a graded COO(M)-module, but that structure is not preserved by these subsets.) We thus can define the (quotient) R-vector space
The R-vector space Hk(M) is called the kth de Rham cohomology group of M. If a is a closed k-form on M, its cohomology class in Hk(M) will be denoted by [a]. Note that Hk(M) is an R-vector space for A-manifolds M as well as for R-manifolds M.
8.9 Theorem. IfM is anA-manifold, then the map [a] ~ [BMa], Hk(M) is an isomorphism ofR-vector spaces.
--4
Hk(BM)
Proof From [S.7] we know that this map is a well defined R-linear map, so only injectivity and surjectivity remain to be proven. To do so, let U be an atlas as in Batchelor's theorem [IV.S.2]. On each chart U E U we can define the Euler vector field X E (S.3), but now the special form of the transition functions for this atlas guarantee that these vector fields coincide on overlaps, i.e., there exists a globally defined Euler vector field X E (Nota Bene. This global vector field X E depends upon the choice of the special atlas U!). The same argument (the special form of the transition functions for our atlas) shows that the decomposition of a k-form a in homogeneous parts a(e) is invariant under a change of coordinates, i.e., each global k-form a decomposes (uniquely) as a = Le a(e) with ate) a global k-form satisfying .c(XE) a(e) = C· a(e). If a is closed, the uniqueness of the decomposition implies that all a(e) are closed, and thusthatC·a(e) =.c(XE)a(e) =dL(XE)a(f). We thus can write (8.10)
da
=0
a = a(O)
+ d (L i
L(XE)a(e») .
/2: 1
Since a(O) does not contain any ~ nor d~, we may identify a(O) with its body part Ba(O). In other words, we have a bijection between k-forms on BM and k-forms a on M satisfying .c(XE)a = O. This proves enpassant the surjectivity of BMclaimed in [S.7-i]. We now are prepared to prove the isomorphism between Hk(M) and Hk(BM). If a is a closed k-form on BM, it can be seen as a closed k-form on M satisfying .c(XE)a = 0, proving that the given map Hk(M) --4 Hk(BM) is surjective. Now [BMa] = 0 is equivalent to saying that a(O) is exact. But then according to (S.lO) the whole form a is IQEDI exact, proving injectivity of the map Hk(M) --4 Hk(BM).
8.11 Remark. The essential ingredient in the proof of [S.9] (and thus in the proof that BM is surjective) is the existence of a special atlas with transition functions as in Batchelor's theorem [IV.S.2]. Since that theorem depends in an essential way on the existence of partitions of unity, it follows that [S.9] depends in an essential way on the existence of partitions of unity.
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Chapter VI
A-Lie groups
Just as differential geometry can be seen as the interplay between analysis and geometry, so can the theory of Lie groups be seen as the interplay between differential geometry and group theory/algebra. An A-Lie group is an A-manifold which happens to be also a group in such a way that the group multiplication is compatible with the A-manifold structure. We have already introduced A-Lie groups in chapter IV, but there they did not playa very important role. On the other hand, this chapter is entirely devoted to the study of A-Lie groups and subjects directly related. Even so, we can only scratch the surface of the theory of A-Lie groups. Any A-Lie group determines an A-Lie algebra, which is the linearized version of the group in the sense that it is the tangent space at the identity equipped with the linearized version of the group multiplication. Once we have the A-Lie algebra associated to an A-Lie group, we can also go backfrom the A-Lie algebra to the A-Lie group with the exponential map. This map is defined in terms of the flow of a vector field and generalizes the usual exponential (series) of matrices. The most important aspect of A-Lie groups is that an A-Lie group is completely determined by its A-Lie algebra up to coverings. More precisely, there is, up to isomorphisms, a unique simply connected A-Lie group G associated to each (finite dimensional) A-Lie algebra 9 and any A-Lie group with the same A-Lie algebra is a quotient of G by a countable discrete subgroup. This fact means that a lot of global geometric properties of an A-Lie group can be translated into algebraic properties of its associated A-Lie algebra. For instance, ifwe define an A-Lie subgroup of an A-Lie group G as an A-Lie group H that admits a smooth injective homomorphism into G, then connected A-Lie subgroups of G are in bijection with the A-Lie subalgebras of its associated A-Lie algebra and normal connected A-Lie subgroups correspond to ideals. Once we have groups and subgroups, it is natural to consider homogeneous spaces, i. e., the quotient of a group by a subgroup. A homogeneous A-manifold is an A-manifold M with a transitive action of an A-Lie group G. Homogeneous A-manifoldsfor a given 265
266
Chapter VI. A-Lie groups
A-Lie group G are completely determined by the proper A-Lie subgroups ofG, where proper A-Lie subgroup means an A-Lie subgroup which is also a submanifold. More precisely, any homogeneous A-manifold is the quotient of G by a proper A-Lie subgroup. Another example where properties related to an A-Lie group can be translated into properties of its A-Lie algebra is the case of invariant vector fields and invariant differential forms. A vector field on a connected A-manifold is invariant under the group action if and only if it commutes with allfundamental vector fields associated to the A-Lie algebra and a differentialform is invariant ifand only ifthe Lie derivative ofthisform in the direction of the fundamental vector fields is zero. In a separate section we prove that any action ofan A-Lie group on an A-manifold can be transformed into a pseudo effective action of a quotient group. This shows that the restriction to pseudo effective actions for structure groups of a fiber bundle is in reality not a restriction at all.
1.
A-LIE GROUPS AND THEIR A-LIE ALGEBRAS
In this section we (re)define an A-Lie group G and we show that topologically it is the direct product ofGw.o.d and the even part of anA-vector space with only odd dimension. We then define the associated A-Lie algebra 9 as the set of left-invariant vector fields on G with the usual commutator of vector fields as bracket; as an A-vector space 9 is isomorphic to the tangent space at the identity of G. We give a formula for the structure constants of9 in terms of the multiplication function in a neighborhood of the identity. We finish with a detailed discussion on how to interpret the A-Lie algebra associated to the A-Lie group Aut(E). Of course this is EnciR(E), but the identification in terms of matrices is subtle and is not what one would think first.
1.1 Definitions. Some of the definitions we will give here were already given in §IV.l. We recall them because they belong rightfully in this chapter. • An A-Lie group is an A-manifold G that admits at the same time a group structure for which the multiplication m : G x G --t G is smooth. Depending on context, we will denote the product m(g, h) also by m(g, h) = go h = 9 . h = gh. The identity element will usually be denoted bye, sometimes also by id. We will show in [1.2] that e has necessarily real coordinates, and in [1.6] that for any A-Lie group taking the inverse is automatically a smooth map. The basic example of an A-Lie group is the group Aut(E) of automorphisms of an A-vector space E (see [IV. 1.3]) . • A smooth left action of an A-Lie group G on an A-manifold M is a smooth map 1> : G x M --t M such that for all m EM: 1>( e, m) = m and such that for all m, g,h: 1>(g,1>(h,m)) = 1>(gh,m). For a fixed 9 E G we will denote the map m 1---4 1>(g, m) by 1>g. If no confusion is possible, we will denote a left action also as 1>(g, m) = 1>g(m) = g(m) = g·m = gm. The evaluation map Aut(E) x E --t E is a left
§ 1. A-Lie groups and their A-Lie algebras
267
action when we equipAut(E) C EndR(E) with the usual composition of endomorphisms as group structure (see [IV.l.3]). • A smooth right action of an A-Lie group G on an A-manifold M is a smooth map 1>: M x G --4 M such that for all m EM: 1>(m, e) = m and such that for all mE M, g,h E G: 1>(1)(m,g),h) = 1>(m,gh). For a fixed 9 E G we will denote the map m f--4 1>( m, g) by 1>g. If no confusion is possible, we will denote a right action also as 1>(m, g) = m· 9 = mg. • A left/right action of an A-Lie group G on an A-manifold M is called transitive if 'rim, m' EM 3g E G: 1>g(m) = m'. • A (homo )morphism ofA-Lie groups is a smooth map p : G --4 H between two A-Lie groups G and H that is at the same time a homomorphism of (abstract) groups. • An isomorphism ofA-Lie groups is an A-Lie group morphism p : G --4 H between two A-Lie groups G and H that is at the same time a diffeomorphism of A-manifolds. It follows that p-1 is also an A-Lie group morphism. • A linear representation of G on E, or just a representation of G is an A-Lie group morphism p : G --4 Aut(E), E being an A-vector space.
1.2 Discussion. Associated to any A-manifold M we have two subsets: Mw.o.d, which is a submanifold, and BM, which is an R-manifold. Moreover, restricting a smooth map to one of these subsets maps it to the corresponding subset in the target space. Looking at BG we thus find that m : BG x BG --4 BG. From the commutativity of the diagram in [111.4.22] (i.e., Bom = moB) we deduce B(e· e) = (Be)· (Be) and thus e = Be, i.e., the identity element has real coordinates. From e = B(g . g-l) = (Bg) . B(g-l) we deduce (Bg)-l = B(g-l), i.e., ifg has real coordinates, then so has g-l. Itfollows immediately that BG is an R-Lie group (BG is an R-manifold and restriction of m to BG remains smooth) and that B : G --4 BG is a homomorphism of (abstract) groups. Looking now at Gw.o.d we can deduce from the fact that inversion Inv is smooth [1.6] that Gw.o. d is an A-Lie group (because Inv then also preserves Gw.o.d). (Note that the proof of [1.6] uses that e has real coordinates, so we have to be careful in what order we prove our statements.)
1.3 Nota Bene. If 1> is a smooth left action of an A-Lie group G on an A-manifold = 1>gh (for a right action we obtain 1>g o1>h = 1>hg), and thus in particular all maps 1>g : M --4 M are bijective with inverse 1>g-1 (because 1>e = id(M). Since the action is smooth, it is in particular continuous. It follows that all maps 1>g are homeomorphisms of M. Moreover, it follows from [III.1.23-g] that if 9 E G has real coordinates (i.e., 9 E BG), then 1>g : M --4 M is smooth. Since (BG)-l = BG we conclude that such a 1>g is a diffeomorphism of M. However, if 9 does not have real coordinates, there is no reason to suppose that 1>g is smooth. Consider for example the smooth left action of Aut(E) on E = E~. According to [11.6.22] and [III. 1.27] a map 1>g = 9 E Aut(E) is smooth if and only if the matrix elements of 9 (i.e., its coordinates) are real.
M, it follows immediately that for g, h E G we have 1>g 0 1>h
Chapter VI. A-Lie groups
268
Even though 9 is in general not smooth, we have defined generalized tangent maps Tg in [Y.3.l9] by the formula
The property 9 0 h = gh then easily gives the property T 9 0 T h = T gh, as if the chain rule were still valid. (For right actions the defining formula for Tg would be Tg(Xm) = T(Xm,Qg), and then we get Tg oTh = Thg, in accordance with the equality 9 0 h = hg.)
1.4 DefinitionIDiscussion. We can interpret the multiplication m : G x G --t G as either a left or a right action of the A-Lie group G on the A-manifold G. If we view it as a left action, i.e., m : G gp x G mfd --t G mfd , the maps mg = m(g, J are usually denoted as mg = Lg and are called left translations of Gover g. In case we view m as a right action, i.e., m : G mfd x G gp --t G mfd , the maps mg = m( _, g) are usually denoted as mg = Rg and are called right translations ofG over g. All left and right translations are homeomorphisms of G; they are diffeomorphisms if and only if 9 has real coordinates (e.g., if Lg is adiffeomorphism, Lg(e) = 9 must have real coordinates by property (A2) of smooth functions).
1.5 Lemma. The generalized tangent maps TLg and TRh commute, i.e., 'rig, h E G: TLgoTR h = TRhoTL g. Proof The associativity of the multiplication says m(g, m(k, h)) = m(m(g, k), h). Applying the tangent map to this identity allows us to compute
TLg(TRh(Xk)) =
= Tm(Qg, Tm(Xk,Qh)) = Tm(Tm(Qg, Xk),Qh) = TRh(TLg(Xk)) . TLg(Tm(Xk,~))
1.6 Lemma. IfG is an A-Lie group, then the map Inv : G --t G, 9 ~ Inv(g) = g-l describing the inverse is a smooth map andfor an arbitrary Xg E TgG we have the equality Tlnv(Xg) = -TL g-1(TR g-1(Xg)) = -TR g-1 (TL g-1 (X g )). In particular
Tlnv(Xe)
= -Xe.
Proof Since h = Inv(g) is the unique solution of the equation m(g, h) = e, smoothness is given by the implicit function theorem. The details are as follows. For go E BG we compute the partial derivatives 8ml8h of m with respect to the second variable at the point (go, g;; 1). Using [III.3.l3] this is exactly the map T Lgo. Since this map is invertible with inverse T L go-1, we can apply the implicit function theorem [111.3.27] to conclude that h = Inv(g) is the unique smooth solution in a neighborhood of go E BG. Since these neighborhoods cover G [111.4.12], Inv is globally smooth [111.4.18].
269
§ 1. A-Lie groups and their A-Lie algebras
To compute TInv, we consider the map 1> : G the chain rule we find
0= T1>(Xg)
--4
G, g
~
m(g, Inv(g)) == e. Applying
= Tm(Xg, Tlnv(Xg)) = Tm(Qg, Tlnv(Xg)) + Tm(Xg,Qg-l)
= TLg(Tlnv(X g))
+ TRg-l(Xg) .
Applying TLg-l gives the announced result (also using [1.5]).
1.7 Proposition. Let G be an A-Lie group which is modeled as an A-manifold on the A-vector space E ofdimension plq. Ifwe denote by F the A-vector space of dimension Olq, then there exists a diffeomorphism Gw.o.d x Fo --4 G, i.e., as an A-manifold G is the direct product ofGw.o.d, which contains only even coordinates, and Fo, which contains only odd coordinates. --4 0' c Eo be a chart around the identity element e E G. In the A-vector space E we not only have the canonical graded subspace Ew.o.d spanned by the even basis vectors (in the equivalence class), but also the graded subspace F spanned by the odd basis vectors. Obviously dim F = Olq and E = Ew.o.d EB F. By a translation over a vector with real coordinates (a diffeomorphism) we may assume that G, mb, E, 1]) = m-y(E, 1]) for which the map m is smooth.
Chapter VI. A-Lie groups
272
Proof Let (Vi) be a basis of TeG = g, which has by definition real coordinates. An elementary calculation shows that the vector field Zl is given by
Zl (x, y, g) = (Q", Qy, EiXi . Vilg) , where the Xi denote the left coordinates of x with respect to the basis (Vi) : x = Ei Xi. Vi. A similar formula holds for Z2, and we find for their commutator the expression
To derive this formula we have used that the vector field Vi has parity C(Vi), and that the commutator of vector fields is bi-additive. That the coefficients (smooth functions on 9 x 9 x G!) Xi and yj come out as they do is because the vector fields Zi do not contain any derivatives with respect to these variables. It thus follows that our bracket is given by (1.14) Since the commutator of vector fields is even, this formula shows immediately that our bracket on 9 is even and bilinear. Since the commutator of vector fields is also graded skew-symmetric and verifies the graded Jacobi identity, it follows easily that our bracket on 9 does so as well. The last part follows immediately from the given formula for [x, y] in terms of the left-invariant vector fields Vi. IQEDI
1.15 Remarks . • It is not hard to show that the vector field W given by the formula = T7r3([Zl, Z2](x,y,g)) E TgG is left-invariant; it is the left-invariant vector field whose value at e is We = [x,y] = T7r3([Zl, Z2](x,y,e))' In this way the bracket [x,y] really is the commutator of two vector fields . • We could have avoided using the A-manifold 9 x 9 x G and the vector fields Zl and Z2 by introducing a basis on 9 = E and defining the bracket on 9 directly by (1.14). The advantage of the given way is that it is intrinsic (avoiding the choice of a basis) and that it introduces a technique we will use more often . • Although it is customary to define the bracket using left-invariant vector fields, nothing prohibits the use of right-invariant vector fields to define a bracket on 9 = TeG. This would mean that one defines the bracket of x, y E 9 as being the value at e E G of the commutator of the right-invariant vector fields whose values at e are x and y respectively. It follows immediately from [1.10] and [V.2.29] that the bracket one obtains that way is the opposite of the bracket defined by means ofleft-invariant vector fields.
Wg
1.16 DefinitionIDiscussion. Let 9 be an A-Lie algebra with basis (Vi)' Then there exist such that constants
cfj
[Vi, Vj]
= EkC~jVk .
If the bracket is smooth, these constants must be real; they are called the structure constants ofg with respect to the basis (Vi). Given that the bracket is bilinear, it is immediate that the bracket is completely determined by these structure constants.
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§ 1. A-Lie groups and their A-Lie algebras
Now if G is an A-Lie group with multiplication map m, it automatically has an associated A-Lie algebra 9 with basis (Oi Ie) relative to some coordinate system (xl, ... ,xn) in a neighborhood U of the identity e. By continuity of m there exists a chart V c U such that m(V x V) c U. Let yi = Xi 0 7rl, yn+i = Xi 0 7r2, 1 ::; i ::; n be the 2n local coordinates on V x V c G x G. Writing the local coordinates of m(g, h) for g, h E V as m k (g, h), 1 ::; k ::; n, we can compute the structure constants of g.
1.17 Lemma. The structure constants of 9 with respect to the basis (Ok Ie) are given by
Proof Since the Vi = Oi Ie have real coordinates, their brackets are the commutators of the associated left-invariant vector fields [1.13]. These are given by the formula
Vilg
om k
0
= TLgVi = (((~, oile)llTm)) = L oyn+Jg, e) ox k Ig . k
Using (V. 1.22) we thus find for the commutator of Vi and
0:
om k 0 ome 0 [ 'Ek oyn+Jx, e) oxk ' 'Ee oyn+j (x, e) oxe lie = om k o2m e 0 'Ek,e oyn+i (e, e) . oykoyn+j ( e, e) oxe Ie k e ( ). o2m _ (_l)(E(xi)jE(xj)) om ( )~I '" +. e, e '" k '" '" e e uyn J uy uyn+t. e, e uX Since m(e, g)
= g,
we have (On+imk)(e, e)
= 8f.
•
The conclusion follows from [III. 3.6]. IQEDI
1.18 Lemma. JfG is an A-Lie group with A-Lie algebra g, then its tangent bundle TG is trivial, a trivialization being given by the map ¢ : G x 9 ---t TG, (g, x) ~ T Lgx = xg. Moreover, if H is another A-Lie group with A-Lie algebra £) and if p : G ---t H is an A-Lie group morphism, then the tangent map Tp is described in this trivialization by (g, x) ~ (p(g), TeP(x))
Proof The map ¢ : G x 9 ---t TG, ¢(g,x) = Tm(Qg,x) is smooth because Tm, the zero section, and the identity 9 ---t Te G are smooth. On the other hand, the map 'ljJ: TG ---t G x 9 given by 'ljJ(Xg) = (7r(Xg),Tm(Q71'(Xg)-l,Xg)) is also smooth, just because Tm, the canonical projection 7r : TG ---t G, and the inverse 9 ~ g-1 are smooth. We finish by showing that they are each others inverse: 'ljJ(¢(g, x)) = 'ljJ(TLgx) = (g, TLg-ITLgx) = (g,x) and ¢('ljJ(Xg)) = ¢(g, TLg-IXg)) = TLgTLg-IXg = X g. The second statement follows from the computation: T p(g, x) = T p(Tma (Qg, x)) = TmH(Tp(Qg), Tp(x)) = TmH(Qp(g), TeP(x)), where we used that p is an A-Lie group IQEDI morphism.
274
Chapter VI. A-Lie groups
1.19 Corollary. Let G be an A-Lie group with A-Lie algebra 9 with basis (Vi). Then r(TG) is a free graded COO (G)-module with basis (Vi). In particular the (vilg) generate the tangent space TgG.
1.20 Discussion. We now attack the question of how to interpret the A-Lie algebra 9 of the A-Lie group G = Aut(E) for an A-vector space E. We have interpreted Aut(E) as consisting of right linear morphisms [IV. 1.3], and we already argued that it is an open subset ofthe even part EndR(E)o [111.2.26]. Since for any U c Fo open in the even part of a finite dimensional A-vector space F we have a natural identification TU ~ U x F [V.1.2], it follows that we have a natural identification T Aut(E) ~ Aut(E) x EndR(E), and in particular In other words, we identify the A-Lie algebra 9 of Aut(E) with the right linear endomorphismEndR(E) on E. But there is more to an A-Lie algebra then just its A-vector space structure: it also has a bracket. Now both 9 and EndR(E) are naturally equipped with a bracket: 9 by the commutator of left-invariant vectorfields, and EndR (E) by the (right) commutator [¢,1/'lR = ¢01/' - (-l)(E(¢)IE(,p))1/'o¢ [1.6.3]. We want to show that our identification 9 ~ EndR(E) respects these brackets. Our first concern is to find a suitable coordinate system for Aut(E) in order to describe the left-invariant vector fields. The obvious choice is the matrix representation MR defined in [11.4.1], which nicely respects the group structure. Unfortunately this choice is not really compatible with more important choices already made: coordinates on (the even part of) an A-vector space are supposed to be left coordinates with respect to a basis. If (ei)i=l is a basis for E, the natural basis for EndR(E) is given by (ei Q9 ej)r,j=l [11.4.1]. Our approach thus imposes the coordinate functions Xij = (eMR)i j defined by
¢= LX i j(¢).eiQge j . i,j
We know that the composition ° corresponds to an elementary (inner) contraction [11.5.2], so we obtain the following composition rule in these coordinates: (1.21) The conjugation \tE(ei)+E(ej) appears because we have to permute the coefficients x j k( 1/') with the ei Q9 ej . Writing for an arbitrary element g E EndR(E) the coordinates as gij = Xij(g), we obtain as multiplication in Aut(E) : (1.22) j
j
The conjugation of h j k gets transformed into the given sign because h is even, and thus the parity of h j k is c( ej Q9 ek ) = c( ej) + c( ek). Now let X be any element of EndR(E)
275
§ 1. A-Lie groups and their A-Lie algebras
with coordinates Xij = Xij(X). In the given coordinates on Aut(E) C EndR(E), the point (h, X) E Aut(E) x EndR(E) corresponds to the tangent vector
Writing z = m(g, h) we now compute, using (1.21) and (1.22), ((Xh I T Lg)) as
=
L
. .
't,),p,q,r
O ( hq r .gP q ) '--1 X'··--. {} 9 h= J {}h'.J ozP r .
L' X'··gP··--1 ° ozP.
·
= "'gP .. \tE(ep)+E(ei)(X i ). ~I h ~ , J {}zp. 9
=
J'
"'(goX)P.
~ J . ),p
. . t,),P
.
J
't,),P
J
9
h
~I h. ozP. 9 J
Similar computations complete the proof of [1.23]. The above computation shows that for a left-invariant vector field X on Aut(E) we have, in the natural identification T Aut(E) ~ Aut(E) x EndR(E), Xg = go Xe. It follows immediately that the vector fields Zi on 9 x 9 x Aut(E) [1.12] are given by
°
Z2(X,Y,g) = "'(goY)pj' ~. ~ ug P . j,p J A simple calculation gives for the commutator
where [X, Y]R denotes the commutator on EndR(E). From this it is clear that the commutator ofleft-invariant vector fields on Au t( E) corresponds to the usual commutator on EndR(E) under our identification 9 ~ EndR(E).
1.23 Lemma. In the natural identification T Aut(E) ~ Aut(E) x EndR(E), the tangent maps T L g , T R g , and TInv are given by
TLg(h,X)
=
(gh,gX)
TRg(h, X)
=
(hg, Xg)
1.24 Remark. Of course, even with 0 as group law, we could have seen Aut(E) as left linearmorphisms. That would have led to a natural identification ofg with End L (E), but an EndL(E) equipped with 0 as composition law, not with o. A careful computation reveals
Chapter VI. A-Lie groups
276
that under this identification the commutator ofleft-invariant vector fields corresponds, in
End£(E), to the bracket defined on homogeneous elements by [X,
Yl = (_l)(E(X)IE(Y)) X
0
Y - Yo
X
= -(X 0
Y -
(_l)(E(X)IE(Y))Y 0
X)
=
-[X, Y1 L
.
It should come as no surprise that this bracket is exactly the bracket obtained from the usual bracket [X, Y1 R on EndR(E) under the isomorphism '1'-1 [1.6.4]. Another variation on the above theme is to consider Aut(E) equipped with the composition law o. In this case the most natural identification for 9 is with End£( E) equipped with 0 as composition. A similar computation as for 0 shows that in this case the commutator ofleft-invariant vector fields corresponds exactly to the usual bracket [X, Y1 L on
End£(E).
1.25 Discussion. We have argued that our approach imposes the left coordinates eMR on Aut(E) instead of the more natural middle ones given by NIR [11.4.1]. A variation upon the given argument is the following. Going from G = Aut(E) to its Lie algebra 9 = Te G = EndR(E) is a kind of derivative (infinitesimal form). Since the map NIR on Aut(E) c EndR(E) is neither left nor right linear, it is not its own derivative (nor its transpose) [111.3.14], and "thus" NIR will not give the coordinates on the Lie algebra. But let us have a look at what would actually happen if we did use the NIR coordinates. The identifications g = Li,j Xi j (g) . ei Q9 ej = Li,j ei Q9 yi j (g) . ej immediately give us the relations
yij(g) =
(_l)(E(e;JIE(e;)+E(e j ))x i j(g)
,
where we have used that the parity of Xij (g) equals c:( ei) +c:( ej) (because g is supposed to be an even endomorphism). These two coordinate systems on Aut(E) thus differ only by some signs. What then could cause such problems that we insisted on the x coordinates? The answer lies in the way one has to identify EndR(E) with tangent vectors at the identity! Let X E EndR(E) be arbitrary, not necessarily even. We have identified it with the tangent vector X c:::: Li,j Xi j (X) . Ox; j' which in the y coordinates gives us
Unfortunately, the combination (-1) (E(e;)IE(e;)+E(ej)) Xij (X) is not the same as yi j( X), unless X is even. In the general case we get
We conclude that there is nothing wrong with using the (more natural) coordinates y on Aut(E), and that the identification of 9 with EndR(E) using the coordinates y also poses no problems, provided we only use even vectors/endomorphisms (see also [2.8]).
§2. The exponential map
277
Yet another way to interpret the problems with the NIR coordinates is the following observation. The yi j (g) coordinates are in between the ei and e j vectors in the tensor product representation 9 = Ei,j ei Q9 yi j (g) . ej . It thus would be natural to put the coordinates of a tangent vector also in between these indices. But the basis vectors ayi j can not be written in a natural way as a product of terms with separate indices i and j. Hence there is no natural way to put the coordinates of a tangent vector between the indices i andj in ayi j •
2.
THE EXPONENTIAL MAP
In this section we start with the definition of the exponential map, which goes from go to G with g being the A-Lie algebra of the A-Lie group G. We then show that the exponential map intertwines an A-Lie group morphism with its associated A-Lie algebra morphism (its tangent map at the identity). We finish with the definition of the Adjoint representation Ad : G --4 Aut(g) and the fact that the derivative of the Adjoint representation is the algebraic adjoint representation: Te Ad = adR : g --4 EndR(g).
2.1 Construction. The usual way to define the exponential map is by following the flows of the left-invariant vector fields. However, as we have seen, for A-Lie groups there are far too few smooth left-invariant vector fields. As before we circumvent this difficulty by looking at all (even) left-invariant vector fields at the same time. We thus consider the A-manifold go x G, on which we define the even smooth vector field ZL by
Since the zero sections and the identity map g --4 TeG are smooth, this defines indeed a smooth vector field on go x G. Obviously this vector field regroups all even left-invariant vector fields on G. Since ZL has no components in the direction of go, its flow has necessarily the form ¢(t, x, g) = (x, 1j;(t, x, g)).
2.2 Proposition. Let W Z L C Ao x go x G be the domain of definition of the map 1j; : W ZL --4 G, then (i) 1j;(t, x, g) = 9 ·1j;(t, x, e) == m(g, 1j;(t, x, e)); (ii) W ZL = Ao x go x G; (iii) 'VA E Ao: 1j;(tA,x,g) = 1j;(t, AX, g).
Proof· Since 1j; is part of the flow of ZL, we have T1j;(atl(to,x,g)) = TL..p(to,x,g)Xe. We then define ¢(t, x, g) = (x, 9 .1j;(t, x, e)) wherever it makes sense. This ¢ obviously
Chapter VI. A-Lie groups
278
satisfies ¢(O,x,g) = (x,g). Moreover,
T¢(Otl(to,x,g») = (Qx, TLgT'l/J(Otl(to,x,e»))) = (Q", TLgTL1f;(t o,x,e)Xe)) =
(Qx,TLg.1f;(to,x,e)Xe) = ZL(¢(ta,x,g)) ,
where in the second equality we used the definition of'l/J. We thus see that ¢ satisfies the conditions of a flow for Z L; by uniqueness of flows it thus must coincide with ¢ on their common domain of definition . • Since W ZL is the maximal domain of definition of ¢, it must contain the domain of definition of ¢. But if (t,x,e) belongs to W ZL , it belongs to hence any (t,x,g) belongs to W, and thus (t,x,g) belongs to W ZL . But then the group law ¢(t, ¢(s, x, g)) = ¢(s + t, x, g), the openness ofWzL , and the fact that x does not move, these all together imply that (for fixed x) all (t, x, g) belong to W ZL' • To prove (iii), consider the A-manifold Ao x go x G on which we define the even smooth vector field Z f by
W
W,
Its flow ¢E has necessarily the form ¢E (t, A, x, g) = (A, x, 'l/JE (t, A, x, g)). We also define ;j;E (t, A, x, g) = 1/;(t, AX, g), and ;fiE (t, A, x, g) = 1/;(tA, x, g). With these we compute
and
T;fiE(Otl(to,g,x,>.»)
A' T1/;(Otl(to>.,x,g») = A . Tm(Q1f;(to>.,X,g) , xe)
by the chain rule
= Tm(Q:i:E(t o,/\,g,x \ )' AXe)
by left linearity of Tm.
=
'P
From these two computations and the uniqueness ofthe flow of Z f, it follows immediately that 1/;E = ;j;E = ;fiE. IQEDI
2.3 Definition. The exponential map exp : go --t G is defined in terms of the flow of the vector field ZL on go x Gas exp(x) = 1/;(1, x, e).
2.4 Proposition. The exponential map exp : go
--t
G has thefollowing properties.
(i) TheflowofZL isgivenby¢(t,x,g) = (x,g·exp(tx)). (ii) 'tis, t E Ao'tlx E go : exp(sx) . exp(tx) = exp((s + t)x).
279
§2. The exponential map
(iii) If X is any even smooth left-invariant vector field on G, its flow is defined on the whole ofAo x G and is given by (t, g) f---> 9 . exp(txe). (iv) To exp = id(fI). (v) Theflow of the even smooth vectorfield ZR(X,g) = (Q", TRgx) on flo x G (the right equivalent of ZLJ is given by (t, x, g) f---> (x, exp(tx) . g). (vi) If X is any even smooth right-invariant vector field on G, its flow is defined on the whole ofAo x G and is given by (t,g) f---> exp(tXe )· g.
Proof • (i) is a direct consequence of [2.2-i,iii], and (ii) follows from the group property of the flow of ZL' • If X is a smooth left-invariant vector field on G, x = xe has real coordinates, and thus the map X: (t, g) f---> g. exp(txe) = 1jJ(t, x, g) is smooth. We then compute, using the left invariance of X, TX(Otl(t,g)) = Tm(Q,p(t,x,g),xe ) = Xx(t,g). Uniqueness of its flow then proves (iii). • To prove (iv), consider the map X: Ao x flo --4 G, (t,x) f---> exp(tx). By the chain rule we find that TX(Otl(o,x)) = Toexp(xlo). Note however that there is a change in interpretation of the x in this formula. The tangent map of the map (t, x) f---> tx transforms the tangent vector Ot at (0, x) into the tangent vector x == xlo E Toflo ~ fI at 0 E flo. Since X(t, x) = 1jJ(t, e, x), we have Tx(otl(o,x)) = TL,p(O,x,e)xe = xe = x (use that 1jJ(0, x, e) = e and e . 9 = g). • According to (i) we have the equality exp(sx) . exp(tx) = exp(tx) . exp(sx). When we see this as maps defined on (s, t, x) E Ao x Ao x flo, we can apply the tangent maps to the vector osl(o,t,x)' Using (iv) we obtain the equality TRexp(tx)Xe = TLexp(tx)xe , We now define Xby X(t,x,g) = (x,exp(tx)· g), and we compute TX(Otl(t,x,g))
= (Q", TRgT1jJ(otl(t,x,e))) = (Q", TRg(TLexp(tx)Xe)) =
(Q", TRg(TRexp(tx)Xe))
=
ZR(X(t, x, g)) .
=
(Q", TRexp(tx).gXe )
Uniqueness of the flow finishes the proof of (v). The proof of (vi) is a variation of that of (iii). IQEDI
2.5 Nota Bene. The restriction to even elements in [2A-ii] and [2A-iii] is essential. One might be tempted to think that for a smooth odd X, i.e., x E Bfll [1.9], its flow is given by (T, g) f---> 9 exp( TX), using that TX E flo (because the time parameter of an odd vector field is odd). In [3.17] we will show that this is the case if x satisfies [x, xl = 0, i.e., the standard condition for integrability. We will also show that this is equivalent to the homomorphism property exp(Tx) . exp(O'x) = exp((T + O')x). In [2.8] we will give an example in which these conditions are not satisfied.
2.6 Lemma. Let G be a connected A-Lie group and U an open neighborhood of e E G. Then G is generated by U, i.e., any element ofG is a finite product of elements of U.
Chapter VI. A-Lie groups
280
Proof Define V = Un U- 1 = {g E U I g-1 E U}, which is an open neighborhood of e becauselnv is adiffeomorphism(~ homeomorphism), and denote by G 1 the (abstract) subgroup generated by V. For any 9 E G 1 it follows that Lg V is an open neighborhood of 9 (because Lg is a homeomorphism) which is contained in G 1 . Hence Gl is open. On the other hand, suppose 9 E G \ G 1 and h E Lg V n G 1 , then 3v E V : 9 = hv- 1 , i.e., 9 E Gl (because V = V-I). Since this contradicts 9 rf. Gl, we conclude that Lg V c G \ Gl, i.e., that G \ Gl is open. We conclude that Gl is open, closed, and non-empty. Since G is connected, we conclude G = G 1 . IQEDI
2.7 Corollary. There exists an open set U C go containing 0 and an open set V C G containing e such that exp : U ---t V is a diffeomorphism. In particular, if G is connected, it is generated by elements ofthe form exp(x) with x Ego.
Proof The first part is a direct consequence of [2.4-iv] and the inverse function theorem [111.3.23]. The second part follows from [2.6]. IQEDI
2.8 Example. Let G be the multiplicative A-Lie group of invertible elements in A, i.e., G = {x + ~ E A I Bx -I O}. It is modeled on an A-vector space of dimension 111 and its multiplication is given by (2.9)
A basis of g = TeG at e = (1,0) is given by the vectors VI = oxle and V2 = 0Ele. The associated left-invariant vector fields are given by
An elementary computation reveals [VI, VI] [ih, V2] = 0 and [V2, V2] = -2Vl. We could also have used [1.17] to obtain these commutators (structure constants): using (2.9) one obtains
0102+1 m = OxOym = (1,0) 0202+1m
= 0Eoym = (0,1)
= oxorym = (0,1) , 0202+2m = 0Eorym = (-1,0). 0102+2m
Inserting the appropriate signs immediately gives
cil
=
ci2
= c Z1 = c~2 = 0 and
C§2 = -2, in accordance with the previously calculated commutators of the Vi. If we denote by (11, P) the left coordinates of an element f E g with respect tot the basis (Vi), we obtain the full bracket in g by (1.14) as
Integrating the vector field ZL(fl, f2, x,~) = fl . VI P is odd), one finds the flow
fl is even and
+ P . V2
(but now f E go, i.e.,
§2. The exponential map
281
which gives for the exponential map (aVI + aV2)
t---+
exp( a, a) :
Combining the even and odd coordinates in a single A-valued "coordinate" on both sides, we can write this expression as exp(a + a) = e a .(1 + a). If we realize that a is odd and thus a 2 = 0, we can see the term 1 + a as the Taylor expansion of e a , i.e., we can write exp(a + a)
= e a +a
.
In this visualization the exponential map of G thus becomes the ordinary exponential map extended to A (see also [3.11]). Once we know the exponential map, it is easy to compute for odd 0-, 7 E Al the product exp(O-V2)' exp(7V2) = exp(O,o-)· exp(0,7) = (1,0-)' (1,7) = (1 +0-7,0- +7). Since this is not equal to exp( (0- + 7 )V2) = (1,0- + 7), we here have an example in which [2A-ii] is not true for odd vector fields (and odd coefficients). We can also consider the map¢: Al x G --4 GdefinedbY¢(7,(x,O) = (x,~)·exp(7v2) = (X+~7,~+X7). A direct calculation gives T¢(OTI(T,X,E)) = XOE I(XHT,UXT) - ~OxI<xHT,E+XT)' which is not equal to v21(xHT,UXT) = (x + ~7)oEI<xHT,UXT) - (~+ x7)oxl(xHT,E+XT)' This shows that [2A-iii] need not be true in the odd case. This example is also perfectly suited to illustrate the truth of [1.25]. The group G can be realized as a group of 2 x 2 matrices equipped with the usual matrix multiplication:
It seems reasonable that the corresponding matrix representation of 9 is given by
And indeed, if I, 9 E 9 are even (meaning 11, gI E A o, p, g2 E AI), a direct computation shows
[/,g]
~ [(j~
in complete agreement with the official bracket. But ... when we take for 1 and 9 the odd element V2, i.e., 1 = 9 ~ (0,1), the official bracket equals -2VI ~ (-2,0), while the bracket of the corresponding (odd!) matrices gives (2,0). As said in [1.25] the origin of this problem lies in the identification between tangent vectors and matrices. Since we use the standard matrix multiplication, this means that we use the coordinates MR. According to the formulas in [1104.1] we thus have
282
Chapter VI. A-Lie groups
i.e., the matrices (~ ~) and ( ~1 ~ the same formulas,
/ = / 1 . VI + / 2 . V2
::::: /1 . (
)
form a basis. For the Lie algebra we thus find, using
° ° + (_° 1
1
)
/2 .
1
A careful calculation reveals that this matrix representation indeed effectively represents the bracket in g; and it corresponds to the previous identification for even elements.
2.10 Definitions. Let Q be an A-manifold, and let G and H be A-Lie groups. A smooth map 1> : Q x G ---) H is called a family of (A-Lie group) homomorphisms from G to H if for all q E Q the map 1>q : G ---) H, 9 ~ 1>( q, g) is a homomorphism of (abstract) groups. If 9 and £) are A-Lie algebras, then a smooth map ¢ : Q x 9 ---) £) is called a family of (A-Lie algebra) morphisms from 9 to £) iffor all q E Q the map ¢q : 9 ---) £), X ~ ¢( q, X) is a morphism of A-Lie algebras [1.6.9] (and thus in particular even). According to [IV.3.17], such a family is equivalent to a smooth map ¢ : Q ---) HomR(g; £))0 such that all ¢(q) == ¢q are A-Lie algebra morphisms.
2.11 Proposition. Let G and H be A-Lie groups, let 9 and £) be their A-Lie algebras, and let 1> : Q x G ---) H be a family of homomorphisms. Then the map T'1> : Q x 9 ---) £) defined by T'1>(q, x) = T1>q(xe) = T1>(~, xe) [V.3.i9] is a family ofmorphisms from 9 to £). in case Q contains a single point, this reduces to the fact that the tangent map at e of a homomorphism between A-Lie groups is a morphism between their A-Lie algebras. Proof First of all note that T'1> is indeed a smooth map, and that it is even and linear in x (because T1> is a smooth even vector bundle map). Since 1> is a family of homomorphisms, the map 1> q sends the identity of G to the identity of H, proving that T'1> has indeed £) = TeH as target space. In order to prove that T'1> preserves brackets, we recall that the bracket on 9 is defined by the commutator of the vector fields zf on 9 x 9 x G (and similarly for £)). We now extend these vector fields to vector fields 2f on Q x 9 x 9 x G by 2f = Q x zf. We also extend the map 1> to a smooth map 1> : Q x 9 x 9 x G ---) £) x £) x H by
1>(q, x, y, g)
= (T'1>(q, x), T'1>(q, y), 1>(q, g)) .
With these ingredients we compute (T1> 0
-
~ (T1> o ZI )(q,x,y,g)
2f) (q, x, y, g) as
-
= T1>(~,Qx,Qy,TLg(xe)) = (QT'(q,x) , QT'(q,y) , T1>q(T Lg(xe))) = (QT'(q,x) , QT'(q,y) , TL(q,g) (T'1>(q, x)) = z[I (T'1>(q, x), T'1>(q, y), 1>(q, g))
§2. The exponential map
283
H
= (Zl
~
oq,)(q,x,y,g),
where, in going from the second to the third line, we have used that q, is a family of homomorphisms from G to H. We thus have proved that the vector fields and are related by the map Cli (for the computations are similar). By [V.2.29] their commutators are also related by Cli, in particular at the point (q, x, y, e) and its image (T'q,(q, x), T'q,(q, y), e) where we have
zfI
2f
2f
,
~
----t
~
~G
~G
(Q, Q, T q,(q, [x, yJ)) = Tq,(Q, Q, Q, [x, yJle) = Tq,([Zl , Z2 J) = [z[i, z£iJ = (Q, Q, [T'q,(q, x), T'q,(q, y)
J) .
IQEDI
2.12 Definition. Let G be an A-Lie group and 9 its A-Lie algebra. We define the map 1 : G x G ----t G by 1 (g, h) = ghg-l. It is elementary to verify that / is a left action of G on itself and at the same time a family of homomorphisms from G to G. We thus can define the associated family T'/ of morphisms from 9 to 9 by T'I (g, x) = T I/ie. Formally the tangent map is left linear, but since it is also even, it is right linear too. We thus can apply [IV. 3.17] to obtain a smooth map Ad : G ----t EndR(g)o, i.e., Ad(g)(x) = T' I(g, x) = T I/ie. According to the definition ofI(g, h), this can also be written as
According to [2.11], each Ad(g) is an A-Lie algebra morphism, i.e., we have the equality Ad(g)([x, yJ) = [Ad(g)(x), Ad(g)(y)J. And even more: the map Ad: G ----t EndR(g) is an A-Lie group homomorphism. To prove this, we compute Ad(h)(Ad(g)(x))
= Th(TIg(xe)) = Thg(xe) =
Ad(hg)(x) .
Since obviously Ad( e) = id(g), this shows that Ad takes its values in Aut(g) and that it is a linear representation of G on g. This representation is called the Adjoint representation ofG.
2.13 Example. Let G be the group Aut(E) of automorphisms of some finite dimensional A-vector space E, and recall that we have identified its A-Lie algebra 9 with EndR(E). We now want to compute the Adjoint representation explicitly. Using [1.23], we find for Ad(g)x = TLg TRg-l xe in thetrivialization T Aut(E) ~ Aut(E) x EndR(E) the expression TLg T Rg-l (e, x) = (e, gxg- 1 ). We thus obtain for the Adjoint representation of Aut(E) on EndR(E) the following formula: I;fg E Aut(E), I;fX E EndR(E) ~ 9
Ad(g)(X)
= go X
0
g-l .
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Chapter VI. A-Lie groups
2.14 DiscussionIDermition. We can apply [2.11] to the morphism Ad : G ---t Aut(g) of A-Lie groups to obtain a morphism Te Ad : 9 ---t EndR(g) of A-Lie algebras. In [2.1S] we will show that this map is the right adjoint representation of the A-Lie algebra 9 defined by adR(x)(y) = [x, yj, i.e., Te Ad = adR. Once we know this, the fact that it is an A-Lie algebra morphism, i.e.,
'v'x,y E g: ad([x,y])
=
[ad(x),ad(y)jR'
is just a reformulation of the graded Jacobi identity.
2.15 Proposition. Let G be an A-Lie group, and let Ad : G ---t Aut(g) be its Adjoint representation. Then the A-Lie algebra morphism Te Ad: 9 ---t EndR(g) is the algebraic adjoint representation adR : (Te Ad(xe))(Y) = [x, yj.
Proof Since Te Ad is an even linear map, it is sufficient to know its values on homogeneous vectors with real coordinates (these contain at least a basis). Similarly, to know the linear map Te Ad(xe) E EndR(g) it is sufficient to know its values on homogeneous vectors with real coordinates. Since Aut(g) c EndR(g)o, we can see Ad as a smooth function with values in the even part of the A-vector space EndR(g). We know from [Y.3.2] that (( xe I T Ad )) = xe Ad E End R (g). Using [1.9] it thus is sufficient to prove the equality (xe Ad)(y) = [x, Yle for smooth homogeneous left-invariant vector fields. So let X and ybe smooth homogeneous left-invariant vector fields on G and let a = c(x) be the parity of x. We define the maps ¢ : Aa x G ---t G and 1> : Aa x G ---t Aa x G by ¢(t,g) = g. exp(tx) and 1>(t, g) = (-t,¢(t,g)). This looks like the flow of the left-invariant vector field but for a general x this needs not be the case (see [2.S] and [2.8] for more details). What we do have is the property (Ot I(o,g) I T¢)) = Xg (but not necessarily for values of t different from zero). Moreover, since ¢(O, g) = g we also have the equalities (V.S.12) and (Y.S.13). As in [y'S.14] and [V.S.1S] we define 'l/; : Aa x G ---t TG by 'l/; = T¢ 0 (Q x y) 01>. Unlike [V.5.1S] we will not compute Ot'l/; for all values of t, but only at t = 0. The computations are completely similar to those of the proof of [V.S.1S] (without the additional s) and we find (Ot'l/; )(0, g) = [x, Yjg. On the other hand, we can use the explicit form ¢(t, g) = Rexp(tx)g to compute 'l/; directly
x,
'l/;(t,g) = T¢(Q_ t , Ygexp(tx)) = TRexp(-tx)'Yaexp(tx) = TLgTLexp(tx)TRexp(tx)-lYe = TLgT1exp(tx)Ye = TLg (Ad(exp(tx)) (y)) , where for the third equality we used that the vector field Y is left-invariant. We thus find that 'l/;(t, e) = Ad(exp(tX))(Y) and thus Ot'l/;(O, e) = (xe Ad)(y), where we used that To exp = id. But restriction to real values and differentiating other coordinates commute [111.3.13], so we find (xe Ad)(y) = [x, YJe. IQEDI
§3. Convergence and the exponential of matrices
285
2.16 Proposition. Let : Q x G ----t H be afamily of homomorphisms from G to H, and let T' be the associated family of A-Lie algebra morphisms from 9 to f). Thenfor all q E Q and all x E go :
(q,expc(x))
= eXPH(T'(q,x)) .
In case Q contains a single point, this reduces to (expc(x)) = eXPH(Te(x)). Proof The exponential maps are defined in terms of the flows of the vector fields ZL. We on 9 x G to a vector field on Q x 9 x G by the formula extend the vector field = Q x Zr. We also extend the map to a map Cli : Q x 9 x G ----t f) x H by Cli( q, x, g) = (T' (q, x), ( q, g)). With these definition we compute
if
zf
zf
if and Zf
We conclude that the vector fields [Y.S.7] and [Y.S.S], the result follows.
are related by the map
Cli.
Combining
IQEDI
2.17 Corollary. Let ¢i : G ----t H be two A-Lie group homomorphisms. If Te¢l and Te¢2 are the same as linear maps 9 ----t f), then ¢1 and ¢2 coincide on the connected component of G containing the identity e.
Proof According to [2.16] we have ¢l(exp(x)) ¢2(exp(x)). The result now follows from [2.7].
= exp(Te¢l(X)) = exp(Te¢2(X)) = IQEDI
2.18 Example. In [III.3.14] we essentially proved that Te gDet = gtrR when we view Aut(E) as a subset of EndR(E)o. From [2.17] we deduce that gDet is the unique (on the connected component) A-Lie group homomorphism Aut(E) ----t A whose associated algebra morphism is gtrR'
2.19 Corollary. Let G be an A-Lie group and let 9 be its A-Lie algebra. For any 9 E G and any x E go we have
9 . expc(x) . g-l
= expc(Ad(g)x)
and
Ad(expc(x)) = eXPAut(g) (adR(x)) .
In particular ifG = Aut(E) we have go exp(X) 0 g-l
= exp(g
0
X
0
g-l).
Proof This is a direct consequence of [2.15] and [2.16]. The particular case G = Aut(E) follows from [2.13]. IQEDI
286
3.
Chapter VI. A-Lie groups
CONVERGENCE AND THE EXPONENTIAL OF MATRICES
In this section we show that the derivative of the exponential map is given by the formula 1 e-ad(x) Tx exp = T Lexp(x) -ad (x) . In order to prove this we have to introduce a notion of convergence of a sequence of smooth functions. We use this notion also to define the exponential of a matrix, and we show that it corresponds to the exponential map defined previouslyfor abstract A-Lie groups.
3.1 ConstructionIDefinition. Let E be an A-vector space of dimension plq, let F be a finite dimensional A-vector space, and let U be open in Eo. Ifwe want to define a suitable notion of convergence of functions in COO(U; F), the usual pointwise convergence will not do, because the non-Hausdorff topology of F prohibits uniqueness. In order to obtain a satisfactory notion of convergence, we decompose a function IE COO(U; F) according to [111.2.23] as q
I(x,~) =
L
where the li1 ... ik (x) are ordinary smooth functions on BU with values in BF. Since these ordinary vector valued functions 1i1 ... ik are uniquely determined by I, it seems natural to define convergence in terms of these components lil ... i k. We thus will say that a sequence In E COO(U; F) converges (pointwise/uniformly on compacta) to I E COO(U; F) if and only if all components (fn)il ... ik E COO(BU; BF) converge (pointwise/uniformly on compacta) to the corresponding component lil ... ik E COO(BU; BF), using any suitable norm on BF to define these notions of convergence in Coo (B U; BF). Note that with this definition we have uniqueness of convergence: if In converges to I and to g, then I = g. This is an immediate consequence of the bijection between a function I and the set of its components li1 ... ik'
3.2 Discussion. Our main application of the notion of convergence will be in the construction of functions on EndR(E)o, with E an A-vector space of dimension plq. On EndR (E) we will use left coordinates X = Xi j . ei Q9 e j . However, in order to simplify notation, we denote the p2 + q2 even ones among the Xi j by Xi, and the 2pq odd ones by ~i. Finally we define the smooth functions gn : EndR(E)o ---t EndR(E)o by
Li.i
gn(X)=xn=~
.
n times
In terms of coordinates, these functions decompose into components
§3. Convergence and the exponential of matrices
287
Each (gn)i, ... ik takes it values in B EndR(E), i.e., in the space of (p + q) x (p + q) matrices with real entries. It is immediate from matrix multiplication that each matrix entry of (gn)i, ... ik is a homogeneous polynomial of degree n - k in the Xi (provided n ;::: k, else it is zero). Nearly as immediate is the estimate
This estimate will permit us to define functions on EndR(E)o by means of converging power series.
3.4 Lemma. Let I(z) = L~=o anz n be a convergent power series on the whole of C, and let E be a finite dimensional A-vector space. Then the sequence offunctions Ii : EndR(E)o -+ EndR(E)o defined by li(X) = L~=o anxn converges uniformly on compacta to a smooth function I : EndR(E)o -+ EndR(E)o (slight abuse of notation). Proof In terms of the coordinates x and~, the functions Ii obviously decompose as
li(X,~) = L k
L ~il i , : G x H -4 H such that = i 0 1> [4.11]. Uniqueness of 1> proves that it is a family of homomorphisms from H to H. Applying [2.16] we find eXPH(T'1>(g,y)) = 1>(g,exPH(y)), valid for all y E £)0. Composing with i, using [2.16] and [2.19], gives us
Since T ei(T'1>(g,y)) and Ad(g)Tei(y) are smooth in (g,y), it follows that ifyissufficiently close to 0 E £)0, both arguments of expo lie in the neighborhood on which expo is bijective. We deduce (by linearity) that Ad(g)Tei(y) = T ei(T'1>(g, y)) for all y E £)0, i.e., Ad(g) (y) E fj for all y E fjo. In order to prove that this is also true for all y E fj, we denote by 7r the canonical projection 7r : fj C 9 -4 g/fj and we define the smooth map Ad' : G -4 HomR(fj, g/fj) by Ad' (g) = 7r 0 Ad(g). We thus have shown that Vg E G: Ad/(g)I~.o = O. By [111.2.31] we conclude that Vg E G: Ad'(g) = 0, i.e., Ad(g)(£)) C fj as wanted. To finish the proof of (i), consider the smooth map ¢ : G x fj
-4
fj C 9 defined
by¢(g,y) = Ad(g)y. ComputingthederivativeT¢(xe,Qy) for x E g, Y E fj, we find ad( x)y. Since ¢ takes values in fj, this belongs to fj C g, i.e., fj is an ideal of g.
297
§4. Subgroups and sub algebras
To prove (ii), let X E go and YE£)o be arbitrary. Using [2.16], [2.19], and [2.15] we find exp(x). i(exp(y))· exp(-x)
= exp(Ad(exp(x))Tei(y)) = exp(ead(x) Tei(y))
.
Since Tei(£)) is an ideal, ad(x)Tei(y) = [x, Tei(y)] E Tei(£)). From this and [3.9-i] we deduce that ead(x) Tei(y) belongs to Tei(£)), say ead(x) Tei(y) = Tei(fj). It follows that exp( x) . i( exp(y)) . exp( -x)
= exp(Tei(fj)) = i (exp(fj)) .
Since elements of the form exp(x) generate G and elements of the form exp(y) generate H [2.7], it follows immediately that H is a normal A-Lie subgroup of G. IQEDI
A6
4.14 Example. Consider the set G = { (x, y,~, 7]) E x Ai I B(x 2 + y2) =I- o} with its obvious structure of anA-manifold of dimension 212. We endow G with the structure of an A-Lie group by giving its multiplication law:
(Xl, yl,
e,
(x 2, y2, e, 7]2) = (x 1X2 - y1y2 + ee 7]1) .
x 1y2 + y 1x 2 + e7]2 + 7] l e , x 1e - yl7]2 + ex2 - 7]ly2 , Xl 7]2 + y 1e + ey2 + 7] l X2) . 7]17]2,
Attentive readers will recognize this A-Lie group as A C *, the multiplicative group of invertible elements (x +~) + i(y + 7]) E A EB iA. An elementary calculation shows that a basis for the left-invariant vector fields is given by
Vx
= xOx + YOy +
vE = -~ox
~oE
+ 7]0.,., - 7]Oy + xOE + yo.,.,
Vy
= -yox + xOy - 7]0E + ~o.,.,
v.,.,
=
7]Ox - ~Oy - YOE
+ xo.,.,
.
Another elementary calculation shows that the only non-zero commutators among these basis vectors are
Of course these commutators could also have calculated by means of [1.17]. For instance, the structure constants eE.,., are given by 0El 0.,.,2 m+O.,.,l 0E2 m = (0, -1,0,0) +(0, -1,0,0), giving as above [vE' v.,.,] = -2vy. Yet another elementary calculation, but this time a rather tedious one, shows that the exponential map exp : go --4 G is given by exp( a, b, a, (3)
= (e a cos(b), ea sin(b), e a cos(b )a-ea sin(b )(3, e a cos(b )(3+ea sin(b)a)
,
where (a, b, a, (3) denote the coordinates in go of dimension 212 with respect to the basis v x , vy, vE, v.,.,. Given this formula, the reader can ascertain that the expression [2A-i]
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Chapter VI. A-Lie groups
verifies the conditions of the flow of ZL' If we identify G with the multiplicative group of A EB iA and flo with A EB iA, this exponential gets the much nicer form
Looking at the commutators (4.15) we see that the graded subspace Qgenerated by the two vectors Wo = Vy and WI = vE + Vry is a subalgebra of dimension 111. In order to find the A-Lie subgroup (i, H) that corresponds to this sub algebra, we compute the exponentials exp(x) with x belonging to Qo. Our formula gives us
exp(a·
Wo
+ a· WI) =
e ia ·(1
+ (1 + i) . a)
.
The product of two of such elements is given by
(e ia .(1
+ (1 + i) . a)) . (e ib ·(1 + (1 + i) . (3))
= ei (a+b+2aj3) ·(1
From this we "deduce" that H is the A-manifold (GS I )
X
+ (1 + i) . (a + (3))
.
Al with multiplication
and with embedding i : H --4 G given by i(e ia , a) = eia (l + (1 + i)a). It is elementary to verify that this is indeed an embedding and that i( H) eGis the submanifold defined by the equations x 2 + y2 = 1 and y. (~ + 'f}) + x· (~- 'f}) = O. The most general subalgebra Q C fI of dimension 111 is generated by the vectors 2 2 Wo = aVE + bvry and WI = (a - b )v x + 2abvy, (a, b) E R2 \ {(O, On. Except for the cases a = ±b, all the corresponding A-Lie subgroups are isomorphic to A as A-manifolds. The case a = -b is the complex conjugate of the example treated above; the corresponding A-Lie subgroup thus is the complex conjugate of H, i.e., as A-Lie group it is H, but with embedding (a, a) ~ e- ia (l + (1 - i)a).
5.
HOMOGENEOUS A-MANIFOLDS
In this section we are concerned with actions of A-Lie groups on A-manifolds. We therefore introduce the notion of fundamental vector field on an A-manifold associated to an element of the A-Lie algebra. We also introduce the notion of a proper A-Lie subgroup which replaces the notion ofclosed subgroup in the non-graded situation. With these definitions we prove that ifH is a proper A-Lie subgroup ofG, then G/H is an A-manifold. Moreover, if mo EM has real coordinates and if H is the isotropy group at m o , then H is a proper A-Lie subgroup and the map G/ H --4 M is an injective immersion.
299
§S. Homogeneous A-manifolds
5.1 Definition. Let 1> : G x M -4 M be a smooth left action of an A-Lie group G on an A-manifold M. For each x E 9 we define a vector field x M on M by the formula
where T1>m denotes the generalized tangent map [V.3.19]. This vector field is called the fundamental vector field on M associated to x E g. If x has real coordinates, i.e., x E Bg, the vector field x M is smooth (because T1> is smooth and e has real coordinates). A particular case of this definition is when G acts on itselfby multiplication, i.e., 1> = mc : G x G -4 G. Comparing definitions shows that xC is the right-invariant vector field on G whose value at e EGis -x (note the minus sign!). Comparing this with [1.10] shows that xC is exactly the vector field T Inv 0 0 Inv.
x
5.2 Lemma. The notion offundamental vector field enjoys the following properties. (i) T1> gx M lm = (Ad(g)x)MI(g,m) ({=} T1>g 0 x M = (Ad(g)x)M 0 1>g ). Cii) Vx E 9 : T1> 0 (xc x Q) = x M 0 1>, or roughly in words, the right-invariant vector field xC on G and the fundamental vector field x M on M are related by 1>. (iii) For x, y E Bg we have [xM, yM] = [x, y]M. (iv) The flow ¢ of the even vector field ZM on go x M defined by the formula ZM(X, m) = (Qx, xMlm) is given by ¢(t, x, m) = (x, 1>(exp(tx), m)); it is defined on the whole of Ao x go x M.
Proof For (i) we use that 1> is a left action of G on M, i.e., 1>g o1>m = 1>m 0 Lg and 1>m 0 Rg = 1>(g,m)' This allows us to make the following computation: T1> 9 (T1>mxe) = T1>m(TLgxe) = T1>m(TRg (Ad(g)x)) = T1>(g,m) (Ad(g)x). For (ii) the computations are similar. Property (iii) is a direct consequence of (ii), [1.10], and [Y.2.29]. For (iv) we note that, by (ii), the vector field ZM on go x M is related to the vector field Zc on go x G x M defined as Zc(x,g,m) = (~,xClg,Qm) by the map id(go) x 1>. Since the flow of Zc is given by (x, g, m) ~ (x, exp( -tx)g, m) [2A-v], the result follows from IQEDI [Y.S.S], taking g = e. 5.3 Corollary. For x E Bgo the flow ofx M on M is given by (t,m) ~ 1>(exp(tx),m).
5.4 Discussion. If 1> : M x G vector field becomes
-4
M is a right action, the definition of a fundamental
without the minus sign. For the natural right action m : G x G -4 G of G on itself by multiplication, the fundamental vector field xC is exactly the left-invariant vector field X. With this notion of fundamental vector field, the results [S.2-ii] and [S.2-iii] remain unchanged; [S.2-i] changes to T1> gx M lm = (Ad(g-l)x)MI(m,g) and [S.2-iv] changes to ¢(t, x, m) = (x, 1>(m, exp(tx))).
Chapter VI. A-Lie groups
300
The reason to introduce the minus sign for fundamental vector field associated to left actions is twofold. In the first place, it makes that [5.2-iii] comes out without a minus sign, making the map x ~ x M from the A-Lie algebra 9 to the set of vector fields on M a morphism of A-Lie algebras. In the second place, if 1> : G x M --t M is a left action of G on M, the map III : M x G --t M defined by III (m, g) = 1>(g-l, m) is a right action of G on M. The use of the minus sign for fundamental vector fields associated to left actions (and its absence for right actions) makes that the two fundamental vector fields corresponding to these left and right actions are the same (this follows from [1.10]).
5.5 DefinitionIDiscussion. Let G be an A-Lie group and (i, H) an A-Lie subgroup. H will be called a proper A-Lie subgroup if i : H --t G is an embedding, i.e., if i( H) is a submanifold of G. According to [V.2ol2], an A-Lie subgroup (i, H) is proper if and only ifi : BH --t BG is an embedding ofR-manifolds. Moreover, for R-Lie groups an R-Lie subgroup is embedded if and only if its image is closed [Wa, Thm 3.21]. We conclude that (i, H) is a proper A-Lie subgroup of G if and only if i(BH) is closed in BG. On the other hand, for R-Lie groups any closed abstract subgroup is a Lie subgroup [Wa, Thm 3.42]. A similar statement does not hold for A-Lie groups because the topology ignores the odd coordinates completely. [5.8] gives a characterization of proper A-Lie subgroups in the context of A-Lie groups.
5.6 Example. Let G = (A1)2 be the abelian additive group of dimension 012. The A-Lie subgroup H defined as {(~, 0) I ~ E Ad is a proper A-Lie subgroup, but it is neither closed nor open in G.
5.7 Discussion. In the sequel of this section we will often introduce a variant of the exponential function associated to a graded subspace of the A-Lie algebra. The idea is the following. Let G be an A-Lie group, 9 its A-Lie algebra, and let £) C 9 be a graded subspace. Since £) is a graded subspace, there exists a supplement 5 C 9 for £) (see [11.6.23]). Using the decomposition 9 = 5 EB £) we define the map exp : 90 --t G by exp( s, z) = exp( s) exp( z) for s E 50, Z E £)0. By [2.4-iv] T(o,o) exp = id and hence exp is a diffeomorphism from a neighborhood of (0, 0) E 50 EB £)0 = 90 to a neighborhood of e E G. Each time we need this variant of the exponential map, we will give the definition adapted to the circumstances, but we will no longerjustify the existence ofthe supplement, nor will we justify the fact that it is a diffeomorphism in a neighborhood of (0, 0). And obviously we will never say explicitly that exp Iso = exp Iso or that exp I~o = exp I~o
5.8 Lemma. Let G be an A-Lie group. (i) Let H C G be an abstract subgroup of G, let £) C 9 be a graded subspace,
let 5 be a supplement to £), and let exp be as in [5.7J. Suppose there exists an open neighborhood U of 0 E 90 such that exp : U --t V = exp(U) is a diffeomorphism and such that exp(U n £)0) = V n H. If in addition BH C H,
§S. Homogeneous A-manifolds
301
then H is a sub manifold of G and the canonical embedding i : H -4 G turns (i, H) into a proper A-Lie subgroup ofG with A-Lie algebra (isomorphic to) £). (ii) Let (i, H) be a proper A-Lie subgroup with A-Lie algebra £), let 5 be a supplement to 6 = Tei(£)) c g, and let the map exp : 60 EB 50 -4 G be defined as in [5.7J: exp(s, z) = exp(s) exp(z). Then there exists an open neighborhood U of 0 E go such that exp : U -4 V = exp(U) is a diffeomorphism and such that exp(U n 60)) = V n i(H). Proof • Let hI E H be arbitrary, and denote ho = Bhl E H. Since ho has real coordinates, Lh o is a diffeomorphism, and thus W = Lh o (V) is an open neighborhood of ho = Bh 1, and thus of hI. We claim that ¢ : W -4 U C go defined by ¢ = exp-l 0 L-;:1 is a chart satisfying ¢(WnH) = Un£)o. Suppose x E ¢(WnH),thenexp(x) = h;;l.'h for some hEW n H. But then exp(x) E V n H = exp(U n £)0) and thus x E £)0. This proves ¢(W n H) c Un £)0; the other inclusion being obvious, it follows from [III.S.1] that H is a submanifold of G. Hence the restriction of the multiplication ma to H is smooth, making H into an A-Lie group . • Since i : H -4 G is an embedding, i(H) is a submanifold of G. In particular there exists a chart ¢ : V' -4 We go around e E G (G is modeled on g) and a graded subspace F of g such that ¢(V' n i(H)) = W n Fo [111.5.1]. According to [Y.2.16] the graded subspace F must be isomorphic to £). By taking a smaller V' and W if necessary we may assume that there exists an open neighborhood U ' of 0 E go such that exp : U ' -4 V'is a diffeomorphism. Since i is a homomorphism, we deduce that exp(U' n 60) c v' n i(H) [2.16], and thus (¢ 0 exp)(U' n 60) c W n Fo. Since ¢ 0 exp is a smooth injective map and since F and 6 are isomorphic, it follows from [V.2.14] that ¢(exp(U' n 60)) is open in W n Fo. Since W n Fo has the induced topology from Wand since ¢ is a diffeomorphism, there exists an open V C V' such that ¢(exp(U' n 60)) = ¢(V) n Fo = ¢(V n i(H)). Taking U = exp-l(V) C U' we find ¢(exp(U' n 60)) = ¢(exp(U) n i(H)) and in particular U' n 60 c U, i.e., U' n 60 = Un 60. Hence exp(U n 60) = V n i(H). IQEDI
5.9 Theorem. Let G be an A-Lie group and (i, H) a proper A-Lie subgroup. Then: (i) the coset space G/H admits the structure ofan A-manifold modeled on that the canonical left action : G x G / H -4 G / H is smooth;
g/6 such
7r: G -4 0/ H with the natural right action of H on G becomes a principal fiber bundle with structure group H; (iii) ifH is also normal, then G/H is an A-Lie group with A-Lie algebra g/6 and 7r : G -4 G / H is a morphism of A-Lie groups.
(ii)
Proof • As a topological space we equip G/H with the quotient topology. By surjectivity of 7r, any subset 0 of G/H is of the form 7r(V) for some subset V of G. According to the quotient topology, a set 7r(V) is open if and only if 7r- 1 (7r(V)) = V . H is open in G. Since right translations are homeomorphisms [1.3], V . H = UhEH Rh(V) is open whenever V is open in G, i.e., we have proven that the canonical projection 7r : G -4 G / H is an open map.
Chapter VI. A-Lie groups
302
Let S be a supplement to 6 and let exp : So EB 60 -4 G, exp(x, y) = exp(x) exp(y) be as in [5.7]. Now let U and V be as in [5.8-ii] and denote by 7rs : 9 -4 S the associated projection. Our first goal is to find an open neighborhood 0 E Ua C So such that the map X : Ua x H -4 G, X(s, h) = exp(s) . i(h) is a diffeomorphism onto its (open) image. Therefore we consider the smooth map ms : So x So -4 G defined by ms(s, z) = exp( -s) exp(z). Let Wbe an open neighborhood of(O, 0) E So XSo such that ms(W) C V, and define w : W -4 So by w = 7rs 0 exp-l 0 ms. Since w(s = 0, z) = z, (ow / oy)( s = 0, z = 0) = id and hence by the implicit function theorem [111.3.27] there exists a (local) function / : So -4 So such that w(s, z) = 0 = w(O,O) is equivalent to z = /(s). But since we know that z = s is such a local function, we conclude that there exists an open neighborhood Ua of 0 E So with Ua x Ua C W such that 'Vs, z E Ua C So : w(s, z) = 0 { = } s = z. Since ms(W) C V, exp-l(ms(W)) C U. Hence on Ua x Ua the equationw(s, z) = 0 is equivalentto exp-l(ms(s, z)) E Un 60' By definition of exp this is equivalentto exp( -s) exp(z) E exp(U n 60) = V n i(H). Hence we have proven that (5.10)
'Vs, s'
E
Ua
C So, 'Vh,
h'
E
H : exp(s) . i(h) = exp(s') . i(h')
{=}
s = s' .
By taking a smaller Ua if necessary, we may assume that Ua C Un So. It is this Ua that we will use. In order to prove that X is a diffeomorphism from Ua x H onto its (open) image, we will use [V.2.14]. We first compute T(s,x) exp for (s, x) E U C So EB 60 :
T(s,x) exp(y, z) = TRexpxTs exp y + TLexpsTx exp z = TRexpx(Ts expy + TLexps Ad(expx)M(x, z)) . Since (i, H) is a normal A-Lie subgroup and since x E 60' z E 6, it follows from [4.13] and [3.15] that Ad(expx)M(x, z) remains in 6. Bijectivity of T(s,x) exp then shows that (Ts exp)(s) is a supplement for (TLexp(s))(bJ. We next compute TCs,h)X for (s,h) E Ua x H:
Since Ad(i(h))Tei(TLh-1Z) is in 6, we conclude from the bijectivity ofTCs,x) exp that T(s,h)X is bijective. Since we have proven injectivity of X in (5.10), we conclude by [V.2.14] that X is a diffeomorphism from Ua x H onto its open image. To finish our preparations, we define for an arbitrary ga E BG the smooth map Xg o : Ua x H -4 G by Xg o = Lgo 0 X (and thus X == Xe). Since for such ga the map Lgo is a diffeomorphism of G, it follows that Xg o is a diffeomorphism from Ua x H onto Vgo = Xgo(Ua x H), which is open in G. We have now all ingredients needed to start the construction ofthe A-manifold structure on G/H. Since 7r is an open map, Ugo = 7r(Vgo) C G / H is open. Moreover, the map Xgo : Ua -4 Ugo defined as Xgo(x) = 7r(Xgo(x,e)) is a homeomorphism. Bijectivity is immediate, U' C Ua is open if and only if U' x H is open in Ua x H, and (to finish) Xgo(U' x H) = 7r- 1 (XgJU')). We conclude that the map CPgo = X;} : Ugo -4 Ua is a
303
§S. Homogeneous A-manifolds
chart for G/H. Moreover, denoting by verifies immediately that
7r1 :
Ua x H
-4
Ua the canonical projection, one
-1
X90
Ua x H
+---~
X90
l~
~11
(5.11)
Ua
Vao
X90
~
+----
Ugo
90 is a commutative diagram, proving that 7r is a smooth map on these charts. To compute a change of charts, let (Ugo ' cPgJ and (Ugl> CP91) be two charts. Since CPgo is a homeomorphism, there exists an open Ug091 C Ugo such that cPgJUgogJ = Ugo n Ug1 . Since (CP91 a cP ;01)(X) = (7r1 0 X;/ 0 XgJ (X, e) is clearly a smooth map, we find that the charts (Ugo ' CPgo) and (U91 , CP91) are compatible. Since they cover G/H we conclude that G/H is a proto A-manifold. Since Ua is an open subset of .50, which is isomorphic to g/fj as an A-vector space, this proto A-manifold is modeled on gjfj. Remains to prove that B ( G / H) is a second countable Hausdorff space. Since all maps are smooth, they commute with the body map, hence B(G/H) = (BG)/(BH). Since G and H are A-manifolds, BG and BH are second countable Hausdorff spaces. By definition of the quotient topology (BG/BH) is also second countable. Consider 1 BG x BG with the subset R = (BH) where in : Be x Be -4 Be is the smooth map (g, h) f--+ g-l . h. Since BH is closed in BG [5.5], R is closed in G x G. Moreover, by definition of cosets, 7r(g) -I 7r( h) if and only if (g, h) rJ. R. Hence if 7r(g) and 7r( h) are distinct points in BG/BH, there exist open sets U, V in BG such that (g, h) E U x V c (e x e) \ R. It follows that 7r(U) n 7r(V) = ¢. Since 7r is an open map, it follows that BG/BH is Hausdorff. The last item of (i) that remains to be proven is that the canonical left action of G on G/H is smooth. Set theoretically this action 1> : e x e / H -4 e / H is defined as 1>(g, m) = 7r(g. 7r- 1 (m)). It follows that on the local chart Ugo we can write 1>(g,m) = 7r(g. Xgo(cpgo(m),e)). Since the right hand side is composed of smooth functions, we deduce that 1> is smooth . • To prove (ii), we first note that 7r- 1 (UgJ = Vgo by definition of Xg o. We then define the map 1/Jgo : 7r- 1 (UgJ = Vgo -4 Ugo x H by the equation 1/Jgo (g) = (m, h) {=} 9 = Xgo(cpgo(m), h). It follows from (5.11) that 1/Jgo is a trivializing chart for 7r. Moreover, an elementary computation shows that it is compatible with the right actions of H on G and Ugo x H, i.e., 1/Jgo(g . h) = 1/JgJg) . h, with (m, h') . h = (m, h' . h). Finally, for two trivializing charts one easily finds, using (5.11), that we have the equality (1/J91 0 1/J;;,1 )(m, h) = (m, ;j91go (m) . h), where the smooth map ;j91go : Ug1 n Ugo -4 H
m-
is defined by the equation (1/J91 0 1/J;;,1 ) (m, e) = (m, ;J91go (m)). We conclude that the map 7r : -4 H is a principal fiber bundle with structure group H acting in the natural way on the right on G. • To prove that G/H is an A-Lie group it suffices to show that the multiplication is smooth. To that end, let Si : Ui -4 e, i = 1,2 be two local smooth sections of the
e
e/
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Chapter VI. A-Lie groups
principal bundle 7r : G -4 G / H. The definition of the group structure on G/H shows that for Xi E Ui the multiplication is defined as
It follows immediately that the multiplication mal H is smooth on UI x U2 . Since the domains of such local sections cover G/H, the multiplication is globally smooth. To prove that the A-Lie algebra of G/H is isomorphic to g/rj, we first note that 7r is a morphism of A-Lie groups and hence that T e 7r is a morphism of A-Lie algebras. Since 7r: G -4 G/ His a fiber bundle, Te 7ris surjective. Since7r(expa(x)) = eXPoIH(Te7r(x)), it follows that rj c ker Te 7r. A simple dimension argument then shows that rj = ker Te 7r. It then follows from [11.3.11] and [11.6.24] that the A-Lie algebra of G/H is isomorphic to g/rj as an A-vector space (which we already knew), but because T e 7r is a morphism of A-Lie algebras, this isomorphism is also an isomorphism of A-Lie algebras. IQEDI
5.12 Definition. Let : G x M -4 M be a smooth left action of an A-Lie group G on an A-manifold M, and let ma E M be arbitrary. Then H = {g E G I (g, ma) = ma} is called the isotropy subgroup at ma.
5.13 Proposition. Let : G x M -4 M be a smooth left action of an A-Lie group G on an A-manifold M, and let ma E BM be arbitrary. Then the isotropy subgroup Hat ma is a submanifold of G and the canonical embedding i : H -4 G is a proper A-Lie subgroup ofG with A-Lie algebra £) = {x E 9 I xMlm o = O}. Proof Note first that H is an abstract subgroup of G (because is a left action) and that £) = ker(Te1/'), where 1/' : G -4 M is defined as 1/'(g) = (g, ma)' Since 1/' and Te1/' are smooth it follows that £) is a graded subspace of g. We thus can choose a supplement 5 C 9 for £). The proof now proceeds in two steps. We first show that exp(£)o) C H, and then that there exists an open neighborhood U of 0 E 50 EB £)0 = go such that exp: U -4 V = exp(U), exp(s, x) = exp(s) exp(x) [5.7] is a diffeomorphism and such that exp(U n £)0) = V n H. Applying [5.S-i] finishes the proof. For the first step we define the vector fieldZMon £)0 x Mby ZM(X, m) = (Q", xMlm), i.e., ZM is the collection of all fundamental vector fields on M with x E £)0' We then consider the two smooth maps Ii: Ao x £)0 -4 £)0 x M defined by JI(t,x) = (x,ma) and Jz(t,x) = (x,1/'(exp(-tx))). Obviously TJIoat = 0 = ZM(x,ma) = ZMoJI, but also T Jz ° at = Z M ° Jz because fz is essentially the flow of Z M [5.2-iv]. Since we also have JI(O,x) = (x,m a) = Jz(O,x) we conclude by [V.5.2] that JI = Jz, i.e., 1/'(exp( -tx)) = ma. We thus have proven that exp(£)o) c H. To prove that there exists an open neighborhood U with the desired properties, we first choose any U such that exp : U -4 V = exp(U) is a diffeomorphism. We now focus our attention on the map III : 50 -4 M defined by llI(y) = 1/'(exp(y)). By definition of £) the map To III is injective. It then follows from [111.3.30] that there exists a neighborhood 0 E U ' C 50 such that IlIlul is injective. By taking a smaller U C 50 EB £)0
§5. Homogeneous A-manifolds
305
we may suppose that III is injective on Un 50. We claim that such an U satisfies our conditions. Obviously exp(U n £)0) c exp(U) n H because exp(£)o) c H. So suppose h E exp(U) n H, i.e., h = exp(s) exp(x) E H for some s E 50 and x E £)0. But then exp(s) = h .exp( -x) E H and thus llI(s) = IlI(O). Injectivity of III on U n50 then proves s = O. And thus h = exp(x) E exp(U n £)0). IQEDI
5.14 Proposition. Let 1> : G x M be a smooth left action of an A-Lie group G on an A-manifold M, let mo E BM, and let H c G be the isotropy subgroup at mo. Then the canonical injection III : G / H --4 M defined by llI(gH) = 1>(g, mo) is an injective immersion, equivariant with respect to the G-actions. In particular if the action of G is transitive, then III : G / H --4 M is a diffeomorphism.
Proof First note that H is a proper A-Lie subgroup of G [5.13], and thus G/H is an A-manifold [5.9]. Since III is injective by definition of H, it remains to show that III is smooth and that Till is injective at all points. Let ¢go : Ugo --4 Uo C 50, 90 E BG be a local chart as in the proof of [5.9] with 5 a supplement to £) in g. In terms of this chart, we have (Ill 0 ¢~ol )(x) = 1>(go . exp(x), mo) = 1>(go, 1>( exp(x), mo)). This shows that III is smooth. Moreover, since £) = ker(T(e,m o )1», since 5 is a supplement to £), and since 1>(go, J is a diffeomorphism, it follows that Tx III is injective. Being equivariant with respect to the G-actions means that llI(g· (gH)) = 1>(g, llI(gH)), which is an immediate consequence of the definition of a left action. The last part of the statement follows from [V.2.14]. IQEDI
5.15 Corollary. The structure of G/H given in [5.9J is uniquely determined (up to diffeomorphisms) by the stated properties.
Proof Denote by M the set G/H equipped with some structure of an A-manifold such that the canonical left action 1> : G x M --4 I'll is smooth. Then by [5.14] we obtain a diffeomorphism between G/H with the structure given in [5.9] and M. IQEDI
5.16 Corollary. Let p : G --4 H be a morphism of A-Lie groups, then ker(p) is a proper A-Lie sub group ofG, and the induced morphism p : G / ker(p) --4 H is an A-Lie subgroup of H.
Proof If we consider the left action of G on H defined as 1> (g, h) = p(g) . h, then ker (p) is the isotropy subgroup at e E H. Hence it is a proper A-Lie subgroup by [5.13]. Since it is also a normal A-Lie subgroup, we conclude by [5.9] that G / ker(p) is an A-Lie group. To prove that the induced map p : G / ker(p) --4 H is an A-Lie subgroup, it suffices to prove that p is smooth, because it is an injective homomorphism by construction. On a neighborhood U C G/ker(p) with a smooth section s : U --4 G the map p is defined as P( x) = p( s( x)). Hence pis smooth on U. Since such U cover G / ker(p), the conclusion follows. IQEDI
Chapter VI. A-Lie groups
306
6.
PSEUDO EFFECTIVE ACTIONS
In this section we prove that every action can be transformed into a pseudo effective action: ifG acts on M, there exists a proper A-Lie subgroup G1/1o ofG such that G1/1o acts as the identity on M and such that the induced action of G I G1/1o on M is pseudo effective.
6.1 Discussion. In chapter IV we defined fiber bundles with a structure group and we required the action of the structure group G on the typical fiber F to be pseudo effective. In the remaining part of this section we will show that we can transform any smooth (left) action into a pseudo effective action. More precisely, we will show that if : G x M ---> M is a smooth left action of an A-Lie group G on an A-manifold M, then there exists a proper normal A-Lie subgroup G1/1o C G such that (i) all elements of G1/1O act as the identity on M, and (ii) the induced action of G IG1/1O on M is smooth and pseudo effective. Forgetting for the moment the smoothness conditions, the natural approach to obtain an effective action would be the following. One would first define GO c G as the set of all elements of G that act as the identity on M, i.e., g E GO if and only if "1m EM: (g, m) = m. Obviously GO is a normal abstract subgroup of G and GIGo acts effectively on M. When one tries to prove that GO is an A-Lie subgroup, it is natural to think that its A-Lie algebra gO consists of those x E g whose associated fundamental vector field x M is identically zero. The next logical step would be to apply [5.8] to really prove what one wants. The problem with this approach is that it is hard (if possible at all) to prove that gO is a graded subspace of g. The definition of gO is in terms of equations, but these equations depend upon even and odd parameters (the local coordinates of M), and we do not have much control over them. Our approach will be to define subsets G1/1O C GO and g1/1 o C gO described by smooth families in GO and gO respectively. Then our equations are "parameterized" by smooth functions over which we have complete control.
6.2 Definitions. • Let K be an A-manifold and X C K a subset. We will say that k E K is part of a smooth family in X if there exists an A-manifold N and a smooth map 1jJ : N -> K such that k E 1jJ(N) c X . • Let : G x M ---> M be a smooth (left) action of an A-Lie group G on an A-manifold M and let g be the A-Lie algebra of G. The subsets G1/1O C GO c G and g1/1 O C gO C g are defined as :
GO = {g
E
Gig acts as the identity on M }
G1/1o = {g
E
Gig is part of a smooth family in GO }
gO
I x M is identically zero } {x E g I x is part of a smooth family in gO} .
= {x
g1/1 o =
E g
6.3 Remark. With hindsight we now can say that the action of G on M is pseudo effective if and only if G1/1O reduces to the identity element of G.
§6. Pseudo effective actions
307
6.4 Discussion. If g EGis part of a smooth family 1/J : N ----; G in GO, then obviously each element g' E 1/J(N) is part of a (the same!) smooth family in GO. In other words, 1/J(N) C G1/1O. It follows that G1/1O is the union of images of smooth maps into GO. Similarly, if x Egis part of a smooth family 1/J : N ----; 9 in gO, then 1/J(N) C g1/1 O. Since is a left action, it is immediate that GO is a normal (abstract) subgroup of G. Moreover, since multiplication and inverse are smooth operations in G, G1/1O is also a normal abstract subgroup. On the other hand, it is not clear whether GO or G1/1O are A-Lie subgroups of G. Similarly, linearity of the tangent map proves that gO is a graded submodule of g. Continuity of the module operations then shows that g1/1 Oalso is a graded submodule. But again, it is not obvious whether they are subspaces, nor is it completely obvious that they are stable under the bracket operations.
6.5 Lemma. Let E be an A-vector space and let (PI, ... , rPk E B*E be afinite number of smooth homogeneous left linear maps. Then £,1> = {e EEl VI :::; i :::; k : (( e I rPi)) = 0 } is a graded subspace of E in the sense of [Il.6.23]. Proof Let G = {gl, ... , gk} be a set of k elements with parity map c : G ----; Z2 given by C(gi) = C(rPi), and let F = F(G,c) be the free A-module (i.e., an A-vector space) on these generators. With these we define the even (left) linear map : E ----; F as (( e I A, g f---+ - ~~;: (g, X = x r , ~ = 0), 1 ::; r ::; k, whose parity is given as c( mjr) + c(er ). In order to combine these functions in a single even map, we introduce an A-vector space F of total dimension k whose basis vectors fr have parity cUr) = c(m jr ) + c(er ). With this A-vector space we define an even map X : V ---> Fo by k
X(g)
=-
L
ojr oer (g,x
= xr,~ = 0)·
fr .
r=l
The fact that the functions rP% (x r ) form k independent elements of *g is equivalent to saying that the Jacobian of X at e has maximal rank k (note that 0e Ogi and Ogi 0e only differ by a sign (_1)(€(gi)I€(e»). Said in yet another way, using thatg>P o is the null space of the equations rP7,. (x r ) = 0, this is equivalent to the fact that g>p o is the kernel of TeX. It follows that the Jacobian at 0 E .50 of the composite map.5o ---> G ---> Fo, s f---+ exp(s, 0) = exp(s) f---+ x(exp(s)) is invertible. Hence if we take the neighborhood V of e E G sufficiently small, we may assume that exp is a diffeomorphism between U = exp-l (V) C go and V and at the same time that X 0 exp is a diffeomorphism between Un.5o and its image in Fo. Now let g E G>P o n V be arbitrary. By definition of G>P o there exists a smooth map 'IjJ : N ---> G>P o such that g E im('IjJ). By taking a smaller N we also may assume that im( 'IjJ) C V, i.e., im( 'IjJ) C G>Po n V. Composing 'IjJ with X, using that taking derivatives with respect to the ~ variables does not interfere with what happens with the g coordinates [111.3.13], and using the definition of G>Po as acting as the identity on M, we deduce that X( 'IjJ( n)) is constant equal to X( e). Since 'IjJ( n) lies in V, there exists (s, z) E U C .50 x gt o such that 'IjJ(n) = exp(s, z). Since is a left action and since exp(g>P O ) C G>Po acts as
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Chapter VI. A-Lie groups
the identity on M, it follows as above that x(1,b(n)) = x(exp(s) exp(z)) = x(exp(s)). Hencex(exp(s)) = x(e), sos = 0 becausexo exp is a diffeomorphism betweenUn.5o and its image in Fo. This proves that G,po n V is included in exp(gt o n U). Since we already know that exp(ggo n U) is contained in G,po n V, we can apply [5.8] to conclude that G,po is a proper A-Lie subgroup of Gwith A-Lie algebra g,po. • To show that G,po is normal, we first note that GO is normal. We then consider the smooth map 1,b : G x G,po ---> GO defined by 1,b(g, h) = ghg-l. Since this is smooth, we conclude by definition of G,po that the image of 1,b lies in G,po, i.e., G,po is normal. • The induced action ind of GjG,po on M is defined by iPind(gG,po, m) = (g, m). This is well defined because the elements of G,po all act as the identity on M. Since 7r : G ---> GjG,po is a locally trivial (principal) fiber bundle [5.9], we choose a local smooth section s : V ---> G for some tri vializing chart V c G j G,po. It follows that the restriction ind : V x M ---> M is given by ind(Z, m) = (s(z), m), which is smooth. Since being smooth is a local property, we conclude that ind is globally smooth. • To prove that iPind acts pseudo effectively, we consider an arbitrary smooth map 1,b : N ---> G j G,po such that all 1,b( n) act as the identity on M. As above we choose a local trivializing chart V and a smooth section s : V ---> G. We then consider the smooth map X : 1,b-I(V) ---> G defined by x(n) = s(1,b(n)). By definition of 1,b, all x(n) act as the identity on M. By definition ofG,po this means that all x(n) lie in G,po. But that means that all1,b(n) are the identity element in GjG,po for n E 1,b-I(V). Since the local trivializing charts cover G jG,po, we conclude that 1,b is constant the identity element. This IQEDI means that ind acts pseudo effectively.
6.8 Lemma. Let G be an A-Lie group and (i, H) an A-Lie subgroup. If there exists a neighborhood V of the identitye E G such that V n i(H) = {e}, then i(H) is a closed discrete subgroup of BG C G. Ifin addition G is connected and H is normal, then i(H)
is contained in the center ofG (and in particular H is abelian). Proof • Since H is an A-Lie subgroup we have g E i(H) =? Bg E i(H). Hence for any g E i(H) we have (Bg)-l . g E V, and thus g = Bg, i.e., i(H) C BG. Now if g E i(H), then Lg is a diffeomorphism and Vg = Lg V is a neighborhood of g such that Vg n i(H) = {g}, i.e., i(H) is a discrete subgroup. Since inversion and multiplication are smooth operations, there exists a neighborhood
W C V of the identity such that W- l . W C V. If g E BG n i(H) \ i(H), then Lg W n i(H) -# ~, i.e., :JXI E BW :Jh l E i(H) : gXI = hI. Since g rJ. i(H), Xl -# e and thus there exists a neighborhood U C W of the identity such that Xl rJ. U (because BG is Hausdorff!). But then again LgU n i(H) -# ~, and thus :JX2 E BU :Jh 2 E i(H) : gX2 = h 2. Butthen hllh2 = x 1l x2 E W-I . Un i(H) c V n i(H) = {e}. Butthis contradicts Xl -# X2 (because X2 E U and Xl rJ. U). This proves that i( H) is closed in BG. • If H is normal, we fix h E H and we consider the smooth map f : go ---> G defined by f(x) = exp(x)· i(h). exp( -x). Since H is normal, f takes values in i(H). Since f is smooth (i(h) E BG), there exists a neighborhood U of 0 E go such that f(U) C Li(h) V. But i(H) n Li(h) V = {i(h)} (i(H) is discrete), hence f is constant i(h) on U. If G
§7. Covering spaces and simply connected A-Lie groups
311
is connected, exp(U) generates G [2.7], and thus we have proven that for all g E G necessarily g. i(h)· g-1 = i(h), i.e., thati(H) is contained in the center ofG IQEDI
6.9 Remark. Unless A does not contain nilpotent elements, a closed subgroup ofBG C G is never closed in G, simply because, by definition of the DeWitt topology, each closed set of G containing e also contains B- 1 { e }.
6.10 Corollary. Let If> : G x M --; M be a smooth left action of an A-Lie group G on an A-manifold M and let G1/1O and 91/1 0 be as in [6.2J.
(i) The action If> is pseudo effective {==} G1/1O = {e} ===} 91/1 0 = {O}. (ii) If 91/1 0 = {O}, then G1/1o is a closed discrete subgroup of BG c G contained in the center ofG. 0
Proof • If 91/1 0 were not {O}, exp(9g ) would contain elements different from the identity acting as the identity on M [6.7], which contradicts that the action is pseudo effective . • Assume 91/1 0 = {O}. According to [6.7] G1/1O is a proper normal A-Lie subgroup, and thus by [5.8] there exists a neighborhood V of e E G such that V n G1/1O = {exp(O)}. The conclusion then follows from [6.8]. IQEDI
7.
COVERING SPACES AND SIMPLY CONNECTED
A-LIE
GROUPS
In this short section we prove that a morphism of A-Lie algebras determines a unique morphism ofthe associated A-Lie groups, provided the source group is simply connected. We thus start with a brief review of covering spaces, universal coverings, and simply connected spaces, and we prove that the simply connected cover ofan A-Lie group has a unique structure of an A-Lie group such that the projection is a morphism of A-Lie groups.
7.1 Definitions. Let X and Ybe a topological spaces. A continuous surjectionp : Y --; X is called a (topological) covering ofX if every x E X admits an open neighborhood U such that p-1(U) = UiE1Ui such that
(i) each Ui is open in Y, (ii) the (Ui)iEI are pairwise disjoint, and (iii) p: Ui --; U is a homeomorphism. It is easy to show that if p : Y --; X is a covering of X and if q : Z --; Y is a covering of
Y, then po q : Z --; X is a covering of X. A covering of X is called connected if Y is connected (and thus X has to be connected too). A connected covering p : Y --; X is called universal if Y is connected and iffor any
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connected covering f : Z ----; X there exists a covering 9 : Y ----; Z such that p = fog. A connected space is called simply connected if the identity map is a universal covering. It follows immediately from the definition of a universal covering that it is unique up to homeomorphisms and that it is simply connected. Theorem [7.2] collects the main results concerning covering spaces that we will need; its proof can be found in most textbooks on algebraic topology (e.g., [Spa]).
7.2 Theorem. Let X be a connected, locally path connected, second countable space admitting an open cover consisting of contractible sets. (i) If X and X' are simply connected, thq; X x X' is simply connected. (ii) There exists a universal covering p : X ----; X.
(iii) Ifp: Y ----; X is a connected covering of X, then Y is second countable. (iv) Let p : Y ----; X be a connected covering of X, and let gi : Z ----; Y be two continuous maps such that po gl = po g2 : Z ----; X. If Z is connected and if there exists a point z E Z such that gl (z) = g2 (z), then gl = g2. (v) Letp: Y ----; X beacovering, f: Z ----; X a continuous map, and(zo,Yo) E ZxY such that f(zo) = p(Yo). IfZ is simply connected, then there exists a (unique) continuous lift 9 : Z ----; Y such that f = po 9 and 9 (zo) = Xo.
7.3 Definitions. In the context of A-manifolds we will call a map p : N ----; M a covering if it is a topological covering [7.1] such that p is smooth and such that the restrictions p : Ui ----; U in condition [7.1-iii] are diffeomorphisms. These additional conditions on a covering exclude maps such as p : Ao ----; A o, x r--+ x3, which is a smooth bijection, but not a diffeomorphism in any neighborhood of O. In view of the inverse function theorem [111.3.23], the additional conditions on a topological covering can be rephrased as saying that p should be smooth and TnP should be a bijection for every n E N, i.e., p is everywhere a local diffeomorphism. The definitions of a universal covering and of simply connected remain the same (but with the changed notion of covering).
7.4 Lemma. Let M be a connected A-manifold. (i) If p : N ----; M is a connected topological covering, then N has a unique structure
of an A-manifold such that p becomes an A-manifold covering. (ii) There exists a universal A-manifold covering p : M ----; M. (iii) Let p : N ----; M be a covering, f: L ----; M a smooth map, and (£0' no) E Lx N such that f (£0) = p( no). If L is simply connected, then there exists a unique smooth lift 9 : L ----; N such that f = po 9 and g(£o) = no. Proof First note that if M is a connected A-manifold, then it is in particular a connected, locally path connected, second countable topological space with an open cover consisting of contractible sets, just by choosing contractible charts. We thus can apply [7.2].
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313
• To prove (i), let p : N ----> M be a topological covering. For any n E N we choose a chart tp : U ----> 0 around m = p(n) in M satisfying condition [7.1-ii] (this can always be done by shrinking U if necessary). Since n lies in a unique Ui , we define tp 0 p : Ui ----> 0 to be a chart for N. We leave it as an exercise for the reader to prove that two charts for N defined in this way are compatible. We thus have constructed an atlas for N, i.e., we have made N into a proto A-manifold for which p satisfies the conditions of an A-manifold covering. With this structure it is elementary to show that Bp : BN ----> BM is a connected covering of BM. Since BM is second countable and Hausdorff, it follows easily from the definition that any covering must also be Hausdorff. We conclude from [7.2-ii] that N is an A-manifold. To prove uniqueness of this structure, it suffices to note that, if p is a smooth local diffeomorphism, then tp 0 p : Ui ----> 0 is a diffeomorphism, and hence a chart for N [IIIA.20]. • Let p : M ----> M be the topological universal covering of M. According to (i) we may assume that it is an A-manifold covering, so it remains to prove that it has the right properties. Therefore, let I : N ----> M be a connected A-manifold covering. It then is in particular a topological covering, and thus there exists a topological covering g : M ----> N such that p = log. It now suffices to prove that this g is smooth. Since p and I are local diffeomorphisms and g a local homeomorphism, we can locally say that g = 1-1 0 p. But this proves that g is locally a diffeomorphism, and in particular globally smooth. • In (ii) we have seen that the universal A-manifold covering is the same as the universal topological covering. This means in particular that the notion of simply connected does not depend upon whether we use A-manifold coverings or topological coverings. Applying [7.2-v] we thus find a continuous lift g : L ----> N with the desired properties. To prove that g is smooth, we choose f!. ELand a chart U :;) I (f!.) of M satisfying the conditions of a covering. Since the Ui are disjoint, there is a unique Ui such that g(f!.) E Ui • Since p : Ui ----> U is a diffeomorphism, it follows that g restricted to l-l(U) coincides with p-l 0 I (use [7.2-iii] if needed). Hence g is smooth in an open neighborhood of f!.. IQEDI
7.5 Discussion. It is elementary to show that if p : N ----> M is a covering (in the sense of A-manifolds), then Bp : BN ----> BM is a covering (in the sense of R-manifolds). On the other hand, if q : Z ----> BM is a covering in the sense of R-manifolds, then one can show (it is elementary but not immediate) that there exists a unique covering p : N ----> M such that Z = BN and q = Bp. In this way one obtains a bijection between coverings of M and coverings of BM. This implies in particular that M is simply connected if and only if BM is simply connected.
7.6 Lemma. Let G be a connected A-Lie group, p : G ----> G its universal covering, and let e E Gbe such that p(e) = e E G. Then Gadmits a unique structure of anA-Lie group such that e is the identity element and p :
G ----> G a morphism
ofA-Lie groups.
----> G is the smooth map defined by I(x, y) = m(p(x),p(y)), then [7.4] there exists a unique smooth map in : G x G ----> G such that p 0 in = I and such by
Proof If I : G x G
Chapter VI. A-Lie groups
314
that m(e, e) = e. By construction p(m(x, y)) = m(p(x),p(y)),i.e., pis amorphism of defines a group structure. To construct the inverse, A-Lie groups once we know that we consider the map Inv 0 p : 0 --; G. By [7.4] there exists a unique smooth map Inv : 0 --; 0 such thatpolnv = Invop. The map h : 0 --; 0, h(x) = m(x,Inv(x)) is such that h(e) = e and po h is constant e E G. Since the constant map e is a lift of this map, by [7.2] we have h(x) = e, i.e., Inv is indeed the inverse and e the identity in (0, m). To prove associativity, we consider the smooth map a: 0 x 0 x 0 --; G defined by a(x, y, z) = p(x) . p(y) . p(z). We have two lifts of this map to 0 : m(x, m(y, z)) and m(m(x, y), z). Since both send (e, e, e) to e, they must be the same by [7.2-iii]. This is associative. The other properties of an abstract group are proved in the proves that same way. IQEDI
m
m
7.7 Discussion. If p : 0 --; G is the universal covering of a connected A-Lie group G, then 71'1(G) == ker(p) is a normal (abstract) subgroup of O. By definition of a covering, there exists a neighborhood V ofe E 0 such that 71'1 (G) n V = {e}. By [6.8] it follows that 71'1 (G) is a discrete central subgroup of O. Since 71'1 (G) is the first homotopy group of G, this proves that the first homotopy group of an A-Lie group is abelian.
7.8 Lemma. Let p : G --; H be an A-Lie group homomorphism and let H be connected. Then p is a covering ifand only ifTeP : 9 --; f) is a bijection. Proof If p is a covering, there exist neighborhoods U of eH E Hand Ui of ec E G such that p : Ui --; U is a diffeomorphism. Hence by the inverse function theorem [III.3.23] TeP is a bijection. If we assume that TeP is a bijection, it follows from [1.18] and [Y.2.l4] that P is everywhere a local diffeomorphism; it follows from [111.3.23] that there exist neighborhoods D :3 eH and V :3 ec such that P : V --; D is a diffeomorphism. Since P is a homomorphism and H connected, it follows from [2.6] that P must be sUljective. We now consider the (smooth) map f : G x G --; G, (9, h) r-+ 9h-1. By continuity of f there exists an open neighborhood V c V of ec such that f(V x V) c V. We finally define D = ker(p) and U = p(V) c D. With these ingredients we can prove that p is a covering. Let h E H and 9 E p-1(h) be arbitrary, then the set Lg(V) is an open neighborhood of 9 and L h (U) is an open neighborhood of h. Moreover, since p is a homomorphism, we have the equality p = Lh 0 po L g-1. Since p is a homeomorphism from V to U and because L g-1 and Lh are (global) homeomorphisms, p = Lh 0 P 0 L g-1 is also a homeomorphism from Lg(V) to Lh(U), Since p is everywhere a local diffeomorphism, it is a diffeomorphism from Lg(V) to Lh(U). Fixing 90 E p-1(h) we claim that the decomposition p-1(L hU) = UdEDLgodV satisfies the conditions of a covering. To prove equality, we choose 9 E p-1(L hU) and then 3g E V : p(g) = h . p(g), which is equivalentto 9;1 9g-1 = d E D. Hence 9 = 90dg, i.e., 9 E LgodV. Since obviously UdEDLgodV C p-1(L hU), we thus have equality. To show that they are mutually disjoint, suppose 9 E Lgod 1 V n Lgod 2 V {:=} 391,92 E V : 90d191 = 9 = 90d292, hence
§S. Invariant vector fields and forms
1 1 d2" d 1 = g2g1 . But f(V X V) c 1d must have d2" 1 = ec and thus d 1
315
V and thus = d2 .
1 d2" d 1 E
V.
Since p(d2"1dd = eH, we
IQEDI
7.9 Proposition. Let G and H be A-Lie groups with A-Lie algebras 9 and ~ respectively and G simply connected. Ifr: 9 ---4 ~ is an A-Lie algebra morphism, there exists a unique A-Lie group morphism p : G ---4 H such that TeP = r. Proof Uniqueness follows from [2.17], so we only have to show existence. Therefore we consider the A-Lie group G x H with its A-Lie algebra 9 x ~ and the canonical projections 71"1 : G x H ---4 G and 71"2 : G x H ---4 H. Inside 9 x ~ we have the sub algebra .5 = {(x,r(x)) I x E g}. According to [4.7] we thus have an associated connected A-Lie subgroup j : S ---4 G x H. Now Tej is an isomorphism from TeS to the subalgebra.5, hence Te(71"1 oj) : TeS ---4 TeG is a bijection. Since 71"1 oj is an A-Lie group morphism, it is a covering [7.8]. Since S is connected and G simply connected, 71"1 0 j must be a diffeomorphism, i.e., an isomorphism of A-Lie groups. We now define P = (71"20 j) 0 (71"10 j)-1 : G ---4 H, which obviously is a homomorphism satisfying TeP = r. IQEDI
8 . INVARIANT VECTOR FIELDS AND FORMS In this section we define the notions of invariant vector field and invariant differential form on an A-manifold on which an A-Lie group acts smoothly. This generalizes the notion of left/right-invariant vector field on an A-Lie group. The main results of this section are that on a connected A-manifold a vector field is invariant ifand only ifit commutes with the fundamental vector fields and that a differentialform is invariant ifand only if the Lie derivative in the direction ofthefundamental vector fields is zero. To prove these results we generalize [V.5.15] and [V.7.27], which are essentially the case ofthe action of a I-dimensional A-Lie group.
8.1 Definition. Let : G x M ---4 M be a smooth left action of an A-Lie group G on an A-manifold M and let Y be a (not necessarily smooth) vector field on M. Extending the notion of a (left/right) invariant vector field on an A-Lie group [1.8], we will say that Y is invariant under the G-action if it satisfies the condition T 0 (Q x Y) = Yo . Using the generalized tangent map T g' g E G [Y.3.l9], we can reformulate this definition as Vg E G : Tg 0 Y = Yo g, which means that for g E G and m E M we have Tg(Ym) = Yg(m)' If: M x G ---4 M is a smooth right action, then we will say that Y is invariant under the G-action if it satisfies the condition T 0 (Y x Q) = Yo . In terms of the generalized tangent map Tg this also reads as Tg(Ym) = Yg(m), but here the map 9 is different from the one in the case of a left action. And of course the
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Chapter VI. A-Lie groups
left/right invariant vector fields on an A-Lie group G are special cases of these definitions when viewing the multiplication m : G x G ----> G as left/right action of G on itself.
8.2 DiscussionINotation. We intend to show that for a connected A-Lie group G with its A-Lie algebra g, the vector field Y is invariant under the G-action if and only if Y commutes with the fundamental vector fields. The actual proof is a bit long, but the idea behind it can easily be explained. For x E 9 we have the fundamental vector field xM, whose flow is given by exp(tx). Using [Y.S.lS] we deduce that Y is invariant under the action of the subgroup formed by the exp( tx) if and only if Y and x M commute. Since the elements of the form exp( x) generate G the result follows. A first problem with this idea is that vector fields must be smooth to be integrable, which restricts attention to x E Bg. However, the main problem is that for odd elements x there is no guarantee that the odd vector field x M satisfies [xM, xM] = 0, a condition necessary for x M to be integrable. Since the even elements in Bg do not generate g, we can not reach the whole group G. To overcome this problem, we note that [V.S.lS] concerns the flow of a vector field. And the flowcPx ofa vector field X can be seen as the action of the I-dimensional A-Lie group A.s(x) on M (apart from the fact that the domain W x need not be the whole of A.s(x) x M). We thus generalize this result to the setting of general group actions: the flow cP x will be replaced by the group action , the time parameter t will be replaced by a group element g E G and the vector field X will be replaced by a fundamental vector fieldx M . In order to prepare the actual statement, we use the generalized tangent map to form the G-dependent vector field 1/Jy on M defined by
Ifwe introduce the function (1) : G x M ----> G x M, (g, m) t--> (g-1, g(m)) (note the analogy with the flow of a vector field), then we can write the definition of 1/Jy as
1/Jy = T (Q x Y) 0
0
(1) :
G
xM
---->
TM .
It is immediate that Y is invariant under the G-action if and only if 1/Jy is independent of g, i.e., 1/Jy(g, m) = Ym . If Z is a smooth vector field on G, we can form the vector field Z x Q on G x M. By abuse of notation we will denote this vector field also as Z. It is immediate from the definition that 1/Jy (g, m) E T mM. And thus the map 1/Jy and the vector field Z satisfy the requirements of [V.3.1O]. Hence it makes sense to talk about the derivative of 1/Jy in the direction of Z, i.e., about the map Z1/Jy : G x M ----> T M. Now recall that for x E 9 we have defined a corresponding fundamental vector fieldx Mon M and that in the same vein we have the right-invariant vector fieldx c on G [S.l]. Moreover, the right-invariant vector field xC and the fundamental vector fieldx M are related by : Tm xClg = xMIg(m) [S.2]. Since x M and xC are smooth if x E Bg, the following statement makes sense.
§S. Invariant vector fields and forms
317
8.3 Proposition. Let If> : G x M ---> M be a smooth left action of an A-Lie group G with associated A-Lie algebra 9 on an A-manifold M, let x E Bg and let Y be a smooth vector field on M. Then
Proof This proof is a close copy of the proof of [V.S.lS]. We first note that the result is additive in the vector field Y and the A-Lie algebra element x, so we may assume that Y and x are homogeneous. Copying the approach of [V.S.lO], we choose a point (g, h, m) E G x G x M and we imagine that 9 is close to the identity e E G. If gr are coordinates around e E G, there exist AT such that xCle = LT ATOgrie = -x. Since x E Bg, the A are real (and thus even); since x is supposed to be homogeneous we have c(Ogr) = c(x) for all Ogr contributing to the sum. Finally we choose local coordinates systems x~ around m and xl around M be a smooth right action of an A-Lie group G with associated A-Lie algebra 9 on an A-manifold M, let x E Bg and let Y be a smooth vector field on M. Then
x 1/Jy =
1/J[x M
,YJ .
§S. Invariant vector fields and forms
319
Proof For a right action 1/Jy is defined as for a left action: 1/Jy (g, m) = T g-1 Y 9 (m)' but now 9 : M ---> M is defined as 9 (m) = (m, g). If is a right action, the map 111 : G x M ---> M defined by 111 (g, m) = ( m, g-1) is a left action of G on M. Looking at the definition of a fundamental vector field, it is immediate that the fundamental vector fieldxM, associated to x E 9 forthe(right) -action is the same as the fundamental vector fieldxM,w for the (left) 1l1-action [5.4]. Defining ~ (g, m) = T1l1 g-1 YWg(m) we have by [8.3] xC;j;y = ;j;[xM, Y] but also ;j;y (g, m) = 1/Jy (g-1 , m), i.e., 1/Jy = ;j;y 0 (Inv x id). Leaving it to the reader to prove that 1/Jy makes sense, i.e., that the condition of [V.3.10] is satisfied, we compute it according to the official definition:
x
(x 1/Jy )(g, m)
= 7ra(T(g,m)1/Jy(xg)) = 7ra(T(g-1,m);j;y(T(g,m) (Inv x id)(xg, Qm))) ~
C
= 7ra(T(g-1,m)1/Jy(xg-1,Qm))
~
= 1/J[xM,y](g
-1
,m)
= 1/J[xM,y](g,m)
,
where for the third equality we used [1.10] and the fact that xC is the right-invariant vector IQEDI field on Gsatisfying x~ = -x [5.1].
8.11 Corollary. Let : G x M ---> M be a smooth left action of an A-Lie group G on an A-manifold M and let Y be a smooth vector field on M. (i) IfY is invariant under the G-action, then "Ix E Bg : [xM, Yj = O. (ii) If G is connected and if "Ix E Bg : [xM, Yj = 0, then Y is invariant under the
G-action. Proof It is obvious from the definition that 1/Jy is zero if and only if Y is zero. Now if Y is invariant, then 1/Jy is independent of the G-coordinates, and thus xC 1/Jy = 0 for all x E Bg. By [8.3] we conclude that [xM, Yj = 0, proving the first part. For the second part we first invoke [8.3] to conclude that x C1/Jy = 0 for all x E Bg. Since the vector fields xC with x E Bg span the tangent space TgG at each point g E G, we conclude by [Y.3.20] that there exists afunction f : M ---> T M such that 1/Jy = f 0 7r M. Since 1/Jy (e, m) = Ym , it follows that f equals Y. In other words, Y is invariant under IQEDI the G-action.
8.12 Definition. Let : G x M ---> M be a smooth left action of an A-Lie group G on an A-manifold M and let w be a (not necessarily smooth) k-form on M, i.e., a section of I\k *TM. Using the generalized tangent map Tg, g E G [V.3.19], we define the generalized pull-back ;w by the same formula (V.7.20) as for smooth k-forms and smooth maps. For Xi, ... ,Xk E TmM the k-form (;W)m is defined by
By definition of an action we have e(m) = m. It then follows directly from [111.3.13] that Tme = id. Hence we always have ;w = w. If : M x G ---> Mis a smooth right action, we define the generalized pull-back ;w by the same formula, but, as for invariant vector fields, it is the definition of the map 9 that changes.
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320
The k-forrn W is said to be invariant under the action ofG if for all g E G we have ;w = w. As a particular case we mention that a k-form W on G itself is said to be left/right-invariant if it is invariant under the naturalleft/right action m : G x G ....... G of G on itself. Recalling that (forleft actions) Tm g(X) = T(g,m) (Qg, X), it is elementary to see that w is invariant under the G-action if and only iffor all (not necessarily smooth) vector fields Xl, ... , X k on M we have
(8.13) where 7r M G x M ....... M denotes the canonical projection. In fact, in order to be invariant, (8.13) need only be verified for a set of vector fields that span the tangent space at each point m E M. Multilinearity then does the rest. For example it is sufficient to verify (8.13) for smooth vector fields (even if w is not smooth). Yet another way to say the same is to note that the above definition is a particular case of the generalized pull-back given in [V.7.23]. This means that we look at the map 1/Jw : G x M ....... N *T M by 1/Jw(g, m) = (~W)m. And then the definition of w being invariant becomes the condition 1/Jw = w 0 7rM. For right actions (8.13) changes in the obvious way; the defining formula of 1/Jw does not change, but as before it is 9 that changes.
8.14 Lemma. Let w be a smooth k-form. dw.
If w is invariant under the G-action, then so is
Proof Using [V.7.22], the equality (Q x X)(f07rM) = (Xf)o7rM and the equality [Q x X, Q x Y] = Q x [X, 11, it follows immediately from (8.13) and the definition of the IQEDI exterior derivative that dw is invariant ifw is.
8.15 Lemma. The map w t---> wefrom left-invariant k-forms on an A-Lie group G to /\k * 9 = /\k *Te G is a bijection. Moreover, w is smooth if and only if We has real · . We E B/\k * g. coordmates, I.e.,
Proof. Suppose w is left-invariant and We
= 0, then for Xl, ... , X k
E TgG we have by
left -invariance: ~(XI'
... ' Xk)wg = ~(XI' ... ' Xk)(L;-lW)g = ~( M be a smooth left action ofan A-Lie group G on an A-manifold M and let w be a smooth k-form on M. (i) Ifw is invariant under the G-action, then "Ix E Bg : .c(xM)w = o. (ii) If G is connected and if "Ix E Bg : .c(xM)w = 0, then w is invariant under the G-action.
9. LIE's
THIRD THEOREM
In this section we prove that for each finite dimensional A-Lie algebra 9 there exists a unique (up to isomorphism) simply connected A-Lie group G with 9 as its A-Lie algebra. For this proof we need to introduce the notion differential forms with values in an A-vector space, a notion that will be generalized and studied in more detail in chapter VII.
9.1 Definitions (forms with values in an A-vector space). Let M be an A-manifold and E an A-vector space. We have seen that a smooth k-form on M can be interpreted as a left k-linear (over C=(M)) skew-symmetric map from vector fields on M to (ordinary, smooth) functions on M, i.e., an element of HomLk(r(TM)k; C=(M)). In analogy, we define a smooth k-form with values in E or smooth E-valued k-form as being a left k-linear (over C=(M)) skew-symmetric map from vector fields on M to smooth functions on M with values in E, i.e., an element ofHomLk(r(T M)k; C=(M; E)). Playing around with the various identifications [1.5.5], [1.8.8], [III. 1.24], [V.7.1], this space is isomorphic to *(N r(T M» 9 such that 7r 0 a = id(I)), where 7r : 9 ----> g/c = I) is the canonical projection. Such maps certainly exist as can be seen by using a basis (see [11.6.24] and [11.6.23]). Associated to this a is a left bilinear map : I) x I) ----> c defined by I) is isomorphic (by Ti) to
L(X, y) = a([x, y]~) - [a(x), a(Y)]g . indeed takes its values in c because 7r is an A-Lie algebra morphism. By definition this is smooth, bilinear, even, and graded skew-symmetric (because the brackets in I) and 9 are). Given this and the A-Lie algebra I), we can reconstruct the A-Lie algebra 9 in the following way. On the A-vector space I) ED c we define the bracket
[(x, a), (y, b)] = ([x, yh, -L(X, y) 9 by rf;(x, a) = a(x) + a. It is easy to show that this rf; is an isomorphism of A-vector spaces, but also that rf; preserves brackets:
rf;([(x, a), (y, b)]) = rf;([x, y]~, -L(X, y)
c defined as
(9.17) In order to prove that these are identically zero, we want to apply [V.3.20]. Therefore we compute (Qh, Xk)Xi = ~((~, Xk) )dXi for an arbitrary left-invariant vector field x.
(Qh,Xk)Xi = -xhdi
+ Lj
(_l)(€(x)I€(vi)+€(Vj» Ad(h-I)i j
= -~(Xhk)~(vf)n + Lj (_l)(€(x)I€(vi)+€(Vj»
.
xdj
Ad(h-I)i j
.
~(xk)~(vf)n
= -~(x, Ad((hk)-I )Vi) + ~(Xk)~( (Lj Ad(h- I )ijvf))n =
=
-L( (x, Ad( (hk) -1 )Vi)) + ~(x, Ad(k- I ) (Ad( h -1 )Vi))
°
because Ad is a homomorphism.
We conclude by [1.19] and [Y.3.20] that Xi(h, k) is independent of k. But if we take k = e, we find Xi(h, e) = 0, and thus Xi is identically zero. This proves that the terms in (9.16) add up to zero, which proves that dK = 0, and thus that K is a constant function. Since K (e, e, e) = 0, we have shown that K is identically zero, i.e., that F satisfies the IQEDI relation (9.11).
9.18 Comments on the proof of [9.6]. In order to put some of the items of the proof of [9.6] in a wider perspective, we give some remarks for the interested reader. • The graded skew-symmetric bilinear function is a 2-cocycle in A-Lie algebra cohomology, the cocycle condition being given by (9.9). Changing the section a changes this cocycle with the coboundary of a 1-cochain. The reconstruction of 9 as the A-Lie algebra I) EDc is part ofthe standard isomorphism between cohomology classes in dimension 2 of A-Lie algebra cohomology and equivalence classes of central extensions of I) by c. • The function F is a 2-cocycle in A-Lie group cohomology, the cocycle condition for this cohomology being given by (9.11). • The functions Ii can be seen as a generalization of a momentum map known from symplectic geometry; here n plays the role of the symplectic form and the vf' the role of the fundamental vector fields of the group action on the symplectic manifold. • The functions Ii - Ii (e) can be put together to form a function on H with values in *1) 0 c. The fact that the functions Xi in (9.17) are identically zero then says that this new function can be seen as a 1-cocycle on H with values in the H-module * I) 0 c. More details can be found in [So, Thm 11.17].
9.19 Examples. An A-Lie algebra of dimension 111 has a basis VO,VI in which V€ has parity c. Since the bracket is even, we have [vo, vol = 0, [vo, VI] = AVI, and [VI, VI] = f-lVo, where A, f-l are real numbers because the bracket is supposed to be smooth. The graded Jacobi identity applied to Vo, VI, VI tells us that Af-l = 0. We conclude that, up to rescaling, there exists three A-Lie algebras of dimension 111 : an abelian one (A = f-l = 0),
333
§9. Lie's third theorem
°
one with [vo, vol = [VI, VI] = and [vo, vd = VI (A = 1, f..L = 0), and a third with [vo, vol = [vo, VI] = and [VI, VI] = Vo (A = 0, f..L = 1). We intend to apply the construction of the proof of [9.6] to find the corresponding A-Lie groups. • In the abelian case we find C = g, and thus ~ = {O}. The corresponding simply connected A-Lie group H is obviously {e} (of dimension 0). Since on an A-manifold of dimension 0 there are no non-zero k-forms with k > 0, we have a = 0, Ii = 0, W = 0, and hence F = O. We conclude that G = Co with group law
°
me (a, b) = a + b . In other words, G is the additive abelian group Co ~ A. • For the second case with [vo, vol = [VI, VI] = and [vo, VI] = VI we find adR(avo + bvd : Vo f-+ -bVb VI f-+ aVl' It follows that c = {O}, and that the image adR(g) c EndR(g) is given by
°
adR(g)
= { ( ~b ~) I a, bE A} .
Using [3.6] and the proof of [4.7], the corresponding A-Lie subgroup G of Aut(g) can be found to be G={ ~) I a E A o, a E A o, Ba > O} .
(!
As an A-manifold this is an open subset of go ~ A, but whose group law is given by = (ab, a + a(3). This can be interpreted as the a~ + a group of affine transformations of the odd affine line AI. • The third case with [vo, vol = [VO, VI] = and [VI, VI] = Vo presents the most interesting application of [9.6]. It is easily seen that adR(avo + bvd : Vo f-+ 0, VI f-+ bvo. It follows that c is the graded subspace generated by Vo of dimension 110, that ~ = gj c is the abelian A-Lie algebra of dimension Oil with single basis vector VI, and that the image adR(g) c EndR(g) is given by
me((a, a), (b, (3))
°
adR(g) = {
(~ ~) I b E A} .
Again using (the proof of) [4.7] and [3.6], the corresponding A-Lie subgroup H of Aut(g) can be found to be
H
= {( ~
nI
a E
Ad .
In other words, H = Al with the usual addition as group operation. Using the section a : ~ ---; 9 defined by a(vl) = VI, we find for the map :
Denoting by by
~
the odd coordinate on H, we find that the left-invariant 2-form
n = -~d~ 1\ d~ = d(-~~d~)
.
°
n is given
°
Hence a = - ~~ d~, which satisfies indeed the condition a e = (because e = in this group). The right-invariant vector field associated to VI = ae Ie is the vector field ae, hence
334
Chapter VI. A-Lie groups
= -d~, and thus f(~) = -~ is a solution. Together with the left-invariant l-form on H we find, using coordinates (~, ry) on H x H, for Wthe l-form
~(8e)n d~
For F we thus find the function F(~, ry) = -~~ry; for the group G = Al gives us the multiplication mc((a, a), ((3, b))
= (a + (3, a + b -
X
Ao ~ A this
~a(3) .
This group is the simply connected covering of the A-Lie subgroup H = GS I X Al discussed in [4.14]. The difference in constants is explained by the fact that there the scaling is such that [WI, WI] = -4wo. In fact, the covering map is given by the morphism p: G --; GS I X AI, (a,a) f-+ (e- 4ia , a). We have also encountered this A~Lie algebra as the A-Lie algebra of the multiplicative group A* = {a E A I Ba =I- O} discussed in [2.8]. This is a non-connected A-Lie group, whose connected components are simply connected. The covering map from G to the connected component containing the identity of A * is given by p : G --; A *, (a, a) f-+ e- 2a +o: = e- 2a + e- 2a a, which is actually an isomorphism because both are simply connected.
9.20 Remark. The three groups of dimensions 111 are exactly the three special cases considered in [MS-V] in a more general approach to integration of (non-homogeneous) vector fields.
Chapter VII
Connections
In a direct product with the two projections on the separate factors, we know what horizontal and vertical directions are: those that project to zero under the tangent map of one ofthese two projections. Afiber bundle 7r : B ----+ M with typical fiber F and structure group G is locally a direct product, but only one of the two projections is independent of such a local trivialization: the one corresponding to the bundle projection 7r. By convention the directions in B that project to zero under the tangent map T7r are called vertical. It follows that on a fiber bundle we do not have a natural definition of what horizontal directions are; the local idea of horizontal directions is not independent of the choice of the local trivialization. A connection on a fiber bundle is an additional structure which provides the notion of horizontal directions. This additional structure can take various forms. The most natural one is to define exactly the horizontal directions, i.e., a subbundle 11. C T B which is a supplement to the subbundle of vertical directions V = ker T7r C T B. In this form it is called an Ehresmann connection. But otherforms for the additional structure are sometimes useful: a connection I-form on a principal fiber bundle, a covariant derivative on a vector bundle, orparallel transport. The notion of an Ehresmann connection is too general for most purposes. A much more interesting subclass of connections is formed by FVF connections, whose form is determined, in a sense to be made precise, by the fundamental vector fields of the structure group on the typical fiber. The connection Ilorm, the covariant derivative and linear connections allfall in this subclass. Moreover, for the subclass offiber bundles concerned (principal/vector), they are equivalent to FVF connections. In this chapter we define the above mentioned notions of a connection and we show how they are related. On principal fiber bundles the FVF connection is also described as the kernel ofthe connection Ilorm, whereas on vector bundles the covariant derivative ofa section describes how far the section is from being horizontal. Moreover, a (vector) bundle B can be seen as associated to a principal fiber bundle P: the structure bundle. Sections of B then can be seen as a special kind of functions on P and the covariant derivative gets 335
336
Chapter VII. Connections
transformed into the exterior covariant derivative on P associated to its FVF connection. This correspondence can be generalized to differentialforms with values in an A-vector space or in a vector bundle. And then a covariant derivative and the exterior covariant derivative can be seen as generalizations of the usual exterior derivative of (ordinary) differential forms. The last aspect of connections that is treated here is the notion of curvature. An Ehresmann connection 11. on a bundle B is in particular a subbundle ofTB. As such one can ask whether 11. is afoliation, i.e., is involutive. In general the answer will be negative, but there are several cases in which one can measure to what extent it is not involutive. For principal fiber bundles with a connection I -form w this is done by the curvature 2-form n = Dw, the exterior covariant derivative of the connection I form. For vector bundles with a covariant derivative V' this is done by the curvature tensor R. In these cases the statement is that the FVF connection is involutive if and only if the curvature is zero. Moreover, we show that nand R correspond under the identifications which link connections on principal fiber bundles with those on associated vector bundles.
1.
MORE ABOUT VECTOR VALUED FORMS
In this technical section we generalize operations concerning A-vector spaces (composition, evaluation, bracket, etcetera) to vector valued differentialforms. We prove some elementary but useful formula? and we introduce the all important Maurer-Cartan I form e Me on an A-Lie group.
1.1 Definition. Let E, F, and G be three A-vector spaces with homogeneous bases (ei), (Ij), and (gk) respectively, and let : Ex F --; G be an even smooth bilinear map. With these ingredients we define the q,-wedge product 1\, which associates to an E-valued p-form a and an F-valued q-form (3, a G-valued (p + q)-form a 1\ (3, all on an Amanifold M. The construction is as follows. The forms a and (3 are uniquely determined by ordinary differential forms a i and (3j according to a = Li a i 0 ei, (3 = Lj (3j 0Ij [VI.9.l]. And then a 1\ (3 is defined by (1.2)
(2: a i
i
0 ei) 1\
(2: (3j 0 Ij) = 2: a
i
1\ I[,,(e i ) ((3j)
0 ( ei,fj) .
i,j
j
Lk
Introducing matrix elements for by q,( ei, Ij) = 7j gk. the G-valued (p + q)-form 1\ (3 is defined by the ordinary (p + q)-forms (a 1\ (3)k given by
a
(a 1\ (3)k =
2: a
i
1\ I[e(e i ) ((3j)
. 7j .
i,j
It is elementary to check that the definition of a 1\ (3 is independent of the choice of the bases for E and F, thus guaranteeing a correct definition of the -wedge product.
§ 1. More about vector valued forms
337
1.3 Notation. Each map If> has its associated If>-wedge product which we denoted as 1\. However, specific maps If> have their own notation for the associated wedge product. We will need the following four specific maps with the associated notation. • Multiplication by scalars: in this application the A-vector space E is A, F = G, and If> is (left) multiplication: If>(a, v) = a· v In this case the If>-wedge product 1\ is simply denoted as 1\ • • Applying a linear map to a vector: here F and G are arbitrary A-vector spaces, E = HomR(F; G), and If> is the evaluation map: If>(A, v) = A(v). In this case the If>-wedge product 1\ is denoted as ~. • Composition of linear maps: here E, F and G are all equal to End R ( C), the set of (right linear) endomorphisms of an A-vector space C, and If> is composition: If>( A, B) = A 0 B. In this case the If>-wedge product 1\ is denoted as {} .
• The bracket in A-Lie algebras: here E = F = G = 9 are all equal to an A-Lie algebra 9 and If> is the bracket: If>(x, y) = [x, y]. In this case the If>-wedge product of a and (3 is denoted by [ a Ii- (3]. As is usual with notation, there is sometimes more than one way to write things. Here the exceptions all occur when either a or (3 is a O-form. The most obvious case is in the first case when g is a O-form on M, i.e., an ordinary function, and (3 an F-valued k-form. In that case the F-valued k-form g 1\ (3 is the same as g . (3. This is a direct consequence of the similar fact for ordinary k-forms [V.7.l]. Less obvious is the similar situation in the second case when A is a HomR(F; G)-valued O-form, i.e., a smooth function A : M ----; HomR(F; G), and (3 an F-valued k-form. Then it is customary to write A 0 (3 instead of A ~ (3. The idea is that at each point m EMit is the composition of the map (3lm from (TmM)k to F with the map Am from F to G. Similarly in the third case: if A is a HomR(F; G)-valued O-form and (3 a HOffiR(F; G)-valued k-form, then it is customary to write A 0 (3 instead of A {} (3. Coming back to the second case, if a is a HomR(F; G)-valued k-form and g an F-valued O-form, i.e., a smooth function s: M ----; F, then it is customary to write a(g) or a· g instead of a ~ g, the idea being that for fixed mE M and Xi E TmM it is the action of the homomorphism~(Xl"'" Xk)a m on the vector g( m).
1.4 Lemma. Let 9 be an A-Lie algebra, let a be an even g-valued Ilorm, (3 an even g-valued 210rm, and let X, Y, and Z be homogeneous vector fields on M. Then: ~(X, Y)[
a Ii- a]
=
-2[ ~(X)a, ~(Y)a]
and
~(X,Y,Z)[(3li-a] =
[~(X,
In case 9
Y)(3, ~(Z)a]
+ (_I)(€(Xll€(y)+€(Zll [~(Y, Z)(3, ~(X)a] + (-1) (€(Zll€(Xl+€(Y» [~(Z, X)(3, ~(Y)a]
= EndR(E) we also have the equality [ a Ii- a] =
2a {} a.
.
338
Chapter VII. Connections
Proof For the first equality we compute for homogeneous I-forms a and 'Y
ij
L (( _l)(€(X)I€(Y)+€(a)+€(vi» ~(Y)ai ~(Xhj
=
ij
- (_l)(€(Y)I€(a)+€(vi» ~(X)ai ~(yhj)
o (-1) (€(vi)I€(Vj)+€(-y»
[Vi, Vj]
= (-l)(€(X)I€(Y)+€(a»[~(Y)a,~(Xh]-
(-l)(E(y)I€(a»[~(X)a,~(Yh].
The special case follows immediately from this result because for even a we have the equality [~(Y)a, L(X)a] = -( _l)(E(X)IE(Y» [~(X)a, ~(Y)a]. For the second equality we compute for homogeneous a and (3: ~(X,Y,Z)[(3f,\a] =
=L
~(X,
Y, Z)((3i
1\
a j ) 0 (_l)(€(vi)I€(vj)+€(a» [Vi, Vj]
ij
=
L( (-1) (€(X)I€(y)+€(Z)+€(,6)+€(Vi» ~(Y, Z)(3i ~(X)aj ij
=
+ (_l)(€(Z)I€(X)+€(Y»
(_l)(€(Y)I€(vi)+€(,6»
+ (-1) (€(Z)I€(Vi)+€(,6»
~(X, Y)(3i ~(Z)aj) 0 (-1) (€(vi)I€(Vj )+€(a» [ Vi, Vj ]
(-1)(€(Z)I€(,6» [L(X, Y)(3, ~(Z)a]
+ (_l)(E(Y)I€(,6» For 9
= EndR(E)
~(Z,
X)(3i
~(Y)aj
+ (_l)(€(X)I€(Y)+€(Z)+€(,6» [~(Y, Z)(3, L(X)a]
(-1) (€(Z)I€(X)+€(Y» [~(Z, X)(3,
~(Y)a]
.
we have the canonical basis ei 0 ej and we compute:
ijpq
o (-1) (€(ei)+€(ej)I€(ep)+€(e q») ei 0
ej
0
- (-1) (€(ei)+€(ej )1€(Y)+€(ep)+€(e q))) ~(X)ai j ~(Y)aP q) ei
0
=L
ep 0 eq
(( -1) (€(Y)I€(ep)+E(eq») ~(X)aP q ~(Y)aij
ijpq
= (_l)(€(X)I€(Y» ~(Y)a 0 ~(X)a
ej
0
ep
0
eq
- ~(X)a 0 ~(Y)a = -[ ~(X)a, ~(Y)a] .
We thus have shown that ~(X, Y)(a {} a) = -[ ~(X)a, ~(Y)a] for all homogeneous X, Y. Combining this with the first result finishes the proof. IQEDI
§ 1. More about vector valued forms
339
1.5 Example (the Maurer-Cartan I-form). Let G be an A-Lie group, 9 its A-Lie algebra,
(Vi) a basis ofg, and (iV) the associated left-dual basis of*g = *TeG. Using [VI.S.I5] we define Wi to be the left-invariant I-form on G satisfying wile = iV. We then define the g-valued I-form 8 MC on G as
which is called the Maurer-Cartan I-form ofG. Since all wi are left-invariant, 8 MC is a left-invariant g-valued even I-form on G (that 8 MC is even follows from the fact that the parity of Wi is the same as that of Vi). Moreover, if x = Lxi. Vi (Xi E A) is a left-invariant vector field on G, the contraction L(x)8 MC yields
L(x)8 MC = LxjL(Vj)w i 0 Vi = Lxi. Vi = x, i,j
i.e., 8 MC is the tautological g-valued I-form on G. Another way to state the tautological nature of 8 MC is the following. Let Xg E TgG be an arbitrary tangent vector, then x = T L g-1 Xg E TeG == 9 satisfies by definition Xg = X g. It follows immediately that L(Xg)8 MC = x. Identifying 9 with the set of left-invariant vector fields on G, we conclude that L(Xg )8 MC is the left-invariant vector field on G whose value at g is the given tangent vector Xg E TgG. We also deduce that 8 MC can be defined by
L(Xg)8 MC = TL g-1Xg. We know that 8 MC is left-invariant, i.e., using the generalized pull-back we have MC for all g E G. To see its behavior under right translations, we first note that by definition we have R;8 MC = Li R;w i 0 Vi. And then we compute:
L;8 MC = 8
L(Xh)R;Wi = L(TRg TLhx)wilhg = L(TLhg TL g-1 TRgx)wilhg = L(Ad(g-l)x)w i . It follows that L(X)(R;8 MC ) = Ad(g-1)(L(X)8 MC ). In other words, using the notation of [1.3], we can write R;8 MC = Ad(g-l) o8 MC . Let us now consider the special case G = Aut(E) with 9 = EndR(E) for some A-vector space E. As explained in [VI. 1.20], we use the basis ei 0 e j for EndR(E). Using the left coordinates gij = eMR(g)i j , i.e., g = Li,j gij ei 0 e j , the Maurer-Cartan form can be written as
8
MC
=
L
dg P q >..~~ 0 er 0 e
S
p,q,r,s
with coefficients >.. that have to be determined. For X E 9 the corresponding leftinvariant vector field X is given by Xg = Li,j,k Xij gk i 8gk j [VI. 1.20]. The condition
(( X I 8 MC))
=
X thus leads to the equations "" Xi j gk i Aks \ir er L i,j,k,r,s
. '\" \ir 1.e., .LJi ,J. ,k Xi j gk i /lks
=
Xr" s lor a11'), r.
,0,
'C/
es = " L " xr s er
,0,
'C/
eS ,
r,s
Since this must be true for all X, we deduce
340
Chapter VII. Connections
that Lk gk i At = 5[ 5~, i.e., At = (g-l)r k 5~ (use (VI. 1.22)). We find for the MaurerCartan form 8 MC : 8 MC = L dg P q (g-lr P 0 er 0 eq = L(g-l 0 dgr q 0 er 0 e q = g-l 0 dg , q,r p,q,r where for the second equality we used (VI. 1.22) and the fact that gi j (and thus dg i j) has parity C(ei) + c(ej). To interpret the last equality, we note that the canonical inclusion Aut(E) ---> EndR(E), g r-+ g can be seen as an EndR(E)-valued O-form (function) on Aut(E), as can be the map g r-+ g-l, Aut(E) ---> EndR(E). The l-form dg thus is an EndR(E)-valued l-form, exterior derivative of the O-form g. The composition g-l 0 dg is the wedge composition of the O-form g-l with the l-form dg, where as usual we have omitted the wedge symbol because the first factor is a O-form [V.7.l], [1.3]. We now go back to the general case and we look at the exterior derivative of the Maurer-Cartan form d8 MC = Li dw i 0 Vi, which is aleft-invariant g-valued 2-form on G. Since the contraction of a left-invariant l-form with a left-invariant vector field is a constant [VI.S.17], the formula for the exterior derivative [Y.7.6] gives us -~(iJi,0)dwk
=
-~([iJi,iJj])wk
=
-C~j ,
where we have used the (real) structure constants C~j of 9 [VI. 1.16]. If we now consider the left-invariant 2-form ()k = ~ Lpq C;qwq A wP (beware of the order of the indices), we can compute q q ~(iJ· iJ)()k = ~(iJ·)(~(iJ)()k) = 1. ' " ck (5 5P _ (_1)(E( vi)I€(vj»5 5P ) = ck. " J 'J 2 L pq J , 'J 'J ' pq where we used that CJi = -( _l)(€(vi)I€(vj»c~j due to graded skew-symmetry of the bracket on g. Since the values of a basis ofleft-invariant vector fields generate the tangent space at each point [VI.1.1S], we deduce that dw k = ~ Lij c']iwi Aw j . In terms of d8 MC this gives us the formula
d8 MC
= ~
Lw i Aw j 0CJiVk ijk
= -~
Lw i Awj 0 (-l)(E(vi)I€(vj»[vi,Vj]. ij
Comparing this expression with the definition of the wedge Lie bracket shows that we can write this equality as d8 MC = -~ [8 MC Ii- 8 MC ] (remember that c( Vj) = c(w j )); it is called the structure equations of G.
1.6 Lemma. The Maurer-Cartan I-form 8 MC on an A-Lie group G satisfies the equation d8 MC = -~[ 8 MC Ii- 8 MC ].
Proof The proof of this result has already been given in [1.5]. Here we give another proof using [104] and (V.7.6). For homogeneous x, y E 9 we have the equalities
-*f,y)d8MC
= X(~(YJ8MC)
- (-1)(€(x)I€(Y»Y(~(x)8Mc)
- L([x,y])8 MC =
-~(
-------;
[x,y] )8 Mc = -[x,y]
and L(X, Y)[ 8 MC Ii- 8 MC ] = -2[ ~(X)8MC, ~(Y)8MC] = -2[ x, y].
§2. Ehresmann connections and FVF connections
2.
EHRESMANN CONNECTIONS AND
341
FVF
CONNECTIONS
In this section we introduce the notion ofan Ehresmann connection on an arbitrary fiber bundle and we show that there is a natural way to transport an Ehresmann connection to a pull-back bundle. We then introduce the more restrictive notion ofan FVF connection, which can be described by local I jorms r a with values in the A-Lie algebra of the structure group of the fiber bundle. We show that transporting an FVF connection to a pull-back bundle still gives an FVF connection and that it is described by the pull-backs ofthe local I-forms ra. Wefinish by showing that an FVF connection is integrable ifand only ifthe local 2jorms d r a + ~ [ra Ii- r a1 are all identically zero.
2.1 Discussion. If A and B are two sets, a function f : A ---> B is constant if and only if the image f (A) consists of a single point: f (A) = {b}. If we have a differentiable structure, f is locally constant if and only if its tangent map T f is zero [y'3.2l]. Thinking in terms of bundles, these elementary facts obtain a new formulation. If 7r : A x B ---> A denotes the projection on the first factor, there is a bijection between functions f : A ---> B and sections s : A ---> A x B of the (trivial) bundle 7r : A x B ---> A, the identification given by s( a) = (a, f (a)). In the direct product A x B it is customary to call the subsets {a} x B vertical and the subsets A x {b} horizontal. The reason for this choice is that it is customary to draw the target space A as a horizontal line and the source space A x B as a rectangle above it. Given the map 7r, the vertical subspaces {a} x B can also be described as 7r- 1 (a). If A and B are A-manifolds, we can also talk about vertical and horizontal directions: a tangent vector (X, Y) E TaA x nB ~ T(a,b)(A x B) [Y.2.2l] is vertical if X is zero, it is horizontal if Y is zero. Again using the map 7r, the vertical directions can be described as those tangent vectors that map to zero under T7r. The set of all horizontalJvertical directions forms a foliation whose leaves are the horizontal! vertical subsets (as long as they are submanifolds). In terms of these definitions, a section s : A ---> A x B is constant if and only if its image is a horizontal subset; it is locally constant if and only if its tangent map T s maps vectors X E TaA to horizontal vectors. Under the identification s(a) = (a, f (a)) this corresponds exactly to constant and locally constant functions f : A ---> B. We now generalize the above picture to a fiber bundle 7r : B ---> M with typical fiber F. Above a local trivializing chart U c M the bundle is isomorphic to the direct product U x F with projection on the first factor. As such we can speak about horizontal and vertical subsets and about horizontal and vertical directions. And as before, the vertical subsets can be described as 7r- 1 (m) and the vertical directions as those tangent vectors that map under T7r to zero. Obviously the notion of a vertical direction does not change when we change the local trivializing chart; it can be described intrinsically by the projection map 7r. On the other hand, there is no reason to think that what is horizontal in terms of one trivialization remains horizontal in another trivialization. Said differently, a local section s E ru(B) can be constant in one trivialization and non-constant in another, i.e., the notion of a (locally) constant section is not well defined. The purpose of a connection is to give a definition of what directions will be called horizontal, and thus what sections
342
Chapter VII. Connections
will be called (locally) constant. The fact that one concentrates on horizontal directions instead of horizontal subsets (submanifolds) is because the latter is too restrictive a notion. If the set of horizontal directions is involutive, we can recover the horizontal submanifolds by means of Frobenius' theorem. However, the set of horizontal directions need not be involutive at all. And indeed, the concept of curvature, which measures more or less the lack of involutivity of the horizontal directions, plays an important role in differential geometry and physics.
2.2 Definitions. On any fiber bundle 7r : B ---; M (locally trivial, with typical fiber F) we have the vertical sub bundle VeT B, which is defined as the kernel of T7r: V = ker(T7r). Its elements are called vertical (tangent) vectors. An Ehresmann connection 1-l on B is a subbundle of 11. C T B which is a supplement to V, i.e., 1-l ED V ~ T B [IVA.6]. A different way to characterize an Ehresmann connection is to require that for all b E B the map T7r : Hb ---; Trr(b)M is a bijection. Since T7r is an even linear map, this implies that Hb is isomorphic to Trr(b)M. Elements of 11. are usually called horizontal (tangent) vectors. An Ehresmann connection 11. on a bundle B automatically defines a projection h : TbB ---; Hb as the H-part in the direct sum TbB = Vb ED H b ; h(Y) is called the horizontal part of the tangent vector Y E TbB. Since the map n7r : Hb ---; TmM with m = 7r(b) is an isomorphism, the inverse map (Tb7r)-l : TmM ---; Hb exists. For any X E TmM the image X h = (Tb7r)-l(X) E Hb is called the horizontal lift of X at b. Similarly, if X is a vector field on M, its horizontal lift X h is the (unique) vector field on B such thatXr is the horizontal lift ofXrr(b) atb. In the context of connections some terminology changes: a (smooth) map f : N ---; B is said to be horizontal if it is tangent to 11. [V.6A]. In particular a (local) section s of the bundle B is horizontal ifTs(TmM) = Hs(m) for all m in the domain of definition of s, and a submanifold C C B (with its canonical injection) is called horizontal if TC c H. The connection 11. is said to be integrable or flat if the subbundle 11. C T B is integrable [Y.6.2].
2.3 Proposition. Let (U, 1jJ) be a local trivializing coordinate chartfor the fiber bundle
7r : B ---; M with coordinates Xi, i.e., in particular, 1jJ : 7r-l(U) ---; U x F is a diffeomorphism. If 11. is an Ehresmann connection, then there exist unique smooth functions 'Yi : U x F ---; T F, 'Yi (m,f) E Tf F with Chi) = c( xi) such that the restriction 11. Irr- 1 (U) in terms ofthe trivialization 1jJ is spanned by the dim M tangent vectors (2.4)
Conversely, ifa subbundle 11. C T B is spanned on local trivializing charts by dim M vectors oftheform (2.4), then 11. is an Ehresmann connection on B. Proof In terms of the trivialization, the projection map 7r : B ---; M is given as projection on the first factor 7rl : U x F ---; U. Since T7rl : H(m,f) ---; T mM is a bijection by definition of an Ehresmann connection, it follows that H(m,f) is spanned by vectors
§2. Ehresmann connections and FVF connections
343
ax'
of the form 1m - 'Yi(m,1) for uniquely determined functions "Ii U X F ---> TF, 'Yi(m, I) E TfF. To prove that these "I are smooth, we argue as follows. By definition of a subbundle, 11. is locally spanned by smooth vector fields Xi in a neighborhood of (m, I). Using the decomposition T(U x F) = TU x TF, each Xi can be written as Xi = Lj Xi for smooth functions and smooth maps Xi : U x F ---> T F
xi axj xi with Xi = Lj xi 'Yj. By definition of an Ehresmann connection the matrix xi must
be invertible. Hence, at least in a (small) neighborhood of (m, I), there exists a smooth inverse to it. This shows that "Ii is smooth in such a (small) neighborhood of (m, f). But smoothness is a local property and thus the "Ii are smooth on the whole of U. To prove the converse, it suffices to note that the conditions guarantee that it is locally generated by smooth independent vector fields, showing that 11. is a well defined subbundle, and that T7rl is a bijection between HI (m,f) and TmM, proving that it satisfies the condition for an Ehresmann connection. IQEDI
2.5 Proposition. Let 7r : B ---> M be a fiber bundle, let 11. be an Ehresmann connection on B, and let g : N ---> M be a smooth map. Using notation as in [lV2.2], there exists a unique Ehresmann connection g*H on the pull-back bundle g* B such that for all C E g* B we have Tg (g*1-l)c C Hg(c).
Proof In [IV.2.2] we have seen that for any trivializing atlas U for B, there exists a trivializing atlas V for g* B and an induced map (also denoted by g) from V to U such that the transition functions of g* B are given by 1/J g (a)g(b) 0 g when the 1/Ja(3 are the transition functions of B associated to the atlas U. Moreover, in these trivializations, the induced fiber bundle map 9 : g* B ---> B takes the form (n, I) r--+ (g( n), I). Now let yj be local coordinates on Va E V and let xi be local coordinates on Ug(a) E U. For any connection it. on g* B there exist local functions 'Yj : Va X F ---> T F such that it. is generated in the trivialization of g* B determined by Va by the vectors
The map T 9 maps these vectors to the vectors
in the trivialization of B determined by Ug(a). If we require that this image lies in which is generated by the vectors ax; Ig(n) + 'Yi(g(n), I), then we must have
H(g(n),j),
_ 'Yj(n, I)
=
L,
ag i ayj (n) . 'Yi(g(n), I) .
We conclude thatthe 'Yj are uniquely determined by the condition Tg (g*H)c C Hg(c) and that they are indeed smooth. Since the condition is independent of the local trivialization, we conclude that this condition determines a unique Ehresmann connection g*H on g* B.
IQEDI
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Chapter VII. Connections
2.6 Discussion. A special case of a pull-back bundle is the restriction of B to a submanifold N of M [IV.2.3]. We thus see that [2.5] tells us in particular that an Ehresmann connection 11. on B induces a unique Ehresmann connection on the restriction BIN = 7r- 1 (N) of B to the submanifold N. It is not hard to see that this induced connection is just the restriction 11.lrr-1(N) of11. toBIN. Slightly more general is the case of an immersion 9 : N ---> M, in which case again we have an induced Ehresmann connection on g* B. This application of [2.5] plays a fundamental role in the concept of parallel transport along a curve in M. Without going into details because that is outside the scope of this book, we briefly sketch the idea of parallel transport. We first note that a curve in M is an immersion 9 : N ---> M of a I-dimensional connected A-manifold N in M. Given a fiber bundle 7r : B ---> M we have a pull-back fiber bundle g*7r : g* B ---> N, and if'H is an Ehresmann connection on B, we have an induced Ehresmann connection g*11. on g* B. We now fix no E BN and bo E BBg(no). Parallel transport of bo along the curve is a smooth horizontal map pt : N ---> B with pte n) E Bg(n) and pte no) = bo, and thus in particular Tpt maps TnN into 11.pt(n), i.e., the vectors tangent to pt(N) are horizontal. A sufficient condition for such a map to exist (and then it is unique) is that g*11. is integrable and that its leaves are diffeomorphic to N via the projection map g*7r : g* B ---> N. Since N is I-dimensional, the induced Ehresmann connection g*11. on g* B is a I-dimensional subbundle ofT(g* B), and thus the integrability condition is automatically satisfied when N is even, i.e., of dimension 110. Given these conditions, the map pt is constructed as follows: since g*11. is integrable, there exists a leaf L passing through g-l(b) E (g* B)n (the map 9 is a diffeomorphism between fibers). Since this leafis diffeomorphic with N via g*7r, we can define pt = go (g*7rIL)-l. This map satisfies the given requirements. That the integrability condition alone is not sufficient is shown in the following elementary example. We take M = Ao with the global even coordinate x and the trivial bundle B = M x Ao == (Ao)2 with the global even coordinates (x, y) and the obvious projection 7r : B ---> M. On B we define the Ehresmann connection 11. by
As curve we choose the canonical embedding M ---> M, i.e., we see M itself as a curve in M. Now suppose that pt : M ---> B is a parallel transport map. Then it must be of the form pt(x) = (x, f(x)) for some smooth function f. And then the condition that Tpt maps TxM into 11.pt(x) implies that f must satisfy the condition (axf)(x) = - f(x)2 because Tpt maps ax to ax + (axf) (x) ay. This implies that f is of the form f(x) = (x - C)-l for some c E R. But such a map is not defined on the whole of M, and thus parallel transport along the whole of this curve does not exist (see also [5.10]).
2.7 Remark. In [Eh] C. Ehresmann introduced his general notion of a connection on an arbitrary fiber bundle. The definition he gave is slightly stronger than that of what here is called an Ehresmann connection. He added the requirement that parallel transport should always be defined. The underlying idea is that parallel transport provides an alternative
§2. Ehresmann connections and FVF connections
345
way to define a connection. Since this approach becomes highly unwieldy in the case of A-manifolds, we here only sketch the procedure in the case of R-manifolds, neglecting all questions about smoothness. Let g : [x, y] ---; M be a curve, let b E Jr-l(g(x)) be arbitrary and let 9 : [x, y] ---; M be a horizontal map satisfying Jr 0 9 = g and g(x) = b. Since 9 is uniquely determined by b, we obtain, by varying b, a well defined map Fg : Jr-l(g(X)) ---; Jr-l(g(y)), b = g(x) f--+ g(y). Running through the curve gin the opposite direction shows that Fg is bijective. Taking the derivative with respect to y at y = x gives us back the tangent vector of 9 at b. If parallel transport over all curves exists, we thus can recover the set of horizontal directions at b, i.e., we can recover the (Ehresmann) connection. This analysis also shows that to any curve g we have associated a diffeomorphism Fg between the fibers over the endpoints. Since in general there is no canonical way to compare, in a fiber bundle, fibers over different points, these diffeomorphisms are a useful tool when one wants to do so. The idea of comparing different fibers in this way is one of the main motivations for the introduction of a connection in the form of parallel transport. In the context of A-manifolds we have ignored this approach to a connection because not all points in a connected A-manifold can be connected by a smooth curve.
2.8 DiscussionlDefinition. The maps "Ii in (2.4) depend, obviously, upon the trivializing set U. Most, if not all, types of connections are special cases of an Ehresmann connection, eventually in disguise. They use special features of the bundle to impose restrictions on the form of the maps "Ii. We will restrict our attention to one special form of these maps, a form that will be sufficiently universal to cover all our examples. The idea is quite simple. We have a typical fiber F with a (pseudo effective) left action of the structure group G. We thus can require that all "Ii are fundamental vector fields associated to this action. More precisely, we define an FVF connection (for Fundamental Vector Field) to be an Ehresmann connection such that for each ma E M there exists a trivializing coordinate chart (U, 1/J) containing ma such that the map f f--+ "Ii (m, f) E Tf F is a fundamental vector field for all m E U. The next results show that the notion of being a fundamental vector field has nice smoothness properties and is independent of the choices that can be made.
2.9 Lemma. Let (U, 1/J) be a trivializing coordinate chart with coordinates (xi) such that the map f f--+ "Ii(m, f) E TfF is afundamental vectorfieldfor all m E U. Then there exists a unique map Ai : U ---; 9 such that "Ii(m, f) = Ai (m)f. Moreover, the map Ai is smooth. Proof By definition of a fundamental vector field, there exists an Ai (m) E 9 for each m E U such that "Ii (m, f) = Ai (m) It thus remains to prove that the map Ai is unique and smooth, which will be a consequence of the fact that the structure group acts pseudo effectively. We start with smoothness. In [2.3] we have seen that the maps (m, f) f--+ "Ii(m, f) == Ai(m)f are smooth. With respect to a basis (Vj) for 9 the map Ai takes the form Ai (m) = L j d (m) Vj for functions
f.
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Chapter VII. Connections
cj : U ----; A. Since the map TiJJ f is left linear, we have Ai(m).f = Lj c1(m) (Vj).f. With respect to a local coordinate system on F with coordinates (y, ry), each fundamental vector field (Vj)F has the form (Vj).f = L~~ F Ej (I) fA. Smoothness of Ai being equivalent to smoothness of the coefficients, we thus know that Lj c1 (m )Ej (I) is smooth for all k, and we want to show that this implies that all c1 are smooth. Since the Ej are smooth, we have Ej(y, ry) = LJ EJ,J(y)ryJ [111.3.17]. Taking derivatives with respect to the ry's of the smooth functions LJ(Lj c1 (m)Ej,J(Y) )ryJ, we obtain in particular that Lj c1 (m)EJ,J(Y) is smooth for all k and 1. Following the proof of [VI.6.6] we introduce the functions rP~(y) = LjjV' EJ,J(Y) E *g for real values of the (even) coordinates y. Varying also the local coordinate charts, we know from the proof of [VI.6.6] that there are £ :::; dim 9 independent elements rP~~ (yd, ... , rP~! (YR.), such that all other elements rP~ (y) are linear combinations with real coefficients of these £ elements. Since these independent elements define g,pO, and since g,pO = {O} [VI.6.l0], we conclude that £ = dim g. Changing the basis of 9 if necessary we may assume that the rP~: (Yr ) form the left dual basis, i.e., rP~: (Yr) = rv. It follows that E;''Jr (Yr) = 5j, and thus .
k
cr(m) = Lj c1(m)Ej,'JJYr) is smooth. To prove uniqueness ofthe Ai, suppose that A~ is another solution. Then Ai - A~ is a smooth family for which the associated fundamental vector field is identically zero. Since g,pO = {O}, this implies that Ai = A~. IQEDI
2.10 Lemma. Let (Ua , 1/Ja) be a trivializing coordinate chart with coordinates xi and let A~'x : U ----; 9 be smooth maps such that the Ehresmann connection 1-l is spanned in the trivialization 1/Ja by Oxilm +A~,X(m).f E TmM x TfP. j Ify is another system of coordinates on Ua, then 11. is also spanned in the trivialization 1/Ja by Oyj 1m + A~'Y(m).f E TmM X TfP ,
with Aj'y : U ----; ggiven by Aj'Y(m) = Li(Oyjxi)(m) . A~,X(m). If(Ub,1/Jb) is another trivialization with Ua = Ub, then 11. is spanned in the trivialization 1/Jb by 0Xi 1m + A~'x (m).f E TmM X TfP , with A~'x : U ----; 9 given by
A~,X(m) = Ad(1/Jba(m)) (A~,X(m) - TL,pba(m)-' T1/Jba ax; 1m) (2.11 )
= Ad(1/Jab(m)-l) A~,X(m) + TL,pab(m)-' T1/Jab 0Xi 1m ,
where 1/Jba : Ua = Ub ----; G is the transition function related to the change of trivialization from 1/Ja to 1/Jb. Proof The first part is a direct consequence of the fact that the tangent map is left linear: .
k
O~i 1m + TiJJf A~,X(m) = ~ ~~~ (m) (o~j 1m + TiJJ f (~ ~~j (m) A~'X(m)))
,
§2. Ehresmann connections and FVF connections
347
87.
because Lj axi yj . ayj xk = Since the matrix aXi yj is invertible, the result follows. To prove (2.11), we first note that (1/Jb ° 1/J;; 1)(m, f) = (m, ( 1/Jba (m), f)) by definition of the transition function 1/Jba : U ----; G. We then compute the image of the tangent vector ax, 1m + A~'x (m)r under the map 1/Jb ° 1/J;;1 :
T( 1/Jb ° 1/J;;1) (axi 1m + A~'x (m)n
= aXi 1m + T f T1/Jba aXi 1m + T,pba(m) A~'x (m)r
+ T,pba(m) (A~'X (m)r + T f T L,pba(m)-l T1/Jba axi 1m) axi 1m + T,pba(m) (A~,X(m) - TL,pba(m)-l T1/Jba aXi Im)~
= axi 1m
=
= axi 1m + (Ad(1/Jba(m)) (A~'X(m) - TL,pba(m)-l T1/Jba axi Im)):(,pba(m),f) , where the second equality follows from the equality f 0 L,pba(m) = ,pba(m)of as maps from G to F. This proves the first equality of (2.11). The second equality is obtained by interchanging the roles of a and b and using that 1/Jba (m) ·1/Jab( m) = e for all m E Ua
IQEDI
= Ub •
2.12 Discussion. We learn from [2.9] that being an FVF connection can be expressed in terms of (local) smooth maps with values in g. And then [2.10] tells us that the notion of being a fundamental vector field on a trivialization is independent of the chosen trivialization as well as the chosen coordinate system. Moreover, the explicit dependence of A;,y(m) in terms ofA~,X(m) also shows that the I-formr a with values in 9 on Ua defined as ra(m) = dx i 0 A~,X(m) = dyj 0 A;,y(m)
L
L j
is independent of the chosen coordinate system. This implies that if (Ua, 1/Ja) is a local trivialization, there exists a g-valued I-form r a on Ua such that, if Xi are coordinates on (a part of) Ua, the local functions A~'x can be recovered from r a by
The existence of r a is independent of whether there exists a global coordinate system on Ua or not, and 11. is given in the trivialization 1/Ja by
2.13 Corollary. Let 7r : B ----; M be a fiber bundle with typical fiber F and structure group G, and let ={ (Ua, 1/Ja) I a E I} be a trivializing atlas for B. If11. is an FVF connection on B, there exist unique g-valuedl -forms r a on Ua such that 11. is given in the trivialization 1/Ja by
(2.14)
348
Chapter VII. Connections
Moreover, on overlaps Ua n Ub the 110rms r a and rb are related by (2.15)
where 8 MG is the Maurer-Cartan 110rm on G. Conversely, ifwe have g-valued 110rms r a on Ua that are related on overlaps by (2.15), then (2.14) defines an FVF connection 11.. Proof IfH is given, (2.14) is a direct consequence of [2.12]. To prove (2.15) we choose (local) coordinates (Xi) on Ua nUb' Using (2.11) and [1.3] we obtain
rb(m) =
L dx
i
0 A~(m)
Ldx 0 (Ad(1,bab(m)-l)A~(m) +TL,pab(m)-l T1,bab oxi lm) = Ad(1,bab(m)-l)ora(m) + Ldx 0TL,pab(m)-l T1,babOxilm. i
=
i
To prove thatthe second term equals And then we use [1.5] to compute
1,b~b8 MG,
o
we first note that T1,bab xi 1m E T,pab(m)G.
Im)8 MG = T L,pab(m)-l T1,bab Oxi 1m . we always have a = 2:i dx i 0 L( 0xi )a, (2.15) follows.
~(Oxi Im)1,b~b8 MG = ~(T1,babOxi
Since for any I-form a To prove the converse, we first note that (2.14) obviously is the local expression of an FVF connection. It only remains to be shown that these local expressions coincide on IQEDI overlaps Ua nUb. But this is an immediate consequence of [2.10], (2.11).
2.16 Corollary. Let 7r : B ---> M be a fiber bundle, let H be an FVF connection on B, and let g : N ---> M be a smooth map. Then the unique Ehresmann connection g*1-l on the pull-back bundle g* B such that for allc E g* B we haveTg (g*H)c C Hg(c) [2.5J is also an FVF connection. In particular, if U and V are (trivializing) atlases as in the proof of[2.5 J, then g*H is determined by the local g-valued 110rms fa = g*rg(a) [2.13 J.
Proof Since H is an FVF connection, the maps 'Yi are given as 'Yi(m, f) = Ai(m)f for g-valuedfunctions Ai. Using the arguments and notation as in the proof of [2.5], g*H is determined by the functions "fj given by
"fj(n, f) =
ogi (Ogi )F L, oyj (n) . Ai(g(n))f = L oyj (n) . Ai(g(n)) f ,
This proves that g*H is an FVF connection. In terms of the g-valued I-forms associated to the trivializing atlases, we find:
r_a
=
"'"' ogij (n) . Ai(g(n)) = " , " , ' 0 Ai(g(n)) = g*rg (a) , L dyJ. 0 "'"' L a Lg*dx' j
i
Y
i
where rg(a) = 2:i dXi 0 Ai(m) is the g-valued I-form on Ug(a) associated to the connection H on B. IQEDI
§2. Ehresmann connections and FVF connections
349
2.17 Remark. In the context of general Ehresmann connections one could wonder why it is so easy to define a pull-back connection, because an Ehresmann connection is an object living on the tangent bundle, and for tangent vectors the notion of pull-back is not (directly) defined. A possible explanation is that the combination Li dxi 0 "Ii is independent of the chosen coordinate system, and for i-forms we do have a natural notion of pull-back. But a precise definition of this object is not easy. However, in the context ofFVF connections, the notion of a pull-back connection becomes natural. Such a connection is defined by local g-valued i-forms. And, as we have seen, the pull-back of these i-forms defines the pull-back connection.
2.18 Proposition. Let 7r : B ---> M be a fiber bundle with typical fiber F and structure group G. Let U = {Ua I a E I} be a trivializing atlas for B and let r a be g-valued I-forms on Ua defining an FVF connection 11. according to [2.13]. Then 11. is integrable ifand only if(all) the local2-fonns
dr
a
+ ~ [r a Ii- r a 1
are identically zero. Proof We have to show that 11. is an involutive subbundle. Since the value ofacommutator
of two vector fields at a point depends only upon the behavior of the two vector fields in a neighborhood of the given point, it suffices to verify that 11.17r-1(ua) is involutive for all a E I. But on 7r- 1 (Ua ) the connection 11.(m,f) is spanned by the vector fields 0Xi 1m + A~'x (m).f [2.12]. Using [V.1.19] it follows that it suffices to show that the commutator of two of these generating vector fields belongs to 11.. To compute such a commutator, we first compute the commutator [oxi , A F 1 for a smooth map A : Ua ---> g. To do so, we choose a basis Vj for 9 and (local) coordinates yk on F. It follows that there exist smooth functions Aj : Ua ---> A such that A = Lj Ajvj. There also exist smooth functions ~k : F ---> A such that (Vj)F = Lk ~k Oyk. And then we compute
jk
jk
We thus find for the commutator of two generating vector fields: [oxi
+ (A~,X)F , ox j + (A~'X)F 1 = (oxiA~,X)F _ (_l)(€(xi)I€(xj))(oxjA~'X)F
+ [A~'X,
A~,xlF ,
where we used [VI.S.2-iii]. Since this commutator is tangent to F, it projects to zero on Ua under the projection T7r. Since T7r is a bijection from 11.(m,f) onto TmM, the condition that this commutator belongs to 11. becomes the condition that this commutator must be zero (for all i and j). Looking at the definition of g,pO [VI.6.2], it follows that the image of the smooth functions Fi~'x : Ua ---> 9 defined by Fi~,X(m)
= oxiA~'x
- (_l)(€(xi)I€(xj))oxjA~'X
+ [A~'X,
A~,xl
Chapter VII. Connections
350
belongs to g1l>o. Since the action of G on F is pseudo effective, it follows from [VI.6.1O] that 11. is integrable if and only if all functions Ft/ are identically zero. On the other hand, using [Y.7.6], [104] and ~(Oxi )fa = A~'x [2.12], we find -~(oxi,oxj)(dfa+~[faf,\fa]) =Fi~'x.
Since any 2-form (3 on Ua can be reconstructed from its contraction with the 0Xi by 1 . i (3 = 2" Lij dx J 1\ dx . ~(OX" Ox j )(3, the result follows. IQEDI
3.
CONNECTIONS ON PRINCIPAL FIBER BUNDLES
In this section we show that an FVF connection on aprincipal fiber bundle can be described either as the kernel ofa so called connection I -form w on the bundle or as an Ehresmann connection that is invariant under the right action of the structure group on the bundle. We also show how the connection I -form w can be reconstructed from the local I forms fa defining the FVF connection. The description in terms ofa connection I form allows us to prove quite easily that there always exist FVF connections on a principal fiber bundle.
3.1 Lemma. Let 7r : P ---; M be a principal fiber bundle with structure group G and let U c M be open. Let T denote the map which associates to each local trivialization 1jJ : 7r-l(U) ---; U x G of7r- 1 (U) the local sections E fu(P), s(m) = 1jJ-l(m, e) with e E G the identity element. Then T is a bijection between the set ofall local trivializations of 7r- 1 (U) andfu(P). The inverse of T is given by the formula1jJ-l( m, g) = s(m) . g.
Proof If s is given as s( m) = 1jJ-l (m, e), then by definition of the right action of G on P we have 1jJ-l(m,g) = s(m) . g. Hence the given formula is a left inverse for T. To show that it also is a right inverse and that it indeed defines a local trivialization, we suppose that s E fu(P) is a local smooth section. We then define the smooth map U x G ---; 7r-l(U) by w(m, g) = s(m)· g. If (V, X) is any local trivializing chart for P, we obtain a map Sx : Un V ---; G such that (X ° s)(m) = (m, sx(m)) [IV. 1.20]. It follows that X ow: U n V x G ---; U n V x G is given by
w:
(3.2)
(xow)(m,g) = (m,sx(m) .g).
From this one deduces that Wis bijective, a bundle morphism and a local diffeomorphism. Hence 1jJ = w- 1 is also smooth. And then (3.2) shows that 1jJ is compatible with the IQEDI structure of the principal fiber bundle, i.e., (U, 1jJ) is a local trivialization.
3.3 Corollary. A principal fiber bundle 7r ifand only iff(P) is not empty.
:
P ---; M is (isomorphic to) the trivial bundle
Proof This is the special case U = Min [3.1].
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§3. Connections on principal fiber bundles
3.4 Discussion. In [3.1] we have established a bijection between local sections and local trivializations of a principalfiber bundle 7r : P ---> M. We are thus allowed to speak of the (local) trivialization 1/J determined by the (local) section s. Now suppose that Sa E rUa(P) and Sb E rUb (P) are two local sections determining two local trivializations 1/Ja and 1/Jb. These two trivializations determine a transition function 1/Jba : Ua nUb ---> G by the formula (1/Jb 0 1/J;;1) (m, g) = (m, 1/Jba (m)g). Applying 1/Jb"1 and substituting the definition of 1/Ja and 1/Jb in terms of Sa and Sb gives us (3.5)
This formula gives us a way to determine the transition functions directly from the defining local sections: given Sa and Sb there exists for each m E Ua n Ub a unique 1/Jba (m) E G such that Sa (m) = Sb( m) ·1/Jba (m ),just because the right action of G on the fibers of P is free and transitive. (3.5) then tell us that it must be the transition function determined by the two associated trivializations 1/Ja and 1/Jb.
3.6 Remark. In the physics literature a local section of a principal fiber bundle is often called a (local) gauge and changing a local section S to a local section by the formula s(m) = s(m) . cjJ(m) is called a (local) gauge transformation.
s
3.7 Definitions. Let 7r : P ---> M be a principal fiber bundle with structure group G whose A-Lie algebra is g. In a local trivialization 7r-l(U) S:! U x G the vertical subbundle V is just the tangent space to the second factor TG. It follows that V is spanned by the left-invariant vector fields on G. Since the right action of G on P corresponds to right multiplication on the second factor and since the fundamental vector fields of right multiplication are exactly the left-invariant vector fields [VI.SA], we conclude that V is spanned by the fundamental vector fields of the right action of G on P, independent of the choice of a local trivialization. More precisely, Vp = {x: I x E 9 }. • An FVF connection on a principal fiber bundle P is called a principal connection, or simply a connection. Contrary to the vertical directions, the fundamental vector fields used in the definition of a principal connection on P are right invariant because the action of the structure group G on the typical fiber G is left multiplication . • A connectionl-formon the principal fiber bundle P is an even g-valued I-form won P satisfying the following two conditions. (i) "Ix E 9 : L(XP)W = x. (ii) Vg E G: ;w = Ad(g-l) ow.
Since the meaning of (i) is rather obvious, we concentrate on the precise meaning of (ii). On the left hand side ;w indicates the generalized pull-back as defined in [V.7.23], [VI.S.12]. On the right hand side we have an example of the alternative notation [1.3] for the evaluation-wedge product of an EndR(g)-valued O-form with a g-valued I-form. Writing all definitions explicitly, condition (ii) says that for all (p, g) E P x G and all Xp E TpP we must have (ii)
L({(Xp,Qg) I T(p,g) M be a principal fiber bundle with structure group G. Let 11. be a principal connection on P and let w be the associated connection I-form. Let {(Ua ,1/Ja) I a E I} be a trivializing atlasfor B and let Sa : Ua ---> P be the local section defining 1/Ja [3.1]. Finally, let r a be the g-valued I-fonns on Ua defining the principal connection 11. [2.13]. Then s~w = r a and WI 7r -l(Ua ) can be reconstructed from r a in the trivialization determined by Sa by
(3.12) Proof Since w is completely determined by the conditions ker(w) = 11. and condition (i) of a connection I-form, it suffices to verify these condition in the trivialization determined by Sa. In this trivialization the fundamental vector field x P is given as x [3.7], and thus
by definition of the Maurer-Cartan I-form. Introducing local coordinates Xi on Ua , we know that the connection 11. is generated by the vector fields oxilm + A~,X(m); with
354
Chapter VII. Connections
= ~(oxilm)ra [2.12]. Using the properties of the Maurer-Cartan I-form [1.5] and the factthat A~,X(m)~ = -TRgA~,X(m) [VI.5.1], we then compute
A~,X(m)
~(oxilm
+ A~,X(m);)(eMClg + Ad(g-l) oralm)
=
= Ad(g-l)L(Oxilm)ralm - ~(TRgA~,X(m))eMClg = Ad(g-l)A~,X(m) - ~(A~,X(m))R;eMC = 0 . We conclude that the given expression for W in the local trivialization determined by Sa has the required properties and thus must coincide with w. From this local expression for wand the fact that in this trivialization the local section Sa is given as Sa (m) = (m, e), it IQEDI follows that s~w = ra.
3.13 Discussion. A natural question is whether there always exists a principal connection on a given principal fiber bundle 7r : P ---> M. The answer is affirmative and relies on a partition of unity argument. LetU = {(Ua,1,Ua) I a E I} be a trivializing atlas. For each a E I we choose the I-form r a == O. These choices do not (in general) satisfy (2.15), and thus do not define a global principal connection. But on the restriction Plua = 7r-l(Ua ) they do define a principal connection. According to [3.11] we thus have a connection I-form Wa = e MC on the local trivialization 7r- 1 (Ua ) ~ Ua x G. Let {Pa I a E I} be a partition of unity associated to the open cover U. For each a E I we then have the global I-form Pa Wa. This global I-form obviously satisfies condition (ii) of a connection I-form, but condition (i) is replaced by ~(XP)Pa Wa = pa . x. It follows that LaEI pa Wa is a well defined global connection I-form on the principal fiber bundle P.
4.
THE EXTERIOR COVARIANT DERIVATIVE AND CUR V A TURE
In this section we continue the study of FVF connections on principal fiber bundles. We define the exterior covariant derivative D and apply it to the connection I -form W to obtain the curvature 2-form n = Dw. We then prove the structure equations of Cartan n = dM.J + ~ [w Ii- w] and the Bianchi identities dn = [n Ii- w]. We also show that n is determined by the local 2-forms d r a+ ~ [ra Ii- r a] if the FVF connection is determined by the local I forms ra. Not surprisingly, we can prove that the FVF connection is integrable ifand only if the curvature n is zero.
4.1 Definition. Let 11. be a principal connection on a principal fiber bundle P ---> M and let W be the associated connection I-form. For any k-form a on P with values in an A-vector space E one defines the exterior covariant derivative Da (with respect to the connection I-form w) by the formula
355
§4. The exterior covariant derivative and curvatnre
where h denotes the projection on the horizontal part h : TpP ---> 11. p. In particular the curvature 2-form f2 is defined as the exterior covariant derivative of the connection I-form w: f2=Dw.
4.2 Lemma. For any two vector fields X and Y we have ~(X, Y)f2 = ~([ hX, hY])w. As a consequence, the principal connection 11. on the principal fiber bundle P ---> M is integrable if and only if its curvature 210rm f2 is identically zero. Proof Using 01.7.6) we have for homogeneous X and Y : -~(X, Y)f2 = -~(hX,
hY)dw
= (hX)(~(hY)w) - (-l)(€(X)IE(Y»(hY)(~(hX)w) - ~([ hX, hY])w
=
-~([hX,hY])w.
According to Frobenius, the connection 11. is integrable if and only iffor all X, Y E 11. we have [X, YJ E H. Since 11. = ker w, this means that 11. is integrable if and only if have the implication ~(X)w = ~(Y)w = 0 ~ ~([X, Y])w = O. Using the definition of the projection h : T P ---> 11., this means that 11. is integrable if and only if for all vector fields X and Y we have ~([hX, hY])w = O. According to our previous computation, this is the IQEDI case if and only if f2 = O.
4.3 Lemma. Let w be a connection I-form on a principal fiber bundle P f2 = Dw be its curvature 2-form. Then we have thefollowing identities:
Dw
== f2
=
dw
+ ~ [w Ii- w]
df2=[ f2 li- w ]
--->
M and let
(the structure equations ofCartan) (the Bianchi identities).
Proof Interpreting k-forms as skew-symmetric k-linear maps on smooth vector fields
(IV.5.l5), it suffices to evaluate these identities on smooth vector fields. Since they are 2- and 3-additive, it suffices to show that we have equality when evaluating on homogeneous vector fields. Moreover, since a vector field splits as a sum of a horizontal and a vertical vector field, we may restrict attention to smooth vector fields which are either horizontal or vertical. And finally, since k-forms are linear over C=(P) and since vector fields of the form x P with x E Bg generate the module of smooth vertical vector fields [VI. 1.19], [3.7], it suffices to use this kind of vertical vertical vector field. We start with the structure equations of Cartan, for which we evaluate both sides on two homogeneous vectors X, Y. We distinguish three cases: both vertical, both horizontal, and X horizontal and Y vertical. If both X and Y are horizontal, we have ~(X, Y)f2 = ~(X, Y)dw. Since by [1.4] ~(X, Y) [w Ii- w] = 0, we have equality for two horizontal vector fields.
Chapter VII. Connections
356
If both X and Y are vertical, we may assume, as argued above, that X = x P and Y = yP for some x, y E Bg. It follows that [x P, yP] = [x, y]P is vertical. By definition of 0, ~(X, Y)O = O. On the other hand we have
and ~(xP, yP)[ w f,\ w] = -2[ ~(xP)w, ~(yp)w] = -2[x, y] [104], which shows that for two vertical vector fields we also have equality. If X is horizontal and Y vertical, we may assume that Y = yP for ayE Bg. By definition of a connection l-form we have the equality ;w = Ad(g-l) ow. From [VI.8.20] we know that we can take the derivative of the left hand side in the direction of y and that at g = e we obtain £(yp)w (: = id). To compute the derivative of the right hand side in the direction of y at g = e, we first note that it depends on g only via Ad(g-l). And then [VI.9.l2] gives us Ye(Ad(g-l) ow) = - adR(y) ow, and thus £(yp)w = - adR(y) ow. We then compute ~([X,yp])w
= ~(X)£(yp)w - (-l)(€(X)IE(y»£(yp)~(X)w = -~(X) adR(y) ow = -( _l)(€(X)I€(y» adR(Y)(~(X)w) = 0,
because ~(X)w = O. In other words, if X is horizontal, then [X, yP] is also horizontal. Since Y = yP is vertical, ~(X, Y)O = O. Moreover, again by [104], we have the equality ~(X, Y)[ w f,\ w] = -2[ ~(X)w, ~(Y)w] = O. Finally,
because ~(yP)w is constant and X and [X, yP] are horizontal. We conclude that also in the third case we have equality, i.e., we have proven the structure equations of Cartan. To prove the Bianchi identities, we first note that dO = ~d[ w f,\ w] according to the structure equations of Cartan. With respect to a basis (Vi) of 9 we have ordinary l-forms Wi defined by w = L i Wi 0 Vi. And then we have dw = L i dw i 0 Vi and [w f,\ w] = Lij Wi 1\ wj 0 (_l)(€(vi)I€(vj» [Vi, Vj], and thus:
ij
= [dw f,\ w]-
2:) _l)(€(vi)I€(vj»dw
j 1\ Wi
0 (-[ Vj, Vi])
= 2[ dw f,\ w] .
ij
Again using the structure equations of Cartan we thus have dO
= [dwf,\w] = [Of,\w]-
~[[wf,\w] f,\w]
The result now follows because of the Jacobi identity, [104] and the following computation
357
§4. The exterior covariant derivative and curvatnre
for three homogeneous vectors: ~(X,Y,Z)[[wf,\w]
f,\w]
= [~(X,Y)[wf,\w],~(Z)w]
+ (-1) (€(X)!€(y)+€(Z» [~(Y, Z)[ w f,\ w ], ~(X)w ] + (_l)(€(Z)!€(X)+€(Y» [~(Z, X)[ w f,\ w], ~(Y)w] = -2( -1) (€(X)!€(Z»
(( _l)(€(X)!€(Z» [[
l.(X)w, l.(Y)w], ~(Z)w]
+ (_l)(€(X)!€(y» [[ l.(Y)w, ~(Z)w], ~(X)w] + (-1) (€(Z)!€(y» [ [~( Z)w, ~(X)w ], ~(Y)w ])
=
0.
IQEDI
4.4 Corollary. On any principal fiber bundle with a connection I-form w we have the equality DD == D 2 w = o.
Proof Using [1.4] and the Bianchi identities [4.3] we have ~(X, Y,
= Since
Z)DD = l.(hX, hY, hZ)dD = [~(hX,
~(h W)w
hY)D, l.(hZ)w]
~(hX,
hY, hZ)[ D f,\ w]
+ (_l)(€(X)!€(Y)+€(Z» [~(hY, hZ)D, ~(hX)w] + (-1) (€(Z)!€(x)+€(Y» [~( hZ, hX)D, ~(hY)w]
.
is zero for all W, the result follows.
4.5 Remark. If G is an A-Lie group, we can see it as a principal fiber bundle over a point M = {mo} with (trivial) projection 7r : G ---> {mo}. Since there is no non-zero I-form on a zero dimensional A-manifold, it follows from [3.11] that any connection I-form w on this principal fiber bundle necessarily is the Maurer-Cartan I-form: w = e Me. And then the structure equations of Cartan [4.3] tell us that [1.6] can be interpreted as saying that the curvature of this connection is zero.
4.6 Discussion. Let w be a connection I-form on a principal fiber bundle 7r : P ---> M, let 11. be the associated principal connection, let {( Ua, 1/Ja) I a E I} be a trivializing atlas for P determined by local sections Sa' and let Xi be (local) coordinates on some Ua . In [3.11] we have seen that the local I-forms s~w are the local I-forms r a defining 11., which are related on overlaps Ua nUb by (2.15). Moreover, the local functions A~'x describing the fundamental vector fields 'Yi are given by A~'x = ~(OXi)S~W [2.12]. We intend to give a similar description of the curvature2-form D = Dw. We thus define the g-valued2-forms s~D on Ua and the homogeneous smooth functions Pt/ : Ua ---> 9 by Pi~'x (m) = -~( oxi, ox j )(s~D)m with parity C(Pi~'X) = c(Xi) + c(x j ). Since a 2-form is graded skew-symmetric in its entries, the functions Pi~'x are graded skew-symmetric in their indices: pa.'x = _(_l)(€(x i )!€(x j » pa.'x. With these functions the 2-form s*D can J' 'J a be written as (s~D)m = -~ dx j 1\ dx i 0 Pi~,X(m) .
L ij
358
Chapter VII. Connections
4.7 Remark. The minus sign in the definition of Ftt is conventional. One could say that it is a consequence of our way to identify the dual of an exterior power [1.7.22], [Y.7.14]. In the ungraded case it would allow us to write s~O = ~ Lij dXi 1\ dx j 0 Ftt with the indices in the same order.
4.8 Proposition. The 2-form s~O is detennined by the i-fonn
s~w
= ra
as
in terms of the functions A~'x and Ftt this equality is given as (4.9) 017l'-1(Ua )
can be reconstructedfrom s~O in the trivialization determined by Sa by
(4.10)
O(m,g)
=
Ad(g-l) 0 (S~O)m
= -~ Ldxj
1\
dx i 0 Ad(g-l)F;',t(m) .
ij
On the intersection Ua n Ub of two local trivializations we have (4.11)
Proof The structure equations of Cartan tell us immediately that we have the equality s~O = d(s~w) + ~[s~w Ii- s~w]. Substituting (3.12) in this equation gives us:
ij
j
ij
ij
= _.! " " dx j 2 ~
1\
dxi,o, I6f
{8 ·Aa,x xt
J
(_1)(€(x i )I€(x j »8Xl·Aa,x ~
+ [Aa,X
Aa,X]}
~'J
.
ij
From this (4.9) follows immediately. To prove the local form of 0, we first recall that the Adjoint representation is indeed a representation of g, i.e., [Ad(g)x, Ad(g)y] = Ad(g) [ x, y]. Using the local expression w(m,g) = Ad(g-l)(s~W)m + 8 MC Ig and [1.6] we compute:
O(m,g)
= dw + ~ [w Ii- w] = d(Ad(g-l)(S~W)m) + ~[Ad(g-l)(s~W)m Ii- Ad(g-l)(s~w)m]
+ ~[Ad(g-l)(S~W)m Ii- 8 MC lg] + ~[8MClg Ii- Ad(g-l)(S~W)m] = Ad(g-l)(S~O)m + (dAd(g-l))
1\ (s~W)m
+
[Ad(g-l)(s~W)m
Ii- 8 MC lg] .
359
§4. The exterior covariant derivative and curvature
The last line is a consequence of the proof of [1.4], from which one can deduce that for even g-valued I-forms a and 'Y we have [a Ii- 'Y] = ['Y Ii- a]. We thus have to show that the last two terms cancel. We will do this by evaluating on tangent vectors. Since each term is a product of a term which only acts on vectors in the M-direction and a term which only acts on vectors in the G-direction, we only have to show that for homogeneous X E TmM and Y E TgG we have
For any Y E TgG there exists y E 9 such that Y =
~.
We then compute:
where the minus sign comes from the fact that L(y) is a derivation and has to be commuted with the even I-form Ad(g-l )(s:w)m. Using [VI.9.12] we compute the second term: L(X,y)(dAd(g-l)) 1\ (S:W)m
= (_l)(€(Y)I€(X»(yg Ad(g-l)) . L(X)(S:W)m = -( _l)(€(Y)lo(X» adR(y) Ad(g-l)L(X)(S:W)m = -( _l)(€(Y)I€(X» [y, Ad(g-l)L(X)(S:W)m] = [Ad(g-l)L(X)(S:W)m, y]. 0
This proves that the two terms cancel and thus we have proven (4.10). To prove (4.11), we note that in the trivialization determined by Sa, the section Sb takes the form sb(m) = (m,1/Jab(m)) (a direct consequence of (3.5)). Using the local expression (4.10) we find immediately
4.12 Proposition. Let 7r : P --> M be a principal fiber bundle with structure group G, and let W be a connection I-form on P and D its curvature 21orm.
c M be open and S E ru(p) a local section. Then S is a horizontal section if and only if s*w = 0, and either of these two conditions implies that DI7r-1(u) = 0 and that the horizontal submanifolds in 7r-l(U) ~ U x G are exactly the horizontal sets U x {g}, g E BG (see [2.1]). (ii) D = Oon P if and only if there exists a trivializing atlas U = {(Ua,1,Ua) I a E I} determined by local sections Sa E rUa (P) such that for all a E I we have s:w = O. (i) Let U
Proof The local section s is horizontal if and only ifTs(Xm) E Hs(m) for all m E U and all Xm E TmM [2.2]. Since Hs(m) = ker ws(m)' this is the case if and only if o = L(((XmIITs)))ws(m) = L(Xm)(S*w)m,which is the case if and only if S*W = O. Using the structure equations of Cartan, this implies directly that s*D = o. And then
360
Chapter VII. Connections
(4.10) tells us that nl 7r -l(U) = O. (3.12) shows that WI 7r -l(U) is given in the trivialization determined by the local section s as 8 MC ' This implies that in this trivialization H(rn,g) is given as TrnM x {O}g C T(rn,g)(U x G). And the integral manifolds of this subbundle are obviously the subsets U x {g} as announced. For (ii), if U exists, then by (i) n = O. We thus assume that n = 0, which means that 11. is integrable [4.2]. Now let p E BP be arbitrary, then through p passes a (unique) leaf (i, L) of the involutive subbundle 11. C T P [V.6.9]. Let £ E L be the unique point such that i( £) = p. By definition of a leaf, Ti : TeL ---; Hp is an isomorphism, and by definition ofaconnection, Trr : Hp ---; T7r(p) is a bijection. Using [V.2.l4] we deduce that there exist neighborhoods V oU and Urn of m = 7r(p) such that 7r 0 i is a diffeomorphism from Vto Urn. We then define s: Urn ---; Pby s = i O ((7r o i)lv)-l. Composing on the left with 7r shows that 7r 0 s = id, and thus s is a local section. Since (i, L) is a leaf, s is horizontal, i.e., s*w = O. Since p E BP is arbitrary, we conclude that every m E BM admits a neighborhood Urn on which there exists a local horizontal section s. Since for all m E M we have m E UBrn by definition of the DeWitt topology, these neighborhoods cover M. The corresponding local sections thus define a trivializing atlas as desired.IOEDI
5. FVF
CONNECTIONS ON ASSOCIATED FIBER BUNDLES
In this section we show that there is a natural way to introduce an FVF connection on an associated fiber bundle starting with one on the original bundle. Defining the structure bundle as the principalfiber bundle with the same structure group and transition functions as the original bundle allows us to show a close relationship between FVF connections on general fiber bundles and those on principal fiber bundles. Using this relationship, we show that a leafofan integrable FVF connection is a covering space ofthe (connected) base space. In particular, a fiber bundle B over a simply connected base space admits an integrable FVF connection ifand only ifB is trivial. We end this section by giving an intrinsic description of an associated fiber bundle to a principal fiber bundle, which allows us to give alternative descriptions of some constructions concerning associated bundles.
5.1 Proposition. Let 7r : B ---; M be a fiber bundle with typical fiber F and structure group G. Let H be another A-Lie group with apseudo effective action on an A-manifold E. Let p : G ---; H be a morphism ofA-Lie groups. Finally, let 11. be an FVF connection on B defined by local g-valuedI -forms r a relative to a trivializing atlas U = {( Ua, 1/Ja) I a E I} for B. Then the local ~-valued I -forms ria = TeP 0 r a define an FVF connection HP on the associatedfiber bundle BP,E ---; M [lV.2.l].
Proof Let us denote by 1/Jba the transition functions associated to the trivializing atlas U. Then, according to its definition [IV.2.l], the transition functions of the associated bundle
§5. FVF connections on associated fiber bundles
are given by 1/J~a
= po 1/Jba' fib
361
According to [2.13] we only have to prove that
== Tepo (Ad(1/Jab(m)-l) ofa(m) + (1/J~be~dlm) = Ad(1/J~b(m)-l) ofla(m) + (1/J~beZ.c)lm .
To prove this, we start with the equality po Ig = Ip(g) 0 p as maps from G to H, where Ig denotes, as in [VI.2.12], the map x r--+ gxg- 1 . Taking the tangent map at the identity and using the definition of the Adjoint representation gives us TeP 0 Ad(g) = Ad(p(g)) 0 TeP, This proves that Tepo Ad(1/Jab(m)-l) ofa(m) = Ad(1/J~b(m)-l) ofla(m). Next we note that po Lg = Lp(g) 0 p as maps from G to H. Taking the tangent map of this identity gives us the equality
Comparing this with the equality 1/J~be~c = Li dXi 0 T L..pab(m)-l T1/Jab aXi 1m given IQEDI in the proof of [2.13], we can conclude thatTepo1/J~be~c = 1/J~beZ.C'
5.2 Definition. In [IV.2.14] we have seen that any fiber bundle can be seen as being an
associated fiber bundle to a principal fiber bundle. We formalize this by defining the structure bundle trs : SB ---> M as being this principal fiber bundle, i.e., SB = Bid,G is the (principal) fiber bundle with typical fiber G and structure group G associated to the fiber bundle 7r : B ---> M with typical fiber F and structure group G by the identity representation id : G ---> G. The underlying idea is that both Band SB are defined by the same transition functions 1/Jab associated to a trivializing atlas U.
5.3 Corollary. SBid,F = B and if H is another A-Lie group with a pseudo effective action on an A-manifold E and if p : G ---> H is a morphism of A-Lie groups, then BP,E = SBP,E, i.e., constructing an associated fiber bundle from the original fiber bundle B or from its structure bundle yields the same result.
5.4 Remark. The structure bundle is a generalization of the frame bundle for vector bundles. If 7r : B ---> M is a vector bundle with typical fiber the A-vector space E, we can define for each m E M the set F m of all bases, also called frames, of the 2t-graded A-module Bm. Since two bases are related by an element of Aut(Bm) [11.2.6], the set Fm is isomorphic to Aut(Bm ). In order to give the disjoint union FB = lImEMFm the structure of a (principal) fiber bundle over M, we proceed as follows. We fix a basis (ei) of E. In a local trivialization (Ua,1/Ja) we define a map {m} x Aut(E) ---> Fm by (m, V) r--+ (1/J~1(Vei))1~lE. This gives us an isomorphism X~l between Ua x Aut(E) and U mEUa F m. To see how this isomorphism depends upon the chosen local trivialization, we choose another one (Ub, 1/Jb), which gives us an isomorphismX;l between Ubx Aut(E) and lImEubFm by (m, W) r--+ (1/J;1(Wei))1~lE. For m E Ua n Ub we get the same
362
Chapter VII. Connections
basis ofB m iffor all i we have '!jI;l(Vei) = ~bl(Wei)' Applying ~b and the definition of the transition function, this happens if and only if for all i we have Wei = ~ba(m) Vei, i.e., if and only if W = ~ba(m) 0 V. This implies that Xb 0 X;l is given by the map (m, V) f---+ (m, ~ba (m) . V). We conclude that :FB is a principal fiber bundle with structure group Aut(E) and the same transition functions as B. In other words, the frame bundle of the given vector bundle B is exactly the structure bundle as defined in [5.2].
5.5 Corollary. Let B ---> M be any fiber bundle and let SB ---> M be the associated structure bundle. Then there exists a canonical bijection between the set of FVF connections on B and the set of FVF/principal connections on SB. In particular, any fiber bundle B ---> M admits an FVF connection.
Proof According to [IV.2.14] and [5.2] the fiber bundle B ---> M and the principal fiber bundle SB ---> M are associated to each other by the identity representation. According to [5.1] an FVF connection on one of these two bundles defined by local i-forms r a determines an FVF connection on the other one by the same set of i-forms. Finally, according to [3.13] any principal fiber bundle admits a principa1JFVF connection. IQEDI
5.6 Corollary. Let P ---> M be a principal fiber bundle with structure group G, let H be an A-Lie group with a pseudo effective action on F, let P : G ---> H be a morphism of A-Lie groups, and let B == pp,F ---> M be the associated fiber bundle (associated to P by the representation p). If H P is a principal/FVF connection on P with associated connection I-form w, and if HB is the associated FVF connection on B given in [5.1], then HB is integrable ifand only ifTeP 0 n is zero, where n is the curvature 210rm on P.
Proof According to [2.18] HB is integrable if and only if the local 2-forms TeP 0
(d r a + ~ [ra f,\ r a])
are zero (use that [TeP(x), TeP(Y) 1 = TeP([ x, y]), i.e., TeP is a morphism of A-Lie algebras). On the other hand, using (4.10), we find
Since s~ n = d r a + ~ [r a f,\
r a1, the result follows.
5.7 Corollary. Let HB be an FVF connection on afiber bundle B ---> M, let H S be the associated FVF/principal connection on the structure bundle SB ---> M [5.5J, and let w be its associated connection I-form. Then the following four statements are equivalent: (i) HB is integrable; (ii) H S is integrable; (ii) the curvature 2-form non SB is zero; (iii) there exists a trivializing atlas U = {(Ua, '!jIa) I-forms r a determining H B are zero.
I a E I} for B such that all local
§5. FVF connections on associated fiber bundles
363
Proof The equivalence between (i) and (ii) has been shown in [4.2], and the equivalence between (i) and (iii) is an immediate consequence of [5.6] because SB is related to B by the identity representation. Since Band S B are determined by the same local I-forms IQEDI r a , the equivalence between (iii) and (iv) is a direct consequence of [4.12].
5.8 Proposition. Let 7r : B ---; M be a fiber bundle with typical fiber F, let 11. be an integrable FVF connection on B and let (i, L) be a leaf of 11.. Then 7r( i( L)) is open and closed in M and 7r 0 i : L ---; 7r( i(L)) is a covering (§ VI. 7).
Proof Suppose that mo E M belongs to the closure of7r(i(L)). According to [5.7], there exists a local trivialization (U, 1jJ) containing mo (part of a trivializing atlas) such that the local I-form ron U is zero. Taking a smaller U if necessary, we may assume that U is connected. By definition of closure, there exists a point m E Un 7r( i(L)). Let f!. E L be such that 7r(i(f!.)) = m, then 7r(i(Bf!.)) = Bm E U because 7r 0 i is smooth and U open. Let f E F be such that 1jJ(i(f!.)) = (m, f). Since the local I-form r on U is zero, the local section S : U ---; B, s(m') = 1jJ-1(m',Bf) is a smooth horizontal section. But then U is connected, sis tangentto 11. and i(Bf!.) = Bi(f!.) E s(U) n i(L) and thus by [V.6.9] s(U) C i(L). Since s is a section, this implies that U C 7r(i(L)). We thus have shown that an arbitrary point mo in the closure of7r(i(L)) admits an open neighborhood U contained in 7r( i( L)), and thus 7r( i( L)) is open and closed. To show that 7r 0 i is a covering, we have to find for all m E 7r( i( L)) an open neighborhood with certain properties [VI.7.1], [VI.7.3]. Since 11. is integrable, there exists (as above) a local trivialization (U,1jJ) of B on which r is zero with U connected, contained in 7r( i(L)) and containing m. Let (7r 0 i)-1 (U) = UaEJ Va be the decomposition of (7r 0 i) -1 (U) in connected components. Then by definition the Va are pairwise disjoint. Since (7roi)-1(U) is open in L, it follows from [111.1.3] that each Va is open in L. Moreover, 7r 0 i is smooth and Te( 7r 0 i) is a bijection for each f!. E L because Ti : TeL ---; 11.i(e) is a bijection by definition ofaleaf, and becauseT7r: l-l i(e) ---; T 7r (i(e» is a bijection by definition of a connection. If we can show that 7r 0 i is a diffeomorphism from each Va to U, we will have shown that 7r 0 i is a covering map. Using the projection 7rF : U x F ---; F, we define the map g = 7rF 0 1jJ 0 i : 7r-1(U) ---; F. Since i is tangent to 11. and since r = 0 on U, i.e., T1jJ(11.) = TU x {o}, it follows that Tg = O. By [V.3.21] g is constant on each Va. Since g is smooth, there thus exist fa E BF such that i(Va) C 1jJ-1(U X {fa}). We now define the local smooth sections Sa(m') = 1jJ-1(m',Ja). Since r = 0 on U, this means (as before) that Sa is tangentto 11.. Since i( f!.) E Sa (U) n i( L) for any f!. E L, it follows from [V.6.9] that there exist smooth maps ga: U ---; Lsuchthat Sa = iog a . Since 7r O Sa = id(U), ga(U) C (7roi)-l(U), and since U is connected, ga (U) is contained in one of its connected components V,a. Moreover, again because 7r 0 Sa = id(U), i(V,a) = 1jJ-1(U x {fa}). Since i is an injective immersion, it follows that i : V,a ---; 'ljJ-1 (U x {fa}) is a diffeomorphism (because Land U have the same dimension). But i(Va) C 'ljJ-l(U X {fa}), and thus by injectivity of i we deduce that a = (3 and that i : Va ---; 'ljJ-l (U x {fa}) is a diffeomorphism. Since the same is true for 7r : 1jJ-1 (U x {fa}) ---; U, we have shown that 7r 0 i : Va ---; U is a IQEDI diffeomorphism.
364
Chapter VII. Connections
5.9 Corollary. Let M be a simply connected A-manifold and let 7r B : B ---> M be a fiber bundle. Then B admits an integrable FVF connection if and only ifit is trivial. Moreover, if B is a vector bundle, then triviality is as a vector bundle. Proof If B admits an integrable FVF connection, then by [5.7] the corresponding FVF connection 11. s on 7r : SB ---> M is integrable. By [5.8] any leaf (i, L) of 11. s is a covering of M. Since M is simply connected, this implies that 7r 0 i : L ---> M is a diffeomorphism. But then i 0 (7r 0 i)-l is a global section of SB and thus SB is trivial. Adding this global trivialization ofSB to a trivializing atlas, the corresponding trivializing atlas of the associated bundle B also contains a global trivialization, and thus B is trivial. In particular, if B is a vector bundle, this global trivialization is compatible with the vector bundle structure and thus B is trivial as a vector bundle. Conversely, if'lb: B ---> M x F is a global trivialization, then (T'lb)-l(TM x {O}) is an integrable FVF connection, determined by the (global) i-form r == O. IQEDI
5.10 Remark. Since any i-dimensional connected A-manifold N is simply connected (it is an interval [VA.l]), it follows from [5.8] that any leaf of an integrable FVF connection on N is diffeomorphic to N via the projection map. A particular consequence is that parallel transport along an even curve always exists for FVF connections (see [2.6]).
5.11 RemarkIDiscussion. The definition of an associated fiber bundle and the construction of the FVF connection on an associated fiber bundle both use local trivializations. Even though this works quite well, one would like to have a more intrinsic/global description. Such a more global description can be given for arbitrary fiber bundles associated to a principal fiber bundle. The idea is as follows. Let trp : P ---> M be a principal fiber bundle with structure group G and let P denote the right action of G on P. Let H be an A-Lie group with a pseudo effective action F on an A-manifold F, and let p : G ---> H be an A-Lie group morphism. These data allow us to define an associated fiber bundle 7r B : B ---> M with structure group Hand typical fiber F. The more intrinsic construction of B starts with the observation that the map W : G x F ---> F given by W(g, f) = F (p(g), f) is a left action of G on F (not necessarily pseudo effective). With this action we define an effective right action x of G on P x Fby
The projection 7r pO 7rl : P x F ---> M (with 7rl : P x F ---> P the projection on the first factor) is constant on G-orbits because trp is constant on G-orbits in P. We thus have an induced map 7ro : (P x F)jG ---> M from the orbit space (P x F)/G to M. now define a map 7rx : P x F ---> B in terms of the localtrivializations 'lba for P and 1/Ja for B relative to the same trivializing atlas by
'!Ie
§5. FVF connections on associated fiber bundles
365
or equivalently, still in local trivializations, by
7rx (m,g,f) = (m, iI!(g,f)) . We claim that this is a well defined map, independent of the chosen local trivialization, and that it is constant on G-orbits. Moreover, the induced map 7rx : (P X F)IG ---; B is a bijection verifying 7r B 0 7rx = 7ro, i.e., we have the following commutative diagram
PxF ,/ 7r x
P x FIG (-(- - - - 4
B.
M The verification of these claims is straightforward and is left to the interested reader. We do not say that 7rx is a diffeomorphism between the orbit space (P x F) /G and B because we have not defined how to induce the structure of an A-manifold on an orbit space (if possible at all; the only instance where we have defined the structure of an A-manifold on an orbit space is for homogeneous A-manifolds [VI.S.9]). Either by using the bijection 7rx or by more direct means, one can give the orbit space (P x F)/G the structure of an A-manifold and then the structure of a fiber bundle over M with structure group G and typical fiber F. Once we have this structure, the bijection 7rx becomes an isomorphism of fiber bundles. It follows that we can take the orbit space (P x F) /G, for which one also finds the notation P x G F, as the definition of the associated fiber bundle B. Once we have the description of the associated bundle B as the orbit space (P x F) /G, we can give a global description of the induced FVF connection. To that end, let H P be an FVF connection on the principal fiber bundle P. Then HB == T7r x (H P x {O}) is the FVF connection on the associated fiber bundle B == (P x F)IG defined in [5.1]. Note that the principal fiber bundle P here plays the role of the bundle B of [5.1] and that the bundle B here is the associated fiber bundle, associated to P by the representation p of [5.1]. Another property of associated bundles that now can be given a more intrinsic description is the following. The construction of an associated bundle starts with a trivializing atlas for the initial bundle, and then the associated bundle has the same trivializing sets. Adding more elements to the original trivializing atlas, it follows that for each local trivialization (U, 1/J) of the initial bundle there exists a corresponding local trivialization (U,1/J') for the associated bundle. If the initial bundle is a principal fiber bundle 7r : P ---; M, we also know that a local trivialization of P is completely determined by a local section s. It follows that for each local section s : U ---; P there is a corresponding local trivialization (U, 1/Js) of B. In terms of the intrinsic description ofB given above, this local trivialization IS gIven as (m, f)
r--+ 7rx
(s(m),f) .
366
6.
Chapter VII. Connections THE COVARIANT DERIVATIVE
In this section we introduce the notion of a covariant derivative on a vector bundle, which is a generalization of the derivative of a vector valued functions to sections of the vector bundle. We show that a covariant derivative is determined by local I -forms r a with values in EndR(E), the space of endomorphisms of the typical fiber E. We prove that these local I -forms behave exactly as the local I -forms defining an FVF connection, thus showing that a covariant derivative on a vector bundle is equivalent to an FVF connection on it.
6.1 Discussion. In [2.1] we discussed the idea of (locally) constant (local) sections of a fiber bundle in terms of Ehresmann connections. For vector bundles there is another approach in terms of a covariant derivative which is based more on the derivative of a function being zero than on the section being horizontal. Let M be an A-manifold, E an A-vector space and I : M -4 E a (smooth) function, corresponding to the section s of the (trivial)bundle 7r : M x E -4 M given by s(m) = (m,I(m)). If X is a vector field on M, we have an action of X on f giving a new function X I : M -4 E [V.1.24]. We can transform this action on functions into an action on sections by defining the section X s as being given by (Xs)(m) = (m, (Xf)(m)). This is a simple transcription of the action of vector fields on E-valued functions to sections. We then can say that s is a constant section if and only if X s is the zero section for all vector fields X on M. If we try to generalize this to arbitrary vector bundles, we encounter a problem: a (global) section s is represented by local functions s,p, but there is no guarantee that the new local functions X s,p glue together to form a new global section X s. This corresponds of course to the fact that being horizontal in one trivialization does not necessarily correspond to being horizontal in another trivialization. The idea of a covariant derivative is to extract the essential features of the above definition of the action of vector fields on sections of a trivial bundle, and to use these to define something meaningful on an arbitrary vector bundle. In [6.18] and [7.2] we will see that the approach to define constant sections via a covariant derivative is equivalent to doing it via an FVF connection.
6.2 Definition. Let 7r : B -4 M be a vector bundle over M with typical fiber the A-vector space E. Recall that r(B) denotes the graded COO (M)-module of smooth sections of the bundle B and that r(T M) denotes the graded COO (M)-module of all vector fields on M. A covariant derivative \7 on the bundle B is a map \7 : r(TM) x r(B) -4 r(B) satisfying the following conditions.
(i) \7 is bi-additive and even; (ii) for all I E COO(M), X E r(TM), s E r(B) we have \7(1 X, s) (iii) for homogeneous I, X, s we have
\7(X, Is)
=
= I\7(X, s);
(Xf)s + (_I)(E(X)IO'(l» I\7(X, s) ,
which can also be written as \7(X, sf) = \7(X, s)I + (_1)(0'(8)10'(1» (X f)s.
§6. The covariant derivative
367
Following custom, we will denote \7 (X, s) also as \7x s ; it is called the covariant derivative
of s in the direction of X.
6.3 Example. Let M be an A-manifold and E an A-vector space. In the identification between E-valued functions on M and sections of the trivial bundle M x E (with its canonical trivialization), the action of vector fields on sections (X, s) f-+ X s as in [6.1] is an example of a covariant derivative on this trivial bundle.
: B ---> M be a vector bundle, \7 a covariant derivative on B, and let V cUe M be two open subsets.
6.4 Proposition. Let 7r
= t\u, then\7(X,s)\u = (\7(X,t))\ufor any X E f(TM). (ii) If X, y E r(TM) are such that Xm = Ym for some m E M, thenfor any s E f(B) we have \7(X, s)(m) = \7(Y, s)(m). (iii) There exists a unique covariant derivative \7u on B \u such that for all s E f( B) we have \7(X, s)lu = \7U(Xlu, slu). (iv) (\7 U) V = \7 v . (i) /fs,t E f(B) aresuchthats\u
7r : Blu == 7r-l(U) ---> U [IV.1.13]. Apart from the difference coming from the presence of a bundle, the proof is a close copy of the proofs of [V.lo4], [V. loS], [IV.S.S]. • (i) Without loss of generality we may assume that sand t are homogeneous of the same parity. For any m E U, let p be a plateau function around m in U. It follows that p(s - t) = O. Using the properties of a covariant derivative, we obtain
Proof Recall first that Blu is the subbundle
0= \7(X, p(s - t)) = (Xp)(s - t)
+ p\7(X, (s -
t))
(p is even).
Since p(m) = 1 this gives us (\7x s)(m) = (\7x t)(m) . • (ii) Let (Xi) be local coordinates on a neighborhood W of m, then there exist functions Xi and yi on W such that Xlw = Li Xiaxi and Ylw = Li yiaxi. If p is a plateau function around m in W, then paxi is a global smooth vector field on M and pX i and pyi are global smoothfunctions. Since p is zero outside W, we have the global equalities p2 X = Li(PXi) . (paxi) and p2 = Li(Pyi) . (paxi). Using the properties of a covariant deri vati ve we find
and a similar equation with X replaced by Y. Evaluating these sections at m, and using = 1 and Xi(m) = yi(m), we find
p(m)
\7(X,s)(m) = L:Xi(m). \7(pax i,s)(m) = L:yi(m). \7(paxi,s)(m) i
= \7(Y, s)(m)
.
368
Chapter VII. Connections
• (iii) As for derivations, the main problem is that not every smooth section above U need be the restriction of a global smooth section. So let t E ru(B), Y E ru(TM) and m E U be arbitrary and let p be a plateau function around m in U. It follows that pt is a well defined global smooth section of B and that pY is a well defined global smooth vector field on M. Moreover, t and (pt) Iu are two local sections above U that coincide in a neighborhood of m and Y and (p Y) Iu are two vector fields on U such that Ym = ((PY)lu)m. Now, if\1u exists, we can combine (i) and (ii) with the defining property of \1u to obtain (6.5)
\1u (Y, t)(m) = \1u (pY)lu(pt)lu(m) = \1 (pY, pt)(m) .
This proves uniqueness of \1u, but we can also use (6.5) to define it. To see that (6.5) indeed produces a well defined \1u, independent of the choice for p, suppose p has the same properties as p. It follows that pt and pt coincide in a neighborhood of m, and (pY)m = (PY)m. And thus by the preceding result \1(pY, pt)(m) = \1(pY, pt)(m), i.e., (6.5) is independent of the choice for p. Since sand p( s Iu) coincide in a neighborhood of m, and Xm = (p(Xlu) )m, it follows that
\1(X,s)(m) = \1(p(Xlu),p(slu))(m) = \1 u (Xlu,slu)(m) , i.e., the covariant derivative defined by (6.5) has the desired property. To prove that \1u is a covariant derivative, we first note that it is obviously bi-additive and even, since the same holds for \1. Property (ii) of a covariant derivative is also a direct consequence of the corresponding property for \1, simply because p is even and thus commutes with any function. To prove property (iii), we only need to add the argument that p(m) = l. • (iv) This is a direct corollary of the uniqueness in (ii). IQEDI
6.6 Definition. The covariant derivative \1u on the subbundle Blu == 7r-l(U) is called the induced covariant derivative. As is customary, we will usually omit the superscript u and use the same symbol \1 to denote the induced covariant derivative on the restriction to an open subset U c M. Worse, in most cases we will not even mention that we use the induced covariant derivative.
6.7 Discussion. If (U, 1jJ) is a local trivialization of B, then the structure of a free graded A-module on each fiber is defined by declaring that the map 1jJ : 7r-l(U) --; U x E is even and linear on each fiber. In other words, for a, b E 7r- 1 (m) and A, J-t E A, if 1jJ(a) = (m, e) and 1jJ(b) = (m, f) then 1jJ(aA + bJ-t) = (m, eA + fJ-t). As in §IV.3 and §IY.S we introduce the local sections fi E ru (B) associated to a basis (ei) of E by the formula
In [IV.lo20] we have shown that there is a bijection between local sections s E ru(B) and functions s1/1 : U --; E. Using (left) coordinates with respect to the basis (ei), each
§6. The covariant derivative
369
(smooth) function 8,p defines ordinary (smooth) functions 8 i : U ---> A by the equality 8,p( m) = Li 8i (m) . ei. Using the free graded A-module structure on each fiber, the local sections fi' and the definition of the function 8.p we thus have the equalities
valid for any local section similar looking formula;:
8 :
U
--->
B. The (local) functions 8 i thus define
8
and 8,p by
(6.8)
An explicit example, though slightly hidden, of the use of the local sections fi is given by the local vector fields ai == aXi associated to local coordinates Xi on an A-manifold [V.1.16]. As a consequence, even though it is not said explicitly, the local I-forms dXi, as well as the local k-forms dxil /\ ... /\ dXik are examples of the use of the local sections fi.
6.9 Discussion. Let (U, 1jJ) be a local trivializing coordinate chart for B with coordinates Xi. Using the local sections fi : U ---> 7r- 1 (U) introduced in [6.7] and the covariant derivative \7 (officially we should say the induced covariant derivative \7 u ), we define homogeneous smooth functions f i j k : U ---> A of parity c(fij k) = c(Xi) + c(ej) + c(ek) by \7(aXi, fk)(m) = f i j k(m) . fj(m) ,
L j
where we used that the local sections fi form a basis of the fiber at each point. We can put these functions together in homogeneous maps fi : U ---> EndR(E) of parity c(f i ) = c(Xi) by fi(m)
= ,L , "f,i'J k
.
k
ej 0 e ,
jk where as usual the e k denote the right dual basis. In terms of the matrix representations given in [11.4.1] this means that we use left coordinates eMR: f i j k = eMR(fi)j k. To show that f i is independent of the choice of the basis (ei) of E, let (ej) be another basis with the associated local sections fj and functions i q p defined by the equality
r
q \7 (axi, fp) (m) = Lq ri p( m) . §.q (m).
By definition the basis (ej) is related to the basis (ei) byej = Li eiaij for some real valued matrix (aij) (real valued because we remain in the equivalence class). It follows that the right dual bases are related by e R = Lk aRk? From the definition offj we deduce the relation fj = Li fi aij and thus, using [6.2-ii] and the fact that the a i j are real constant, we obtain
q
jq
Chapter VII. Connections
370
Comparing coefficients of fj gives us the equality 2:k f i j k . a k p which we compute
= 2: q
j
i\qp . a q
with
and thus f i = f i is independent of the choice of a basis. If we change the coordinates xi to ye, we get new functions I'e j k defined by the j equality 'V(fJyi, fk)(m) = 2: j I'e k(m) . fj(m). Since fJyi = 2:i(fJyi xi)fJxi,it follows from [6.2-iiJ that these functions are related to the functions f i j k (m) by (6.10)
fi by I'e = 2:i( fJyi
Xi)
==
I'e j k
. ej ® ek :
EndR(E) are related to the . fi. We conclude that the even EndR(E)-valued I-form f on U
As a consequence, the maps I'e
2: j k
U
---t
defined by
f
=L
j dxi ®fijk' e ® ek
=L
i dx ®
fi
ijk
=
L
dye ®
I'e
e
is well defined. It follows that this I-form exists on U, even if there does not exist a global coordinate system on U. With respect to the basis (ej ® e k ) of EndR(E), the EndR(E)-valued I-form f can be written as (see (VI.9.3)) (6.11)
f
=
L
fj k ®
k (ej ® e )
jk
with ordinary I-forms fj k 2:i dxi . f i j k. As for the EndR(E)-valued I-form f, it follows from (6.10) that these ordinary I-forms are well-defined, independent of the chosen local coordinates.
6.12 Remark. In the particular case that the vector bundle 7r : B ---t M is the tangent bundle B = TM and that the covariant derivative is derived from a metric, in that case the functions f i j k are called the Christoffel symbols associated to the metric.
6.13 Lemma. Let (U, 'Ij;) be a local trivialization, let S E fu(B) be a local section with its associated local function s..p and let X = 2:i Xi fJxi be a vector field on U. Then the localfunction ('Vxs)..p associated to the local section 'Vxs is given by (6.14)
§6. The covariant derivative
371
Proof Since (6.14) is additive in s, we may assume that s is homogeneous. The local section \7X s can then be written as
\7x s
= L(Xsk)~k + LXi(_I)(€(skl!€(xi))sk\7(Oxi, ~k) k
ik
= (L(Xsj) + LXi(_l)(€(Sk)!€(Xil)skrijk) ~j' j
ijk
Using that ek(s,p) = ek (2: p sPe p ) = (_l)(€(sk)!€(e k)) sk, the function (\7x s),p is given by
(\7x s),p = (L(Xsj)
+ LXi(_l)(€(Sk)!€(Xi))skrijk)
=
X (L sj ej)
+ L Xi r ij k ej (_l)(€(skl!€(e k)) sk ijk
j
= XS,p
ej
ijk
j
+ L Xi rijk ej' ek(s,p) = XS,p + L Xi r i · s,p ijk
= X s,p + t(X)r . s,p
.
6.15 Lemma. Let (Ua , 1/;a) and (Ub, 1/;b) be two local trivializations of B with transition function 1/;ab : Ua n Ub ----t Aut(E) and let r a andr b be the EndR(E)-valued I-forms on Ua and Ub associated to the covariant derivative \7. Then r a and rb are related on Ua n Ub by (6.16)
Proof Throughout this proof we will use left coordinates eMR [11.4.1] for endomorphisms A E EndR(E). The left coordinates Aij == eMR(A)i j can be defined by the formula Aej = 2:i Aij ei. This is compatible with the definition of the endomorphisms r i in terms of the functions r i j k [6.9]. The transition function 1/;ab is defined by the equality (1/;a o 1/;b 1 )(m,J) = (m,1/;ab(m)J). Associated to each trivialization we have local sections f:.i and f:.~. On Ua n Ub they are related by:
f:.~(m) = 1/;b 1 (m, ej) = (1/;;;10 1/;a 0 1/;b 1 )(m, ej) = 1/;;;l(m, 1/;ab(m) ej)
= L 1/;ab(m)ij . 1/;;;l(m, ei) = L 1/;ab(m)ij . ~nm) , i
w here the first equality of the second line follows from the fact that the1/; a is compatible with the A-module structure (actually, the A-module structure on each fiber is defined in this way [IV.3.2]). Using coordinates Xi on Ua n Ub we compute:
L ~
r~jk' 1/;ab P j' f:.~
= L j
r~jk' f:.~
= \7(Oxi,
f:.~)
= L i
i
\7(Ox i ,1/;ab k' f'l)
Chapter VII. Connections
372
Comparing coefficients of f~( m) and using that 1/Jba (m) is the inverse of 1/Jab( m) [IY.1.S] as well as (VI. 1.22) for matrix multiplication in terms of left coordinates, we obtain the relation
r ,bj k (m)
P f, j p + "'(_l)(E(xi)IE(et)+E(ek».f, . •f, j L..- o1/Jab ox i k . •'f/ba L..'f/ab i k . rap , £ 'f/ba p . p pi
= '"
Substituting this in the definition of r~ and using (VI.1.21) several times for matrix multiplication in terms of left coordinates for homogeneous but not necessarily even endomorphisms, we compute:
L dx i ® r~j k . ej ® ek ijk i = L dx i ®
rb=
(_l)(E(x )IE(e t )+E(e k »1/Jab £k
. r~p i . 1/Jba j p . ej ® ek
ijkpi P
" . o1/Jab k . k + 'L..-dx' ® ~. 1/Jba Jp' ej ®e ijkp
=L
dx i ®
i
(_l)(E(x )IE(e t )+E(e k »1/Jab i k
. (1/Jba °
rn i . ej ® ek j
ijki
'"
.
.
= L..-dX'®(1/Jbaor~o1/Jab)Jk·ej®e
k + 'L..-1/Jba " . o1/Jab o (dx'®( ox i ))
0k
i
= 1/Jba ora ° 1/Jab + 1/Jba ° d1/Jab
,
where the last equality follows (among others) from the identity df = 2:i dXi . oXi f, valid for any smooth function. Note that the composition symbol ° in the last line (and some in the line before that) is the alternative notation of the ,{} symbol. A faster proof, though less direct and demanding more explanation to justify all steps, is the following computation. Let S be a section, with local representative functions Sa == s"Ij;a and Sb == s"Ij;b' and let X be a vector field. Then Sa and Sb are related by Sa = 1/JabSb. Since a similar relation holds for the local representatives for \7x s, we have, using (6.14),
1/Jab(\7x S)b
=
(\7x s)a
= X Sa + t(X)ras a =
X(1/JabSb)
+ t(X)ra1/JabSb
+ t(X)ra1/JabSb = 1/Jab(XSb + t(X)(1/Jbaora o1/Jab + 1/JbaOd1/Jab)Sb)
=
(X 1/Jab)Sb + 1/JabX Sb
Comparing this with the expression (\7x sh result.
=
XSb
+ t(X)rb Sb
.
also gives the desired
IQEDI
6.17 Proposition. LetU = {(Ua, 1/Ja)} be a trivializing atlasfor the vector bundle Band suppose that on each Ua we have an even EndR(E)-valued I-form ra. If these r a are
§6. The covariant derivative
373
related to each other by (6.16), then there exists a unique covariant derivative \7 on B such that the fa are determined by \7 as in [6.9 J. Proof To prove existence, we construct \7 as follows. For any s E f(B) and any vector field X E f(T M) we define the section t = \7x s piecewise on each trivializing chart Ua by (cf. (6.14))
ta = XSa
+ t(X)fa sa . = 1/Jba (m )ta (m)
These local functions glue together if and only if they satisfy tb( m) (IV.1.21). We thus compute:
tb
=
X tb + t(X)fbtb
=
X( 1/JbaSa)
+ t(X) (1/Jba
0
fa o1/J;;;}
+ 1/J;;b1 d1/Jab) 1/Jbasa 0
1
=
(X1/Jba)Sa +1/JbaXsa +1/Jba t (X)fa sa +1/J;;b (X1/Jab)1/Jba Sa
=
((X 1/Jba)1/Jab + 1/JbaX 1/Jab) 1/Jba sa + 1/Jba (X Sa
+ t(X)fa Sa)
=
1/Jbata ,
where the last line follows because 1/Jba o1/Jab = id, implying that X (1/Jba 0 1/Jab) = O. This proves that t is indeed a well-defined section and thus that we have defined a map \7 : f(T M) x r(B) ---. f(B). That this map has the properties of a covariant derivative follows from the fact that the local defining formula has these properties. Let us show this for the third condition, leaving the others to the reader. For X, f, and S as required we define sections t = \7x sand U = \7x f s. In order to show that we have U = (X j)s + (-1) (E(X)IE(f)) ft, we show this for all local representative functions, using that the local representative functions respect the A-module structure:
U..p
= =
+ t(X)f fs..p = (Xj)s..p + (_l)(E(X)IE(f)) f(Xs..p + t(X)fs..p) (Xj)s..p + (_l)(E(X)IE(f))ft..p .
X(Js..p)
To prove uniqueness, it suffices to note that the action of \7 is completely determined by IQEDI the local f via (\7x s)..p = Xs..p + t(X)fs (6.14).
6.18 Corollary. Let 1[" B : B ---. M be a vector bundle with typical fiber E and let 1[" s : S B ---. M be the associated structure bundle. Then the following four objects (i) (ii) (iii) (iv)
an FVF connection on B, a covariant derivative on B, a (principal) connection on SB, a connection I-form on SB
are four incarnations of a same concept: all four objects are determined by local even EndR(E)-valued I-forms fa associated to a trivializing atlas and satisfying (2.15)/(6.16). Proof The only thing that has to be proven is that (2.15) and (6.16) are the same. But that is a direct consequence of [VI.2.13] and that 8 MC = g-l odg on Aut( E) [1.5]. IQEDI
374
Chapter VII. Connections
6.19 Definition. Just as FVF connections on a principal fiber bundle have the special name principal connection, so are FVF connections on a vector bundle usually called linear connections. For a linear connection one can also find the name affine connection in the literature. However, since there is nothing really affine in such a connection (the I-forms r a take their values in the linear group EndR(E), not in the affine group of the typical fiber E), the name linear connection should be preferred.
7.
MORE ON COVARIANT DERIVATIVES
In this section we study covariant derivatives on vector bundles in more detail. We start by showing that the link between a covariant derivative and an FVF connection is given by the fact that the covariant derivative measures how far a (local) section is away from being horizontal with respect to the FVF connection. Knowing that an FVF connection defines an FVF connection on a pull-back bundle, we show that the associated covariant derivative on a pull-back bundle again measures, in a sense to be made precise, how far away a lift is from being horizontal. In a similar way we can construct a covariant derivative on an associated vector bundle from one on the original bundle. Of this phenomenon we investigate several examples: the dual bundle, the bundle of homomorphisms and the second tensor power.
7.1 Discussion. In order to get a better understanding of the link between a covariant derivative and a linear connection, we have to delve deeper in the (coordinate) structure of a vector bundle. So let E be an A-vector space, e1, ... ,en a basis, and let (ie) and (e i ) be the associated left and right dual bases. As an A-manifold, E is modeled on E~ and it follows easily from [III. 1.26] (see also [Y.3A]) that for any vector vEE its (left) coordinates are given by the values yi(V) = ((vallie)) and yi(V) = (vdie>, where v = Va + V1 denotes the splitting into even and odd parts. And indeed we have
v=L
(vii ie))
. ei = L (va
+ vd ie)) . ei =
L(yi(V)
+ yi(V)) ei
.
To compute the fundamental vector field AE on E associated to A E EndR(E) (EndR(E) is the A-Lie algebra of Aut(E)), we have to be very careful in the use of coordinates. According to the definition ofa fundamental vector field, we have A{f = -(AIITv>, where denotes the left action of Aut(E) on E, and thus v(g) = g . v. According to [VI. 1.20], on Au t( E) we have to use left coordinates gij = eMR(g)i j [I1A.l] defined by g = I : i j gij ei ® ej . And then the tangent vector A E EndR(E) = Te Aut(E) is given j as A ~ I : i j Ai j 09i j ' when A itself is given as A = I : i j Ai j ei ® e . In terms of the coordinates gij on Aut(E) and yi, yi on E ~ Eg, the coordinates of gv are given by (g is even) j
pq
§7. More on covariant derivatives
375
and similarly j
pq
This gives us for the fundamental vector field the result
Now let 7r : B ----t Mbe a vector bundle with typical fiber E, let Hbe a linear connection on B and let \7 be a covariant derivative on B. Suppose furthermore that H and \7 are determined by the same local EndR(E)-valued I-forms a associated to a trivializing atlas U = {(Ua, 'l/Ja) I a E I}. Then H is given in the trivialization 'l/Ja by (see [2.12])
r
With these preparations we can state a more direct relationship between the linear connection H and the covariant derivative \7.
7.2 Proposition. Let 7r : B ----t M be a vector bundle with typical fiber E, let H be a linear connection on B and let \7 be a covariant derivative on B. Suppose furthermore that Hand \7 are determined by the same local EndR(E)-valued I-forms a associated to a trivializing atlas U = {(Ua, 'l/Ja) I a E I}. Thenfor any section S E r(B) and any vector field X on M we have
r
x
(\7 s)(m)
(7.3)
= 7ra( ((Xm I Ts)) - X~(m) ) .
Roughly in words: \7x s measures how far away ((XIITs)) is from being horizontal. Conversely, if Xm or s is homogeneous, then X~(m) is determined by (\7x s) (m) via (7.3). Proof Let Ua be the trivializing set containing m and let Sa : Ua ----t E be the function determining the section s in the trivialization 'l/Ja, i.e., 'l/Ja(s(m)) = (m, Sa(m)). Let furthermore (yi) be the (global) coordinates on Eo with their associated coordinates (It) on El (see [Y.3.4]). Then X~(m) is given in the trivialization 'l/Ja by -h X(m,Sa(m))
=
'( . 8 Xm - " ~ ((t(Xm)ra((sa)o)ll'e)) 8yi ISa(m)
,
+ {t(Xm)ra((sah)llie> ~Isa(m))
,
376
Chapter VII. Connections
and (( Xm I Ts)) is given in that same trivialization by (see [Y.3.4])
((XmII Ts )) = Xm
. 8iISa(m) 8 . +" L-( ((Xm(sa)oll'e> + ((Xm(sahll'e> Y
i
8 ) . ~Isa(m) uy
It follows that (( Xm I Ts > - X~(m) projects to zero under T7r and thus we can apply [Y.3.7]. This gives us in the trivialization determined by 'lj;a
7r{)
7r{)({XmIITs)) -X~(m)) = 2:({Xm(sa)olli e )) + ((Xm(sahll ie >+ i
((t(Xm)ra((Sa)O) Ilie))
+ ((t(Xm)ra((sah) Ilie> )ei
= Xmsa + t(Xm)ras a = (\7x S)a(m) , where the last equality follows from [6.14]. Since this is the local expression of (\7x s)(m) in the trivialization determined by 'lj;a, we have proven (7.3). If we know (\7x s)(m), we know the coefficients ((Xmsa + t(Xm)ras a Ilie>, and thus we know the coefficients (( t(Xm)ras a I ie whereas for X~(m) we need to know
>,
the values of {t(Xm)ra((Sa)a) Ilie)), for a = 0,1. If S is homogeneous, it is obvious that the former determines the latter. When it is Xm that is homogeneous, we can decompose (( t(Xm)ra Sa I ie)) in its homogeneous components which are given by the formula ((( t(Xm)r aSa I ie)) )E(ei)+E(X",)+a = (( t(Xm)r a( (Sa)a) I ie )), and thus again we obtain the desired result. IQEDI
7.4 Discussion. Let 7r : B ---+ M be a vector bundle with typical fiber E and let \7 be a covariant derivative on B. From [6.18] we know that \7 is equivalent to a linear connection H on B. But for FVF connections we have two ways to construct new ones: on pull-back bundles [2.16] and on associated bundles [5.1]. It follows that we have two ways to create new covariant derivatives out of the given one. We first concentrate on pull-back bundles. We thus consider a smooth map g : N ---+ M and the pull-back bundle g*7r : g* B ---+ N and the associated vector bundle map 9 : g* B ---+ B [IV.2.2]. Given the covariant derivative \7 on B and the associated linear connection H [6.18], we apply [2.16] to obtain an induced linear connection g*H on g* B. And thus by [6.18] we have an induced covariant derivative g*\7 on g* B associated to this linear connection. If U and V are (tri vializing) atlases as in the proof of [2.5] and if \7 is determined by the local EndR(E)-valued I-forms r a, then the induced covariant derivative g*\7 is determined by the local EndR(E)-valued I-forms a = g*rg(a). The natural question that arises is whether there is another way to define g* \7 which does not involve the local I-forms ra. According to the definition of a covariant derivative, if S is a section of g* B and X a vector field on N, then we have to define (g* \7) X S as a section of g* B. The natural idea that comes to mind is the following. Find a vector field Y on M related to X by g, i.e., \:In EN: Yg(n) = {Xn I Tg and find a section t of B such that tog = 9 0 s. And then define (g* \7) X S as
t
>,
g((g*\7)XS)
=
(\7y s)(g(n)) ,
§7. More on covariant derivatives
377
which is well defined because 9 is a bijection between (g*7r) -1 and 7r- 1 (g( n)). For a generic smooth map g there is no hope that such Y and t exist, but perhaps we might circumvent this problem. For Y we indeed can: in view of [6.4-ii] we could use any Y such that, at a fixed point n EN, we have Yg( n) = (( X n II T g)) in order to define (g* \7) X 8 at n EN. How such a Ybehaves elsewhere is of no importance for (\7y 8)(g(n)). However, no such trick is available for the section t. If g is injective, t must be given on g(N) by t(g(n)) = g(8(n)), and thus we have to extend this t outside the image g(N). This might not be easy depending on g(N). But when g is not injective, there is no hope to find a section t for a generic section 8 ofg* B. We thus have to be more subtle in our tentative to give a definition of g*\7 without using the local I-forms
ra.
7.5 Remark. In [6.4-iii] we have shown that there exists an induced covariant derivative on the restriction of the bundle to an open subset. But such a restriction is a particular case of a pull-back bundle [IV.2.3]. The fact that the proof of the existence of this induced covariant derivative is not immediate confirms that extending t outside of g(N) (as described in [7.4]) will not be an easy task, iffeasible at all.
7.6 Definition. Let 7r : B ----+ M be a fiber bundle with typical fiber E and let g : N ----+ M be a smooth map. A map (7 : N ----+ B will be called a lift of g if 7r 0 (7 = g, i.e., if we have a commutative diagram
B
(J"/
17r
---+ 9
M.
N
The set of all smooth lifts (7 : N ----+ B will be denoted as Lift g (B). A particular case of a lift is a section: a section 8 : M ----+ B is a lift of th identity map id : M --+ M, and thus r(B) = Liftid(B). If B is a vector bundle, the set Liftg(B) is in a natural way a graded COO(N)-module: for f E COO(N) and(7,T E Liftg(B) we define f '(7+T E Liftg(B) as
(J. (7 + T)(n)
=
f(n) . (7(n)
+ T(n)
.
7r : B ----+ M be a vector bundle with typical fiber E, let g : N ----+ M be a smooth map, and let g*7r : g* B ----+ N be the pull-back bundle with associated vector bundle map 9 : g* B ----+ B. Then the map J : r(g* B) ----+ Liftg(B) defined by J(8) = go 8 is an isomorphism of graded COO (N)-modules.
7.7 Lemma. Let
Proof Let us first show that J is a morphism of graded COO (N)-modules. For that choose f E COO(N) and 8, t E r(g* B). Then:
J(J.
8
+ t)(n) =
(go (J.
8
+ t))(n)
=
g(J(n)· 8(n)
+ t(n))
= f(n) . g(8(n)) + g(t(n)) = (J. J(8) + J(t))(n) ,
378
Chapter VII. Connections
where the second equality follows from the fact that 9 is a vector bundle morphism [IVA.I9]. To prove that 'J is injective, suppose s, t E f(g* B) are different, i.e., 3n EN: s(n) =1= t(n). Since both belong to thefiber (g*7r)-l(n) and since 9 is an isomorphism when restricted to a fiber, g(s(n)) =1= g(t(n)), i.e., 'J(s) =1= 'J(t). Finally to prove surjectivity, let lJ E Liftg(B) be arbitrary. Using the set theoretic definition of g* B, we define s : N ----+ g* B by s( n) = (n, lJ( n)), which indeed belongs to g* B because 7r(lJ(n)) = g(n) by definition ofa lift. Obviously g(s(n)) = lJ(n) by definition of 9 in terms of the set theoretic definition of g* B. To show that this s is smooth, we look at its representative in a local trivialization. Let V be a trivializing atlas for B and let U be an atlas for N such that for each Ua E U there exists Vg(a) E V such that g(Ua) C Vg(a). According to [IV.2.2] we have a local trivialization of g* B given by 'lj;a : (g*7r)-l(Ua) ----+ Ua x E of g* Band Xg(a) : 7r- 1 (Vg(a) ----+ Vg(a) x E of B. In terms of this trivialization the lift lJlua is given as lJlua(n) = (g(n), S(n)) for some smooth functionS: Ua ----+ E. And then slua is given as slua(n) = (n, S(n)), which is obviously smooth. IQEDI
7.8 Proposition. Let 7r : B ----+ M be a vector bundle with typical fiber E, let g : N ----+ M be a smooth map, and let g*7r : g* B ----+ N be the pull-back bundle. Let furthermore \7 be a covariant derivative on B and let H be the associated linear connection defined by the same local EndR(E)-valued I-forms. then the induced covariant derivative g*\7 on g* B is given by thefollowing procedure. For X a (smooth) vector field on Nand s a smooth section of g* B, denote lJ = 'J(s) = go s E Liftg(B) andT = 'J( (g*\7)(X, s)). Thenfor any n E N we have (7.9)
Roughly in words: T( n) measures how far (( Xn I TlJ)) is from being horizontal. Proof We use notation as in the proof of [7.7]. According to that proof, we have (locally) 'lj;a(s(n)) = (n, sa(n)) for some smooth function Sa : Ua ----+ E and simultaneously Xg(a)(lJ(n)) = (g(n), sa(n)). Iffg(a) is the EndR(E)-valued I-form on Vg(a) defining \7 and H, then g*\7 is determined by g*fg(a) :=:: fa [2.16]. Thus in the trivialization (Vg(a), Xg(a») the local expression for (g*\7)(X, s) is given as
((g*\7)(X, s)t(n) = Xnsa
+ t(Xn)fa . Sa(n) = XSa + t(((Xn IITg> )fg(a) . Sa(n)
.
On the other hand, with notation as in the proof of [7.2], we have
((XnIITlJ))
. 87lsa(n) 8 . ifijiISa(n) 8 ) = ((XnIITg)) + " L-( ((Xn(sa)oll'e> + ((Xn(sahll'e)) ;
y
y
and -------
h
((XnIITg))(g(n),Sa(n))
= ((XnIITg)) -
L,
(
. 8 ~t(((XnIITg)))fa((sa)o)II'e)) 8yilsa(m)
+ ((t«XnIITg»fa((sah)lli e> ~Isa(m») and then the result follows as in the proof of [7.2].
,
§7. More on covariant derivatives
379
7.10 Discussion. [7.7] tells us that the sections of the pull-back bundle are a natural generalization of sections of the original bundle. And then comparing (7.9) with (7.3) shows that the induced covariant derivative g*\1 is the natural generalization of \1 to these generalized sections. Having treated induced covariant derivatives on pull-back bundles, we now turn our attention to covariant derivatives on associated bundles. As before we let 1[" : B ----+ M be a vector bundle with typical fiber E and \1 a covariant derivative on B. But now we consider a representation, i.e., a homomorphism of A-Lie groups, p : Aut(E) ----+ Aut(F) for some A-vector space F. If H is the linear connection on B having the same local EndR(E)valued I-forms a as \1, then we know from [5.1] that there exists a linear connection HP on the associated vector bundle 1["P : BP,F ----+ M, associated to B by the representation p. And thus we have a corresponding covariant derivative \1P on BP,F. Moreover, HP and thus \1P is determined by the local EndR(F)-valued I-forms a = Tep(ra) [5.1]. As for the situation with pull-back bundles, the natural question is whether there exists a way to define \1 P which does not involve the local I-forms ra. The answer is positive for A-vector spaces F that are constructed out of E by natural operations such as F = E* or F = EndR(E). However, the answer depends upon the representation p, so we will give several examples to show the idea.
r
t
7.11 Proposition. IfF = A and p the trivial representation, i.e., Vg E Aut(E) : p(g) = 1, then BP,A is the trivial bundle BP,A = M x A and in the identification r(BP,A) ~ Coo (M), the induced covariant derivative is given by \1I s = X s (see [6.3 J).
If F = E* (the right dual of E) and p the natural representation, i.e., Vg E Aut(E) : p(g) = (g-l )*, then BP,E' = B* is the right-dual bundle and, using the identification r(B*) ~ r(B)* [IV.5.14], the induced covariant derivative \1P is the unique covariant derivative on B* such that for s E r(B) and for homogeneous a E r(B*), X E r(TM) we have
7.12 Proposition.
(7.13)
7.14 Proposition. If F = EndR(E) and p = Ad the adjoint representation, i.e., Vg E Aut(E), VA E EndR(E) : p(g)(A) = Ad(g)(A) = gAg- 1 [VI.2.13], then BAd,EndR(E) = EndR(B) is the bundle of right-linear endomorphisms ofB and, using the identification r(EndR(B)) ~ EndR(r(B)) [IV. 5. 9], the induced covariant derivative \1P is the unique covariant derivative on EndR(B) such that for s E r(B) and for homogeneous cP E r(EndR(B)), X E r(TM) we have
(7.15)
Chapter VII. Connections
380
7.16 Proposition. IfF = E !& E and p the natural representation, i.e., Vg E Aut(E) : p(g) = g ® g, then BP,E0E = B ® B is the tensor product of B with itself and, using the identification f(B ® B) ~ r(B) ® f(B) [IV.5.l7], the induced covariant derivative \1P is the unique covariant derivative on B ® B such that for t E r(B) and for homogeneous
X
E
f(TM), s E f(B) we have
(7.17)
Proofs . • [7.11] If U is any trivializing atlas for B (e.g., the full vector bundle structure), it also is a trivializing atlas for the associated bundle. But for the associated bundle all transition functions are constant 1, and thus we have a natural global trivialization of BP,A as BP,A ~ M x A. Since TeP = 0, it follows that all induced local I-fonns r'a = TeP 0 fa are identically zero. This shows that on any local trivializing chart (U, 'Ij;) we have (\1{ s),p = X s,p, and thus the result holds globally . • [7.12] Ifthe transition functions for B are given by 'lj;ab with respect to some trivializing atlas, then the transition functions of BP,E* are given by p( 'lj;ab) = ('Ij;;;b1 ) *, which are exactly the transition functions of the right-dual bundle B* [IVA.16]. To compute TeP we first note that we have identified Te Aut(E) with EndR(E) and (thus) Te Aut(E*)with EndR(E*). The representation P is the composition of the map Inv : Aut(E) ----+ Aut(E) with taking the dual map: * : EndR(E) ----+ EndL(E*). We see that the natural target space of P is the space of left linear endomorphisms of E*, not the right linear ones. However, since g E Aut(E) is even, it is left and right linear and it is equal to its transpose (1.2.16), and thus we can as well define p as p(g) = 'I'(g-l)*. The map A ~ 'I'A* is an even isomorphism from EndR(E) to EndR(E*). Combining [VI.1.6] with [V.3.3] we find:
TeP: A
~
-'I'A* .
Using (6.14) we now compute the local expression of\1{a: in a trivialization (U, 'Ij;): (\1{a:),p
= Xa:,p + t(X)(Tepo r) a:,p = Xa:,p + (TeP(t(X)r)) (a:,p) = Xa:,p - ('I'([t(X)f]*)) (a:,p) = Xa:,p - (_l)(E(X)!E(a»((a:,pII [t(X)f]*))
,
where the last equality follows from (1.2.16) and the fact that f is even and thus that we have c(t(X)r) = c(X). Since coo(Ui E*) is naturally isomorphic to COO(U; E)*, we can evaluate (\1{a:),p E COO(U; E*) on s,p E coo(Ui E), yielding
(\1{a:),p(s,p) = (Xa:,p)(s,p) - (_l)~(X)!E(a» ((a:,pll [t(X)f]*))(s,p) =
(Xa:,p)(s,p) - (-l)(E(X)!E(a»a:,p«t(X)r)(s,p))
= (Xa:,p)(s,p) -
(-l)(E(X)!E(a»a:,p«\1xs),p - XS,p)
= (Xa:,p)(s,p) + (_l)(E(X)!E(a»a:,p(Xs,p) =
- (-l)(E(X)!E(a»a:,p«\1x s ),p)
X(a:,p(s,p)) - (-l)(E(X)!E(a»a:,p«\1x s),p),
381
§7. More on covariant derivatives
where for the second equality we used [1.2.20]. Looking carefully at the identification f(B)* with f(B*) (see also [IVA.5]) shows that we have a(s)lu = a1f;(s1f;) E COO(U). The above computation thus gives us the equality
Since U is an arbitrary trivializing chart, we have shown (7.13). Uniqueness of \!P follows immediately from the fact that via (7.13) we know the action of \!{ a on any S E f( B) and thus we know \!{a itself. • [7.14] The transition functions for B being 1/Jba, those for BAd,EndR(E) are given by A ~ 1/Jba . A· 1/J"ba1. Comparing this with (IVA.17) shows that these are exactly the transition functions of the bundle EndR(B) of right-linear endomorphisms of B. Using (6.14) and [VI.2.15] we now compute the local expression of \!{¢ in a trivialization (U, 1/J): (\!{¢)1f;
= X¢1f; + t(X)(Te Ad =
X ¢1f;
+ t(X)f
0
0
r) ¢1f;
= X¢1f; + ad R (t(X)f)(¢1f;)
¢1f; - (-1) (E(X)IE(¢)) ¢1f; 0 t(X)f .
Since coo(U; EndR(E)) is naturally isomorphic to EndR(Coo(U; E)), we can evaluate (\!{¢)1f; E coo(U; EndR(E)) on s1f; E COO(U; E), yielding (\!{¢)1f;(S1f;)
= (X¢1f;)(s1f;) + t(X)r(¢1f;(s1f;)) = (X¢1f;)(s1f;) + t(X)r(¢1f;(s1f;)) = X(¢1f;(s1f;)) + t(X)r(¢1f;(s1f;)) -
(-l)(E(X)IE(¢))¢1f;(t(X)f(s1f;)) (-l)(E(X)IE(¢))¢1f;((\!x s )1f; - XS1f;) (-l)(E(X)IE(¢))¢1f;((\!x s )1f;) .
Analyzing the identification f(EndR(B)) ~ EndR(f(B)) shows that we have the equality ¢1f;(s1f;) = (¢(s))1f; and thus the above computation gives us ((\!{¢)(S))1f;
= X(¢(s))1f; + t(X)r(¢(s))1f; - (-l)(E(X)IE(¢))(¢(\!x s ))1f; = (\!x(¢(s)))1f; - (_l)(E(X)IE(¢)) (¢(\!xs))1f; .
Since this is valid for an arbitrary trivializing chart (U, 1/J), we have shown (7.15). Uniqueness of \!P follows as for [7.12] . • [7.16] If the transition functions for B are given by 1/Jab, then the transition functions of BP,E0E are given by p( 1/Jab) = 1/Jba ® 1/Jba, which are exactly the transition functions of the tensor product bundle B ® B [IVA.9]. Since the tensor product is indeed a product (as can be seen by computing explicitly the matrix elements of p(g)), the tangent map is given by
TeP : A in a local trivializing chart (U, (\!{ (s ® t))1f;
1/J)
~
A ® id + id ® A .
we thus obtain
= X(s ® t)1/J + t(X)(Tepo r)(s ® t)1f; = X(s ® t)1f; + (t(X)f ® id + id ® t(X)r)(s ® t)1f; .
382
Chapter VII. Connections
The identification r(B ® B) ~ f(B) ® f(B) is such that we have (s ® t),p = s,p ® t,p, and thus we obtain:
+ (t(X)f ® id + id ® t(X)r)(s,p ® t,p) + (_l)(E(X)IE(s))s,p ®Xt,p + (t(X)fs,p) ® t,p + (_l)(E(X)IE(s)) S,p ® t(X)ft,p = ('Vx S),p ® t,p + (-1) (E(X)IE:(s)) s,p ® ('Vx t),p = (('Vxs) ® t),p + (_l)(E(X)IE:(s))(s ® 'Vxt),p .
('VI (s ® t)),p = X(S,p ® t,p) = (XS,p) ®t,p
Since this is true for an arbitrary trivializing chart, we have shown (7.17). Uniqueness of
'V P follows from the fact that the tensor products s ® t generate f( B ® B) (because of the IQEDI isomorphism between f(B ® B) and r(B) ® f(B)).
7.18 Remark. The representation Ad in [7.14] can also be written as Ad(g) = (g-l) * ® g (see [IVA.16]). Combining the ideas of the proofs of [7.12] and [7.16] yields for Te Ad the formula
Te Ad : A
1--+
id ® A - 'I'A * ® id .
A careful study of the identification given in [1.7.12] for the right linear case shows that for homogeneous elements the action of 'I'A * ® B on an endomorphism cP is given by
('I'A* ®B)(cP)
=
(_l)(E(A)IE(B)+E(¢))B°cP°A.
We thus obtain
+ t( X) (id ® f - 'I'f* ® id) cP,p = X cP,p + t( X) (f ° cP,p - cP,p = X cP,p + t(X)f ° cP,p - (-1) (E(X)IE(¢)) cP,p t( X)f .
('VI cP),p = X cP,p
0
r)
0
This provides an alternative approach to the proof of [7.14].
8.
FORMS WITH VALUES IN A VECTOR BUNDLE
In previous sections we have introduced the exterior derivative of differential forms, the exterior covariant derivative on a principal fiber bundle and the covariant derivative on a vector bundle. In order to show how these objects are related, we have to generalize vector valued differential forms to differential forms with values in a vector bundle. This generalization is presented in this section.
383
§8. Forms with values in a vector bundle
8.1 Definition. Let p : B ----+ M be a vector bundle with typical fiber E. A k-form on M with values in B is a section of the bundle (I\k *T M) 129 B, just as an ordinary k-form is a section of the bundle (I\k *T M). In analogy with ordinary k-forms and vector valued k-forms, the set of all smooth B-valued k-forms will be denoted as nk (M; B), i.e.,
As for ordinary differential forms, we define n(M; B) as the direct sum over all k : co
= E9nk(M;B).
n(M;B)
k=O
Let (ei) be a basis for E and let (U, 'Ij;) be a local trivializing chart for B (i.e., a chart for M with coordinates Xi and trivializing for B). According to [6.7] dXil 1\ ... 1\ dxik 129 fj
forms a basis for a local section of the vector bundle (I\k *T M) 129 B, i.e., a (smooth) k-form w with values in B has the local expression
wlu =
(8.2)
j
w
wt, . .
with j = 2:i 1 , ... ,ik Wt, ... ,i k dX i1 1\ ... 1\ dXik and where the ,i k are (smooth) functions on U. The parity of the B-valued k-form w is determined in the usual way: c( w) = c(w j )+c( ej), where c( wj ) is the standard parity of the local k-form wj . Obviously the local k- forms w j on U are defined independently of the local coordinates xi. However, they need not glue together to form global k-forms on M. More precisely, let (Ua , 'lj;a) and (Ub, 'lj;b) be two local trivializations with associated local sections f't and f~ and associated local k-forms w~ and w{ The transition function 'lj;ba : Ua nUb ----+ Aut(E) for the bundle B defines (left) matrix elements 'lj;ba (m) i j == •M R ( 'lj;ba (m)) i j E A by the formula 'lj;ba (m )ei = 2: j 'lj;ba (m )i j ej [11.4.1], [VI. 1.20]. Combining the definition of the transition function with that of the local sections f't and f; gives us the equality
f't(m) =
L 'lj;ba(m)i j f~(m) . j
Since 2:i w~ 129 f't and 2: j also have the equality
wt 129 f; both represent the same section on Ua nUb, we must
(8.3)
This implies that if we want the local k-forms w~ to glue together to form a global k-form j w , then the transition functions 'lj;ba must be the identity. Said differently, they glue together to form a global k-form if the bundle B is trivial. In the general case the (local) k-forms wi depend upon the local trivialization (U, 'Ij;), but they also depend upon the choice for the basis (ei) for E. If e1, ... , en is another
384
Chapter VII. Connections
basis for E, it is related to the former basis by immediately follows that fi
= 2: j
j
Ai fj
ei =
and thus
2:j
wlu
j Ai ej
for (real) constants
= 2:i wi ® fi = 2: j
w
j
Aij.1t
® fj with
(8.4)
This formula looks exactly as (S.3) but its interpretation is completely different: (S.4) concerns a single trivialization (U, 'Ij;) and two bases, whereas (S.3) concerns two trivializations and a single basis. Moreover, the matrix elements 'lj;ba (m )i j in (S.3) depend upon the choice of the basis for E, whereas the matrix elements Ai j in (S.4) do not depend upon the trivialization. A way to hide the dependence of the local k-forms w j on the choice of a basis is to form the local E-valued k-form w'Ij; = 2: j w j ® ej, which is independent of such a choice. We thus have the similar looking formulae (8.5)
wlu
=L
wj
®fj
j
where the first formula is a description of the B-valued k-form restricted to a local trivializing chart (U, 'Ij;) with the trivialization hidden in the local sections fj' and where the second is a local E-valued k-form which depends upon the trivialization 7/J, as indicated by the subscript 'Ij;. This should be compared with (6.S), which can be seen as the special case of a O-form. Using the local E-valued k-forms w'lj; we can give another interpretation of (S.3). If (Ua , 'lj;a) and (Ub , 'lj;b) are two local trivializations, we can form the local E- valued k-forms Wa == w'Ij;a = 2:i w~ ® ei and Wb == W'Ij;b = 2: j wt ® ej. We now interpret 'lj;ba as an even EndR(E)-valued O-form on the intersection Ua n Ub and we want to compute the wedge-dot product 'lj;ba t.-. Wa [1.3-ii]. Since the matrix elements 'lj;ba (m )i j are defined in such a way that we have 'lj;ba = 2:ij 'lj;bai j ej ® e i , we compute according to (1.2): 'lj;ba
t.-. Wa =
L
7/Jbai
j
1\ ([E(ej )+E(ei) (w~) ® (ej ® e i )( ek)
ijk
=
L
'lj;bai
j
.
([E(ej)+E(ei)(w~) ® ej
L w~ ®
W~
'lj;bai
j
r??)ej
0
0k =
oi = L
7/Jba ei .
To obtain the second equality we used that 'lj;bai j is a function (O-form) and thus the wedge product is just the ordinary product; to obtain the third equality we used that 'lj;ba is even, and thus that c('Ij;bai j ) = c(ej) + c(ei). Using the alternative notation for 7/Jba t.-. Wa [1.3] we thus can rewrite (S.3) as (8.6) We conclude that, given a trivializing atlas U = {(Ua , 'lj;a)}, the B-valued k-form W is represented by a system of local E-valued k-forms Wa on Ua satisfying the compatibility
385
§8. Forms with values in a vector bundle
condition (8.6). It is not hard to show that conversely a system oflocal E-valued k-forms Wa on Ua satisfying the compatibility condition (8.6) defines a (global) B-valued k-form w. This description of B-valued k-forms should be compared with the description of sections of (vector) bundles given in [IV .1.20].
{(Ua,1/;a) I a E I} for the bundle Band a basis e1, ... , enfor the typical fiber E, any B-valued k-form W is represented by local k-forms w~, 1 :::; i :::; non Ua which are related on overlaps by (8.3) and which change all j at the same time by (8.4) in case we change the basis (ei) to eej) given byei = j Ai ej.
8.7 Corollary. Given a trivializing atlas U
=
2:
8.8 Discussion. In [VI.9.l] we have defined k-forms with values in an A-vector space. This can be seen as a particular particular case of k-forms with values in a vector bundle as follows. We first construct the trivial bundle B = M x E with its canonical global trivialization 1/; : B ----t M x E. In this (global) trivialization a B-valued k-form is given by n global k-forms Wi as w = 2:i wi~ §.i' From this we can obtain the global E-valued k-form 2:i wi ® ei. Since they depend in the same way on the chosen basis for E, we get a bijection between B-valued k-forms and E-valued k-forms. In this sense an E-valued k-form is a particular case of a vector bundle valued k-form. However, whereas we are allowed to change the (global) trivialization when we consider B-valued k-forms, we are not allowed to change the trivialization for E-valued k-forms. E-valued k-forms correspond to B-valued k-forms in a given fixed global trivialization. Another way to interpret this particularity is to play with the various identifications as in [V.7.l]. This gives us thatr((N *TM)®(MxE)) is isomorphic to Homtk(*f(TM)k; f((MxE))). Using a (global, fixed) trivialization 1/; [IV. 1.20] we obtain an isomorphism between f(M x E) and COO(M; E) given by S ~ s'Ij;. Since the latter space represents the space of E-valued k-forms, we obtain an identification between k-forms with values in the trivial bundle M x E and E-valued k-forms. However, this identification obviously depends upon the chosen global trivialization 1/;. Hence the statement that E-valued k-forms are a particular particular case of k-forms with values in a vector bundle.
8.9 Discussion. By definition, a B-valued O-form is a section of the (vector) bundle (N *TM) ® B = (M x A) ® B, where N *TM ~ M x A comes with a canonically defined trivialization. Using the natural isomorphism A ® E ~ E [1.4.10] we obtain a natural isomorphism (M x A) ® B ~ B [IV.4.23]. It follows that a B-valued O-form can be identified with a section of B. In terms of local representations this amounts to omitting the tensor product symbol: if the B-valued O-formw is locally given by wlu = 2: j w j ®§'j, then the wj are O-forms, i.e., ordinary functions. It is identified with the (global) section of B whose local expression is given by 2: j w j . §'j' Now suppose that B is the trivial bundle B = M x A with its canonical trivialization. Using the same techniques as above, we obtain an identification between I\k *T M and (N *TM) ® (M x A). It follows that k-forms with values in the trivial bundle M x A are
386
Chapter VII. Connections
just ordinary k-forms, i.e., vector bundle valued k-forms are a generalization of ordinary k-forms.
8.10 More definitions. Playing around with the various identifications as in [Y.7.1], we *T M) ® B) is isomorphic to Hornik (*r(T M)k; f(B)) [8.8]. already argued that f( In the same way we can show that it is isomorphic to (N *r(TM)) ® r(B). The first isomorphism allows us to identify B-valued k-forms as skew-symmetric k-linear maps (over COO(M)) from vector fields on M (sections of TM) to sections of B. The second isomorphism allows us to define the contraction of a B-valued k-form w with a vector field X to yield a B-valued (k - I)-form t(X)w. There are at least two ways to define this contraction. The first is to apply [1.6.16], which does not yield a directly applicable formula. The second is to note that the contraction t( X)w is such that, as a (k - 1 )-linear skew-symmetric map, this B-valued (k - I)-form is given by
(N
However, the most useful formula to define this contraction is by using the local expression (8.2) and to note that the contraction is given by the local formula
(t(X)w)lu
=
L t(X)w1 ® f.j . j
Using the compatibility condition (8.3), it is not hard to show that one can use the above local expression to define the global contraction t( X)w, without any reference to the various identifications described above. Moreover, using the local E-valued k-forms w..p (8.5), it is immediate that we have t(X)(w..p) = (t(X)w)..p.
8.11 Remark. In [VI.9.1] we have defined the pull-back of a k-form with values in an A-vector space. This can be seen as a special case of a more general notion of a pull-back of vector bundle valued k-forms involving the pull-back bundle. Since we will never need this more general notion, we do not go into the details of the more general notion.
8.12 Definition. In [1.1] we have defined the -wedge product of vector valued differential forms. We now want to extend this definition to the case of arbitrary vector bundles. So let E, F, and G be three A-vector spaces with homogeneous bases (ei), (Ij), and (gk) respectively, and let : Ex F ---+ G be an even smooth bilinear map. Suppose furthermore that B ---+ M, C ---+ M, and D ---+ M are vector bundles over M with typical fibers E, F, and G respectively. If a is a B-valued p-form and f3 a C-valued q-form, the -wedge product a 1\4> f3 should be a D-valuedp+ q-form, all over M. The idea of the construction is as follows. In any common trivializing chart U c M for the bundles B, C, and D, the forms a and f3 determine ordinary differential forms ai and f3j on U according to
f3lu = Lf3 j
j
® f.j
.
§8. Forms with values in a vector bundle
387
We now introduce matrix elements for by ( ei, ij) = 2:k 7j gk. With these matrix elements, the D-valued p + q-form a !\ (3 should be defined on the trivializing chart U by
(a!\ (3)lu =
(8.13)
La
i
!\ 7j flk .
i,j,k
The main problem is that these local expressions need not glue together to form a globally well defined D-valued p + q-form.
8.14 Lemma. Let U = {Ua I a E I} be ajoint trivializing atlasfor the three bundles B, C, and D, and let 'l/;ba be the transition functions for B, Xba those for C, and cPba those for D. If the map intertwines these transition functions in the sense that for all mE Ua n Ub and all e E E, i E F we have
then the -wedge product a !\ (3 is globally well defined by (8.13). Proof On two trivializing charts Ua and Ub we have, as in [S.l], local p-forms a~ and a~ and local q-forms (3~ and (3{ They satisfy the relations (S.3) a~lm = 2:£ a~lm 'l/;ba(m)£i and (3tlm = 2: s (3~lm Xba(m)J With these we compute:
L
ai,lm!\ 7j gk
i,j,k
=
L
a;lm 'l/;ba(m)£i !\ (ei, ij)
i,j,i,s
=
L
a;lm!\ ('l/;ba (m)£i ei, Xba(m)sj ij)
i,j,i,s
= L a;lm!\ ('l/;ba(m)e£, Xba(m)is) e,s =
L a;lm !\ ((3~lm) ® cPba(m)(e£, is) e,s
= L a;lm!\ is
cPba(m)gr
i,s,r
=
L
a;lm!\
L
j+
+
(-1)
I: iba(m).g) = p(g-l) 0 p('Ij;ba(m)-l) 0
is horizontal and
(2: a~lm ® ej) j
=
p(g-l) 0 p( 'lj;ba(m)-l) 0
(2: a~ 1m p('lj;ba( m) )i j ® ej)
= p(g-l) p( 'lj;ba(m)-l) (2: a~ 1m ® p( 'lj;ba(m)) (ei) ) 0
0
i
= p(g-l)o (2:a~lm ®ei) = aal(m,g)' i
Since ('Ij;b 0 'Ij;;;l )(m, g) = (m, 'lj;ba(m)· g), we thus have shown that ('Ij;b 0 'Ij;;;l )*ab = aa. In other words, the E-valued k-forms 'Ij;~aa and W;ab coincide on overlaps Ua nUb. It follows that there exists a well defined global E-valued k-form a = 'J(a) on P such that al 7r -l(Ua) = 'Ij;~aa. Moreover, this E-valued k-form is horizontal and of type p, i.e.,
=
a E n~or)P; E). A priori the construction of a depends upon the trivializing atlas U. A way to make it manifestly independent of such a choice would be to take for U the whole fiber bundle structure. However, since any chart added to U is compatible with the elements ofU, this will not change a. And thus, even though the construction of a from depends upon an atlas, the result does not.
'J(a)
a
Chapter VII. Connections
398
10.8 Proposition. The map') : nk(M; pp,E) isomorphism of graded COO (M)-modules.
----t
n~or,p(P; E), 0: ~ ')(0:)
== a
is an
Proof Replacing 0: E nk(M; Pp,E) by a sum of two elements, or multiplying it by a function on M will replace a by the corresponding sum or multiply the result by the given function. Hence the map') is a morphism of graded COO(M)-modules. An E-valued k-form a on P is zero if and only if all local k-forms aa = ('ljI;1 )*a are zero, which is the case if and only if all o:~ are zero (l0.7). But this happens if and only if 0: is zero, showing that') is injective. To prove surjectivity, let a be an E-valued k-form on P which is horizontal and of type p. Since it is horizontal, there exist E-valued functions f~ ... ik on Ua x G such that aa == (( 'ljI;1 )*a) I(m,g) = 2:i1... ik dX il 1\ ... 1\ dX ik ® f~ ... ik (m, g) [10.2]. Since a is of type p, we must have fi~ ... ik(m,g) = p(g-1)ft .. ik(m, e). Decomposing fi~ ... ik(m,g) with respect to a basis (ej) of E as f~ ... ik (m, g) = 2: j f~':'.ik (m, g) . ej, we can define the local k-forms o:~ on Ua by
O:~lm =
L
dxil
1\ ... 1\
dX ik . f~,.J.ik (m, e) .
i1···ik
Since p(g) is even, we thus have by constructionaal(m,g) = p(g-1) 0 2: j O:~lm' Comparing this with the construction of'), we see that if the local k-forms 2: j o:~ ® f.j glue together to form a global k-form 0:, then the a we started with is the one constructed from 0:, i.e., ') is surjective. To prove that the local k-forms 2: j o:~ ® f.j glue together, we note that, by construction, we have ('ljIb 0 7jJ;1 )*ab = aa. Since ('ljIb 0 'ljI;1)( m, g) = (m, 'ljIba(m) . g), this implies that the local functions f~ '" ik must satisfy the compatibility condition
ft .. ik (m, 'ljIba(m) . g)
= f::;' ... ik (m, g) .
Together with the type p condition, this implies that we have
and thus o:~ 1m = 2:i o:~ 1m p( 'ljIba (m)) i j . By [8.7] this means that the O:a glue together to IQEDI form a global pp,E-valued k form on M.
10.9 Remark. As before we use the abbreviation B = pp,E. For O-forms the identification') : nO(M; B) ----t n~or,p(P; E), which is an identification between r(B) and COO(P; E), can be stated in a more intrinsic way using the description [5.11] of the associated bundle. Let f : P ----t E be a function of type p (it is automatically horizontal), i.e., f (pg) = p(g-1) f (p), and let s = ,)-1 (J) E f( B) be the associated section of B. Then we have or all pEP the equality (10.10)
7rx
(p, f(p))
= S(7r(p)) .
§ 10. Principal fiber bundles versus vector bundles
399
This formula can also be used to construct the correspondence f f-+ S as follows. If the functionfis horizontal and oftypep, then (pg, f(pg)) = (pg, p(g-l)f(p)) = (p, f(p))·g, i.e., the map p ~ 7rx (p, f (p)) is constant on the fibers 7r- 1 (m). We thus can define the section s: M ----t B by s(m) = 7r x (P,f(p)) for an arbitrary p E 7r-l(m). Conversely, if s : M ----t B is a section, we claim that there is a unique function f : P ----t E of type p satisfying (10.10). Uniqueness of f follows from the fact that if 7rx (p, e) = 7rx (p, e') then e = e' (the pre-images of 7rx are G-orbits in P x E). Existence follows from the surjectivity of7rx . Since 7rx (p,f(p)) = 7rx (pg,p(g-l)f(p)) it follows that f satisfies f(pg) = p(g-l )f(p), i.e., f is of type p.
10.11 Examples. • Let 7r : P ----t M be a principal fiber bundle with structure group G. Denoting by Conn(P) the set of all connection I-forms on P, we know it is not empty [10.5]. The difference w - Wo of two elements W o , W E Conn(P) being horizontal and of type Ad [10.5], we can define a map Conn(P) ----t nJ,.or Ad (P; g) by W ~ W - Wo for a given fixed Wo E Conn(P). This map is obviously injec'tive; it is also surjective because if 0" is a horizontal g-valued I-form of type Ad, then w = Wo + 0" satisfies the conditions of a connection I-form. We conclude that the space of all connection I-forms (the space of all principal connections) on P is an affine space modeled on the graded CCXJ(M)-module nLr Ad(P; g). Since this space is isomorphic to nl(M; pAd,g), the difference w - Wo of two ~onnection I-forms can be seen as a pAd,g-valued I-form. We could have shown this result directly by combining [2.13] and [3.11]: w - Wo is determined by local g-valued I-forms a - r~ which, according to (2.15), are related on overlaps by
r
Here we have deduced this result from the more general identification given in [10.8]. The presence of the term 1/J~b e Me in (2.15), which contains derivatives of the transition function 1/Jab prevents the individual connection I-forms to be interpreted as sections of some associated bundle. Since the space ofFVF connections on a fiber bundle only depends upon the structure group [5.5], we conclude that the set of all FVF connections on a fiber bundle 7r : B ----t M is an affine space modeled on a graded CCXJ(M)-module of sections of a vector bundle (determined by B) . • Coming back to a principal fiber bundle 7r : P ----t M with structure group G and a connection I-form w, we have seen that the curvature 2-form n = Dw is horizontal and of type Ad [10.5]. We thus can see n as a pAd,g-valued 2-form. And, in a similar way as for the difference of two connection I-forms, this result could have been deduced directly from (4.11) . • Twisting our point of view, we now start with a vector bundle 7r : B ----t M with typical fiber E. Combining [5.3] with the first part of [7.14] then shows that we have EndR(B) = SBAd,EndR(E). Now suppose that \7 is a covariant derivative on B. Then it can also be seen as an FVF connection on B or as a connection I-form w on SB [6.18]. When we see it as a connection I-form, we have the associated curvature 2-form
400
Chapter VII. Connections
n, which in turn corresponds to an SBAd,EndR(E)_valued 2-form, i.e., an EndR(B)valued 2-form via the identification 'J: n2(M; EndR(B)) --t n~or,Ad(SB; E). If we now compare [4.8] with this construction (in particular with (10.7)) and with the local description of the curvature tensor R of the covariant derivative \7 as given in [9.10], we see that 'J(R) = n, i.e., the incarnation of the curvature 2-form n on the structure bundle SB as an EndR(B)-valued 2-form is exactly the curvature tensor R of the covariant derivative. 10.12 Proposition. Let 1[" P : P --t M be a principal fiber bundle with structure group G, let P : G --t Aut(E) be a representation and let B == pp,E be the vector bundle associated to P by the representation p. Let furthermore w be a connection I-form on P and let a E n~or,p(P; E) be a horizontal E-valued k-form on P of type p. Then the exterior covariant derivative Da is also horizontal and oftype p and is given by the formula (1O.l3)
Da
= da + (Tepow) f.o.a.
Moreover, if'Jk : nk(M; B) --t n~or,p(P; E) is the identification between B-valued k-forms on M and horizontal E-valued k-forms on P of type p and if \7 is the covariant derivative on B associated to the connection w, then we have for (J E nk(M; B) the equality (10.14)
i.e., the identification 'J intertwines the covariant derivative \7 and the exterior covariant derivative D. Proof To prove (10.13) we proceed as in the proof of [4.3]: we show that we have equality when evaluating on k + 1 smooth homogeneous vector fields X o , ... , X k that are either horizontal or vertical, and using only vertical vector fields of the form yP, Y E Bg. If all Xi are horizontal, contraction ofthe left gives by definition ofthe exterior covariant derivative the value t(Xo, ... , Xk)da. Contraction of the -wedge product (TeP 0 w) f.o. a with X o, ... , X k yields a sum of terms, each involving the contraction ofw with some Xi, which is zero. We conclude that the equality holds when evaluating on k + 1 horizontal vector fields. Whenever one of the Xi is vertical, the left hand side of (10.13) is zero by definition. If at least two are vertical, contraction of (TeP 0 w) f.o. a with X o , ... , X k yields a sum of terms, each involving the contraction of a with k vector fields among X o, ... , X k . Since among these k at least one is vertical, the result is zero. Still assuming that at least two among the Xi are vertical, contraction of da with X o, ... , X k gives a sum of two terms, the first being a single sum, the second being a double sum (V.7.4). Each summand in the first term contains the contraction of a with k elements among X o, ... , X k and thus is zero. Each summand in the second term contains the contraction of a with k - 1 elements among X o, ... , X k as well as contraction with the commutator of the remaining two. If
§ 10. Principal fiber bundles versus vector bundles
401
the remaining two are not both vertical, at least one among the k -1 is vertical and thus the full contraction is zero. If the remaining two are both vertical, i.e., of the form x P and yP for x, y E Bg, then their commutator [xP, yPj = [x, yjP is also vertical, and thus again the full contraction is zero. We conclude that if at least two among the Xi are vertical, then contraction of the right hand side with Xo, . .. , X k yields zero, and thus in that case too we have equality. Remains the case with only one vertical vector field. By skew-symmetry we may assume that it is the first: Xo = yP for some y E Bg and Xi is horizontal for i :::: 1. If X is any horizontal vector field and (3 any (suitable) R-form, then
t(X)((Tepow) ~ (3) = (t(X)(Tepo w)) ~ (3 - (Tepow) ~ t(X)(3 = -(Tepow) ~ t(X)(3 , where the minus sign after the first equality comes from the fact that TeP 0 w is a I-form, and where the second equality comes from the fact that X is horizontal. Since we have t(yP , Xl, ... , X k ) = t(yp) 0 t(X I ) 0 • . • 0 t(X k ), we obtain the equality t(yP,XI"",Xk)((Tepow)~o:)
= (_l)k (Tept(yP)w) ·t(Xl, ... ,Xk)o: = (_l)k TeP(Y) . t(X l , .. . , Xk)o: ,
where the right hand side should be interpreted as the action of TeP(Y) E EndR(E) on the E-valued function t(Xl"'" Xk)o:. As said before, contraction of do: with the vector fields yP, Xl, ... , X k yields a sum of two terms, the first a single summation and the second a double summation. In the single summation, only the term in which 0: is not contracted with yP remains (all other being zero) and gives yP (t( Xl, ... , X k) 0:). Whenever, in the double summation, the vector field yP does not appear in the commutator, contraction with 0: yields zero, and thus only a single summation remains and we find:
(_l)k. t(yP, Xl'"'' Xk)do: = yP (t(X l , ... , Xk)O:) )+ I: (E(Xp)IE(Xj)) + (-1) O,12 \7,366 0, 12 -1-, 61
UJ,3
[ _ f;.. -1,337
[_,_]'28,210 ,(}, 337, 389 t.-., 337, 389 A,40 B, 57, 58, 82, 105, 126, 129 BM,262 CD, 94
C k ,94 Coo, 97 C oo C)o:,97 r, 148
r- 1 ,
l75 (C in]), 1m cfj,272 ([,62 58 c,3,4 e,287 G, 88,106,137 Hk,263 In, 58, 119 L g ,268 £,253 eMs, 68 NIs ,68
of,
Ms,68 m,266 mL,2 mR,2 7r a,221,222
9'\,22 R g ,268 8MC,339
'I', 13 V, 200 Symbols attached to others x M : fundamental vector field, 299 a + ib: complex conjugation, 188 x: left-invariant vector field, 271 t: transpose, 71
Index
2l-graded A-module, 6 commutative ring, 3 commutativity, 3 ring, 3 2l-grading, 3 A-Lie algebra, 270 associated to an A-Lie group, 271 A-Lie group, 142,266 A-Lie subgroup, 292 normal, 296 proper, 300 A-manifold, 128 A-module, 2 A-vector space, 85 action effective, 143 left/right, 142, 266 additive, k-, 2 adjoint representation, 31, 32, 284, 325 Adjoint representation, 283,284 affine connection, 374 algebra 2l-graded commutative, 28 2l-graded Lie, 28 of parity a, 31 associative, 28 of parity a, 32 associated bundle, 151, 364 atlas, 124 adapted to a subbundle, 160
trivializing, 145 automorphism of a module, 11 basis, 46 dual, 67 ordered, 59 orthonormal, 192 Batchelor's theorem, 196 Berezinian, 41, 55, 78 Bianchi identities, 355, 357 bilinear map, 8 bimodule,2 body of A, 57 of a (proto) A-manifold, 126 of a linear map, 82 of a matrix, 58 of a module, 82 of a smooth function, 105, 129 border point, 228 bracket of a Lie algebra, 28 bundle associated, 151, 364 fiber, 145 principal, 155, 301 frame, 361 of morphisms, 170 pull-back, 151 structure, 361 trivial, 147 vector, 156 411
412
Cartan's structure equations, 355, 357-359 center of a Lie algebra, 325 central extension of a Lie algebra, 325 of an A-Lie group, 325 chain rule, 116 chart of an A-manifold, 124 Christoffel symbols, 370 closed differential form, 262 cocycle of a Lie algebra, 325, 332 of an A-Lie group, 325 cohomology de Rham, 263, 327 of a Lie algebra, 325, 332 of an A-Lie group, 325, 332 commutativity, 2l-graded, 3 commuting flows, 236 compatible chart of a fiber bundle, 144 compatible chart of an A-manifold, 124 complex conjugation, 188 connection, 342, 345, 351 FVF, 345 I-form, 351 affine, 374 Ehresmann, 342 flat, 342 integrable, 342 linear, 374 principal, 351 contraction elementary, 74, 274 operator, 12 contraction, elementary, 74, 75 convergence, 286 coordinate (even/odd), 104 cotangent bundle, 248 counter example, see example, counter covariant derivative, 366 exterior, 354 induced, 368 covering, 311-312 Coxeter group, 23 curvature 2-form, 355 curvature tensor, 394
Index
decomposition, 2 degree-a part of a vector bundle, 167 derivation (right/left), 29 determinant, 41, 61, 77, 80, 111 graded, 41, 55, 78-80, Ill, 118,285 DeWitt topology, 93 de Rham cohomology, 263, 327 diffeomorphic, 128 diffeomorphism, 128 differential form, 248 with values in a vector bundle, 383 with values in an A-vector space, 323 dimension differential, 207 evenlgradedlodd/total, 61 of a (proto) A-manifold, 124 direct product of bundles, 153 of A-manifolds, 133 direct sum, 7, 14 of bundles, 164 dual basis, 67 bundle, 170 of a module, 9 ofa morphism, 14 effective action, 143 Ehresmann connection, 342 elementary contraction, 74, 75, 274 embedding, 214 endomorphism, 9 enlarging the structure group, 154 equivalence of bases, 83 of A-Lie subgroups, 293 equivariant map, 305 Euler vector field, 261 evaluation operator, 12 even, 3,4, 56 exact differential form, 262 example, 3, 4, 21, 28, 56, 57, 59, 71, 74,97,110,118,125,132,133, 137, 143, 144, 152, 231, 280,
Index
283, 287, 297, 332, 339, 367, 396,399 counter, 7, 10,60,63,65,66,76,81, 84,87, 103, 123, 130, 148, 163, 191, 192, 216, 221, 224, 244, 300,344 exponential map, 278 of matrices, 287 exterior algebra, 27 covariant derivative, 354 derivative, 249 power, 25 of a vector bundle, 167 family of Lie algebra morphisms, 282 of A-Lie group morphisms, 282 f.g.p, 47-54, 59,172,180,181,196,250 fiber bundle, 145 map, 147 principal, 155, 301 structure, 145 over a point, 145 typical, 145 finite dimensional, 59 type, 46 finitely generated, 46, 48, 50, 51, 196 flat connection, 342 flow, 228 commuting, 236 global, 234 local,228 foliation, 243, 341 form, differential k-, 248 frame, 361 bundle, 361 free 2l-graded A-module, 16 Frobenius' theorem, 243, 244, 342, 355 fundamental vector field, 299 FVF connection, 345 gauge, 351
413
transformation, 351 generator, 16, 46 graded, 56 determinant, 41, 55, 78-80, 111, 118, 285 subspace, 59, 86 trace, 75, 76, 80, 285 transpose, 71 graph, 134, 149 homogeneous, 3, 4 homomorphism, 9 horizontal k-form,395 lift, 342 map, 342 part of a tangent vector, 342 section, 342 submanifold, 342 tangent vector, 342 ideal, 296 identification, 11 immersion, 214 implicit function theorem, 122 independent elements, 46 initial condition, 228 integrable connection, 342 subbundle, 243, 244, 342 vector field, 228 integral manifold, 244 interchanging map, 22 interval, 228 invariance of dimension, 121 invariant k-form, 320 left/right, 320 vector field, 315 inverse function theorem, 121 invertible homomorphism, 11 involutive subbundle, 243 isomorphic fiber bundles, 147 modules, 11
Index
414
isomorphism, 11 of fiber bundles, 147 of Lie algebras, 270 of vector bundles, 157 of A-Lie groups, 267 isotropy subgroup, 304 Jacobi identity, 28, 30-32, 270-272, 284, 326, 332, 356 Jacobian, 116 k-additive, 2 k-form,248 M-dependent,257 with values in a vector bundle, 383 with values in anA-vector space, 323 k-linear map (leftlright), 8 Kronecker delta, 58 leaf, 244, 246, 294 left multiplication, 2 translation, 268 left -invariant vector field, 270 Lie derivative, 253 lift, 377 linear connection, 374 linear map (left/right), 8 local flow, 228 locally finite, 134 Maurer-Cartan I-form, 339, 357 metric on a free graded A-module, 192 on a vector bundle, 194 modeled, an A-manifold on an A-vector space, 124 module, 2-6 momentum map, 332 morphism, 9 of Lie algebras, 30, 270 of vector bundles, 157 of A-Lie groups, 142, 267 nilpotent vector, 81
normal A-Lie subgroup, 296 notational shorthand, 119 odd, 56 ordered basis, 59 orthogonal complement, 193 orthonormal basis, 192 parallel transport, 344, 364 parity, 3, 4 of a linear map, 8 of a section, 158 reversal, 102 shift operation, 32, 102 partition of unity, 95, 128, 135, 136, 159, 167, 175, 194, 195, 197, 199, 263 plateau function, 136, 160, l76, 205, 250, 327 principal connection, 351 fiber bundle, 155, 301 product of bundles, 153 projective, 47 proper A-Lie subgroup, 300 proto A-manifold, 124 pseudo effective action, 143 pseudo metric pseudo metric on a free graded Amodule, 188 on a vector bundle, 194 pull-back bundle, 151 map, 183 of a differential form, 255 E-valued, 324 generalized, 257, 319 of a section, 183 push forward of a section, l78 push forward of a vector field, 218 quotient, 16 bundle, 164 rank, 73
Index
415
of a function, 122 of a matrix, 58 of a vector bundle, 156 reducing the structure group, 154 regular value, 214 related vector fields, 218 representation of a Lie algebra, 30 of an A-Lie group, 267 restriction of a bundle to a sub manifold,
146 right multiplication, 2 translation, 268 right-invariant vector field, 270 ring, 2l-graded, 3 commutative, 3 second countable topology, 128, 245, 247 section, 148 shorthand, notational, 119 signature, 24 simply connected, 312 skew-symmetric, 24 skew-symmetrization operator, 40 smooth A-structure, 124 functions, 94-101 linear map, 85 map between A-manifolds, 128 system, 96 maximal, 97 tree, 96 strong bundle map, 147 structure bundle, 361 constants of a Lie algebra, 272, 273, 280, 297, 328, 340 equations of an A-Lie group, 340 group, 145 subalgebra of a Lie algebra, 28, 293 subbundle, 160 integrable, 243, 244, 342 involutive,243 sub manifold, 130
sub module, 2 2l-graded, 4,6 generated by, 6 sum of, 6 subspace graded, 59, 86 of an A-vector space, 86 sum of submodules, 6 supplement, 15, 63 of a bundle, 165 support of a function, 134 of a section, 159 symmetric,2l-graded, 24 symplectic geometry, 332 tangent bundle, 204 map, 212 generalized, 225, 257 to a subbundle, 244 tensor product, 18 of vector bundles, 166 topology DeWitt, 93 on an A-vector space, 93 second countable, 128, 245, 247 trace, 76 graded, 75, 76,80,285 transition function, 145 transitive action, 267 transpose, 71 graded, 71 of a morphism, 13 transposition operator, 13 trilinear map, 8 trivial bundle, 147,223,224 vector bundle, 158, 223, 224 tri vializing atlas, 145 chart, 144 sections, set of, 160 type p k-form, 396 typical fiber, 145
416
Index
vector bundle, 156 vector field, 2(1) Euler, 261 fundamental, 299 integrable, 228 invariant, 315 left/right-invariant, 270 vertical
wave, 228 wedge product, 26 for vector bundle valued forms, 336, 386 symbol, 26 Whitney sum, 164 without odd dimensions, 106, 136 Yang -Baxter equation, 23
subbundle, 342 tangent vectors, 342
zero section, 158