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LIEALCEBRAS Nathan Jacobson Lie group theory, developed by M. Sophus Lie in the lgth centur.|, ranks among the more important developments in modern mathematics. Lie algebras comprise a significant part of Lie group theory and are being actively studied today. This book, by Professor Nathan Jacobson of Yale, is the definitive treatment of the subject and can be used as a textbook for graduate courses. Chapter I introduces basic concepts that are necessary for an understanding of structure theory, while the following three chapters present the theory itself: solvable and nilpotent Lie algebras, Cartan's criterion and its consequences, and split semi-simple Lie algebras. Chapter 5, on universal enveloping algebras, provides the abstract concepts underlying represerrtation theory. Then the basic results on representation theory are given in three succeeding chapters: the theorem of Ado-Iwasalva, classification of irreducible modules, and characters of the irreducible modules. In Chapter 9 the automorphisms of semi-simple Lie algebras over an algebraically closed field of characteristic zero are determined. These results are applied in Chapter l0 to the problems of sorting out the simple Lie algebras over an arbitrary field. The reader, to fully benefit from this tenth chapter, should have some knowledge about the notions of Galois theory and some of the results of the Wedderburn structure theory of associative algebras. Nathan Jacobson, presently Henry Ford II Professor of Mathematics at Yale University, is a well-known authority in the field of abstract algebra. His book, Lie Algebras, is a classic handbook both for researchers and students. Though it presupposes a knowledge of linear algebra, it is not overly theoretical and can be readily used for self-study.
Unabridged, corrected (1979) republication of the original (1962) Bibliography. Index. ix a 331pp. \s/B x 8/4. Paperbound. A DOVER EDITION DESIGNED FOR YEARS OF USE! We have made every effort to make this the best book possible. Our paper is opaque, rvith minimal shorv-through; it rvill not discolor or beconre brittle with age. Pages are servn in signatures, in the method traditionally used for the best books, and will not drop out, as often happens u'ith paperbacks held together with glue. Books open flat for easy reference. The binding will not crack or split. This is a permanent book.
ISBN 0-486-63832-4
$5.50in U.S.A.
o o o o
a
G C rE F' t
14
xo 5 o o
LIEALGEERAS by
I.TATHAN JACOBSON Henry Ford II Professor of Mathematics Yale University, New Haven, Connecticut
DoverPublications, Inc. New'\brk
Copyright O 1962 by NathanJacobson. All rights reserved under Pan American national Copyright Conventions.
and Inter-
Published in Canada by General Publishing Companf, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable aild Company, Ltd., l0 Orange Street, London WCZH 7EG. This Dover edition, first published in l9?9, is en unabridged and corrected republication of the work originally published in 1962 by Interscience Publishersl a division ofJohn Wiley & Sons, Inc. Int enut ional Standard B ooh Num b er : 0 - 486 -6i $ 2 -i Ubrary of CongressCatalog Card, Number: 79-52005 Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N.Y. 10014
PREFACE The present book is based on lectures which the author has given at Yale during the past ten years, especially those given during the academic year 1959-1960. It is primarily a textbook to be studied by students on their own or to be used for a course on Lie algebras. Besides the usual general knowledge of algebraic concepts, a good acquaintance with linear algebra (linear transformations, bilinear forms, tensor products) is presupposed. Moreover, this is about all the equipment needed for an understanding of the first nine chapters. For the tenth chapter, we require also a knowledge of the notions of Galois theory and some of the results of the Wedderburn structure theory of associative algebras. The subject of Lie algebras has much to recommend it as a .subject for study immediately following courseson general abstract algebra and linear algebra, both because of the beauty of its results and its structure, and because of its many contacts with other branches of mathematics (group theory, differential geometry, differential equations, topology). In this exposition we'have tried to avoid rnaking the treatment too abstract and have consistently followed the point of view of treating the theory as a branch of linear algebra. The general abstract notions occur in two groups: the first, adequate for the structure theory, in Chapter I; and the second, adequatefor representation theory, in Chapter V. Chapters I through IV give the structure theory, which culminates in the classification of the so-called "split simple Lie algebras." The basic results on representation theory are given in Chapters VI through VIII. In Chapter IX the automorphisms of semi-simple Lie algebras over an algebraically closed field of characteristic zero are determined. These results are applied in Chapter X to the problem of sorting out the simple Lie algebras over an arbitrary field. No attempt has been made to indicate the historical development of the subject or to give credit for individual contributions to it. In this respect we have confined ourselves to brief indications here and there of the names of those responsible for the main ideas. It is well to record here the author's own indbbtedness to one of the great creators of the theory, Professor Hermann Weyl, whose lectures at the Institute for Advanced Study in 1933lvl
vl
PREFACE
1934 were truly inspiring and led to the author's research in this field. It should be noted also that in these lectureB Professor Weyl, although primarily concerned with the Lie theory of continuous groups, set the subject of Lie algebras on its own independent course by introducing for the first time the term "Lie algebra" as a substitute for "infinitesimal group," which had been used exclusively until then. A fairly extensive bibliography is included; howevef, this is by no means complete. The primary aim in compiling the bibliography has been to indicate the avenues for further jstudy of the topics of the book and those which are immediately rrelated to it. I am very much indebted to my colleague George $eligman for carefully reading the various versions of the maguscript and offering many suggestions for improving the expo$ition. Drs. Paul Cohn and Ancel Mewborn have also made valuable comments, and all three have assisted with the proofreading. I take this opportunity to offer all three my sincere thanks. May 28, 1961 New Hauen, Connecticut
N.lrs.$.r Jlconsou
CONTENTS Cnlrrpn
I
Basic Concepts 1. Definition and construction of Lie and associative algebras 2. Algebras of linear transformations. Derivations 3. Inner derivations of associative and Lie algebras . 4. Determination of the Lie algebras of low dimensionalities 5. Representations and modules . 6. Some basic module oPerations 7. Ideals, solvability, nilpotency . 8. Extension of the base field .
2 5 9 11 14 t9 23 26
CulprnR II Solvable and Nilpotent Lie Algebras Weakly closed subsets of an associative algebra Nil weakly closed sets Engel's theorem Primary components. Weight spaces Lie algebras with serni-simple enveloping associative algebras 6. Lie's theorems 7. Applications to abstract Lie algebras. Some counter examples . 1. 2. 3. 4. 5.
31 33 36 37 43 48 51
Cn.l.rtsn III Cartan's Criterion and Its Consequences 1. 2. 3. 4. 5. 6. 7.
Cartan subalgebras . Products of weight spaces An example Cartan's criteria Structure of semi-simple algebras Derivations. Complete reducibility of the representations of semi-sinrple algebra lvii l
57 61 64 66 70 73 75
Viii
CONTENTS
8.
Representations of the split three-dimensional simple Lie algebra. 9. The theorems of Levi and Malcev-Harish-Chandra . 10. Cohomology groups of a Lie algebra . 11. More on complete reducibility . . .
83 86 93 96
Cn^l,rtpn IV Split Semi-simple Lie Algebras 1. Properties of roots and root spaces. 2. A basic theorem on representations and its consequencesfor the structure theory 3. Simple systems of roots 4. The isomorphism theorem. Simplicity . 5. The determination of the Cartan matrices 6. Construction of the algebras 7. Compact forms .
.i .
. 10g
. .; . . . . .
. Ilz . 119 .IZT . LZg . 13S . 146
Definition and basic properties . . The Poincar6-Birkhoff-Witt theorem Filtration and graded algebra Free Lie algebras The Campbell-Hausdorff formula . Cohomology of Lie algebras. The standard complex Restricted Lie algebras of characteristic p Abelian restricted Lie algebras .
151 156 163 t67 170 174 185 r92
Cn.c.prpn V Universal Envetoping Algebras
CHlrrnn
VI
The Theorem of Ado-Iwasawa 1. Preliminary results . 2. The characteristic zero case 3. Thecharacteristicpcase.
..
. Z}L .Z}J
,f
.207
CHAPTER. VII Classification of Irreducible Modules 1. Definition of certain Lie algebras .
ix
CONTENTS
.212 .zLs
2. On certain cyctic modulesfor E 3. Finite-dimensionalirreducible modules 4. Existence theorem and isomorphism theorem for semi-simpleLie algebras . 5. Existenceof E, and Ea . 6. Basic irreducible modules
.220 .223 .225
Cn.lprpn VIII Characterg of the lrreducible
Modules
1. Some propertiesof the Weyl group 2. 3. 4. 5.
240 243 249 257 259
Freudenthal's formula Weyl's character formula Some examples . Applications and further results CHlrrnn
IX
Automorphisms Lemmas from algebraic geometry Conjugacy of Cartan subalgebras . Non-isomorphism of the split simple Lie algebras Automorphisms of semi-simple Lie algebras over an algebraically closed field 5. Explicit determirtation of the automorphisms for the simple Lie algebras
1. 2. 3. 4.
.zffi .271 .274 .275 .281
Cru,prpn X Simple Lie Algebras oyer an Arbitrary 1. Multiplication algebra and centroid of a non-associative algebra . 2. Isomorphism of extension algebras . 3. Simple Lie algebras of types A-D 4. Conditions for isomorphism 5. Completeness theorems . 6. A closer look at the isomorphism conditions 7. Central simple real Lie algebras Bibliography. In d e x
Field . 290 .295 . 298 .303 . 308 . 311 . 313 ..319 .329
LIEALGEBRAS
CHAPTER I
Basic Concepts The theory of Lie algebras is an outgrowth of the Lie theory of continuous groups. The main result of the latter is the reduction of "local" problems concerning Lie groups to corresponding problems on Lie algebras, thus to problems in linear algebra. One associates with every Lie group a Lie algebra over the reals or complexes and one establishes a correspondence between the analytic subgroups of the Lie group and the subalgebras of its Lie algebra, in which invariant subgroups correspond to ideals, abelian subgroups to abelian subalgebras, etc. Isomorphism of the Lie algebras is equivalent to local isomorphism of the corresponding Lie groups. We shall not discuss these matters in detail since excellent modern accounts of the Lie theory are available. The reader may consult one of the following books: Chevalley's Theorl of Li,e Groufs, Cohn's Lie Groups, Pontrjagin's Topological Groups. More recently, two other types crf group theory have been aided by the introduction of appropriate Lie algebras in their study. The first of these is the theory of free groups which can be studied by means of free Lie algebras using a method which was originated by Magnus. Although the connection here is not so close as in the Lie theory, significant results on free groups and other types of discrete groups have been obtained using Lie algebras. Particu' larly noteworthy are the results on the so-called restricted Burnside problem: Is there a bound for the orders of the finite groups with a fixed number r of generators and satisfying the relation tr* : I, m a fixed, positive integer? It is worth mentioning that Lie algebras of prime characteristic play an important role in these applications to discrete group theory. Again we shall not enter into the details but refer the interested reader to two articles which give a good account of this method in group theory. These are: Lazard tzl and Higman [1]. The type of correspondence between subgroups of a Lie group and subalgebras of its Lie algebra which obtains in the Lie theory tll
2
YIE ALGEBRAS
has a counterpart in chevalley's theory of linear algebraic groups. Roughly speaking, a linear algebraic group is a subdroup of the group of non-singular n x n matrices which is specified,by a set of polynomial equations in the entries of the matrices. An example is the orthogonal group which is defined by the set of equations Xirl?i:l,\idiiditt:A, i + h , j , k - 1 , . . . t l t t o n t h e e h t r i e se ; ; o f the matrix (a;1). with each linear algebraic group chevalley has defined a corresponding Lie algebra (see Chevalley t2l) hrhictr gives useful information on the group and is decisive in the theory of linear algebraic groups of characteristic zero. I In view of all this group theoretic background it is inot surprising that the basic concepts in the theory of Lie algdbras have a group-theoretic flavor. This should be kept in mindr throughout the. study of Lie algebras and particularly in this chdpter, which gives the foundations that are adequate for the mafin structure theory to be developed in Chapters II to IV. euestioris on foundations are taken up again in Chapter V. These concern some concepts that are necessary for the representation theoryl which will be treated in Chapters VI and VII. 1.
Defrnition and construction of Lie and aasociatiae algebras
We recall the definition of a non-associative algebra (:not sarily associative algebra) ?I over a field, a. This is jirst a space ll over o in which a bilinear composition is defined. for every pair (r, !), r,y in ?I, we can associate a pfoduct and this satisfies the bilinearity conditions
(1) (2)
(xr+xil:h!*rz! a(xy):
necesvector Thus rJ ell
x(y, + !r): rlt * *!z (ax)y - x (a y ) , a€O .
A similar definition can be given for a non-associative algebra over a commutative ring t0 having an identity element (unit) 1. This is a left O-module with a product ry € ?I satisfying (1) iand (2). We shall be interested mainty in the case of algebras over fields and, in fact, in such algebras which are finite-dimensiorlal as vector spaces. For such an algebra we have a basis (er,er,.)..,en)and we can write €r€i:}I=ffure* where the 7's are in A. The zt faiy arte, called the constants of multiplication of the algebra (relative to the chosen basis). They give the values of every prduet e;e1,
I. BASIC CONCEPTS
1,2, . . , , n. Moreover, these productsdetermineevery productin ?I. Thus let r and y be any two elementsof tr and write r: ZEi€;, !:>Tiei, Errti€O. Then, bV (1) and (2), xy - ( T,{'erXl,n iei) : l,(E ie)Qte i) . r,t t
- \E;@{nP)):
\E;nr(e&t) ,
and this is determined by the e&t. This reasoning indicates a universal construction for finite-dimensional non-associative algebras. We begin with any vector space lI and a basis (er) in t. For every pair (l,l) we define in any way we please *gt ds an element of lt. Then if x : LTEee; y - Zlv$t we define
(3)
try: f
Erro@oun)
i, i:L
One checks immediately that this is bilinear in the sense that (1) and (2) are valid. The choice of e1e1is equivalent to the choice of the elements Tux in @ such that e&t:}Taixan. The notion of a non-associative algebra is too general to lead to interesting structurai results. In order to obtain such results one The must impose some further conditions on the multiplication. most important ones-and the ones which, will concern us here, are the associative laws and the Lie conditions. Dprrurrrorv 1. A non-associative algebra lI is said to be associatiue if its multiplication satisfies the associative law
x(vz).
(4)
A non-associative algebra lI is said to be a Li,e algebra if its mu'ltiplication satisfies the Lie conditions
(5)
x,:0,
( x y ) z + ( y z ) x +( z r ) y- 0 .
The second of these is called the Jacobi identity. Since these types of non-associative'algebras are defined by identities, it is clear that subalgebras and homomorphic images are of the same type, i.e., associative or Lie. If lI is a Lie algebra andx,yell, then 0 -.(r* y)': x'+ xy + ltc*!': xl +yr so that
(6)
-yx
4
LIE ALGEBRAS
holds in any Lie algebra. Conversely, if this condition holds then 2tr' :0, so that, if the characteristic is not two, then rz't: 0. Hence for algebras of characteristic * 2 the condition (6) canj be used for the first of (5) in the definition of a Lie algebra. Pnoroslrlox 1. A non-associatiue algebra W,with basis(er,er,. ..,eo) ouer O is associ,atiaeif and only if (ep)er-er(e$) for i,j,k: 1,2, . -.,n. If eaei:\,Tu,e, theseconditionsare equiaalentto (7 )
l T u rT 1 ,, ,
\ T ur T , * r:
i , j , k , s : 1, 2, " ., ? .
The algebra 2l is Lie if and only if e',t:0,
e&i:
- ei01,,
(ete)er * (eie)er * @p)e1 : g
for i,j,k:1,2, (8)
. . ., n.
These conditions
Tur:0 Z(TotTrr,
,
Tux:
are equiaalentt to -
T i t : r, * Ti*,Tru * T*rrTr.rr): 0 .
Proof: If ?I is associative, then (ep)er,: e;(epx). Conversely, assume these conditions hold for the et. lf x: ),E;ev, ! :Zqpt, z : 2($t, then (xy)z : 2E iv{n(e&)er and x(yz) : 2iniCpr(ep*). Hence (ry)z - x(yz) and lI is associative. If €;€1: ET;t&,, then (ep)e*: Xr, eT;irT*cqcand et(ep*) : X", tTirtTirr€t. Hence the linear independence of the ei implies that the conditions (e;b)er,: e;(eq*) are equivalent to (7). The proof in the Lie case is slmilar to the foregoing and will be omitted. In actual practice the general procedure we have indicated is not often used in constructing examples of associative and of Lie algebras except for algebras of low dimensionalities. We shall employ this in determining the Lie algebras of one, two, and three dimpnsions in g4. There are a couple of simplifying remarks that can be made in the Lie case. First, we note that if e?:0 and e;ei : -' ,ru, in an algebra, then the validity of (e;e)en* (ep*)er + @p)et: 0 for a particular triple i, j, k implies (eie;)erc * (e;e)et * (eret)e;: 0, Since cyclic permutations of i, j, k are clearly allowed it follows that the Jacobi identity for (e;,e;')4, is valid for it, j',k', a permutation of. i, j,h. Next let i : j . Then e?e*+ (e;e*)e;* (ene;)e;: 0 * (e;et)er- (e;er)e;- O. Hence e?: 0, ei01: - ri€i or, what is the same thing, x' : 0 in ?I implies that the Jacobi identities are satisfied for e;, e;,,€i. In particular, the Jacobi identities are consequencesof x' :0 if dim A < 2
5
I. BASIC CONCEPTS
and if dim t[ :3, then the only identity we have to check is = 0. (ep)es* (eze)er* (eser)ez 2.
Algebrat
of linear tranaformations.
Deriaations
Actually, it is unnecessary to sit down and construct examples of associative and Lie algebras by the method of bases and multiplication tables since these algebras occur "in nature." The prime examples of associative algebras are obtained as follows. Let lll be a vector space over a field O and let @denote the set of linear trans' formations of Dt into itself. We recall that if. A, BeE and ae0, then A + B, aA and AB are defined by x(A + B) : xA * xB, x(aA): s.(rA), x(AB): (rA)B for r in lll. Then it is well known that € is a vector space relative to * and the scalar multiplication and that multiptication is associative and satisfies (1) and (2). Hence 6 is an associative algebra. It is well known also that if tJt is rz-dimensional, n, 1 @, then O is m'-dimensional over O. lf. (er, €2,...,2^) is a basis for fi over o, then the linear transformations if. r+i, i,i -1, "',/rt, form a .E;l such that etEti:ei,0rE;i:0 basis for O over O. If AeE, then we can write 0;A - }ia;Pt i -- l,- . . t fri, and (a) : (ar) is the matrix of. A relative to the ba' sis (ec). The correspondence A-+ (a) is an isomorphism of 0 onto the algebt? O^ of. m x m matrices with entries ait in O. The atgebra @is called the (associatiae)algebra of linear transform' ations in llt over O. Any subalgebra l[ of 0, that is, a subspace of O which is closed under multiplication, is called an algebra of linear tr ansformations. If lt is an arbitrary non-associative algebra and a e lI, then the mapping ar which sends any x into ra is a linear transformation. It is well known and easy to check tl:pt (a * D)" : aB * bn, (aa)p: aan if U is associative, (ab)": anba. Hence if U is an as' "nd' algebra, the mapping a -> an is a homomorphism of ?I into sociative the algebra @ of linear transformations in the vector space ?I. If lI has an identity (or unit) 1, then a -+ an is an isomorphism of lI into @. Hence 2t is isomorphic to an algebra of linear transform' ations. If lI does not have an identity, w€ can adjoin one in a simple way to get an algebra, ?I* with an identity such that dim lI* : dim ?I + 1 (cf. Jacobson l2l, vol. I, p. 84). Since lI* is isomorphic to an algebra of linear transformations, the same is true for lt. If U is finite-dimensional, the argument shows that
LIE ALGEBRAS
l[ is isomorphic to an algebra of linear transformation$ in a finitedimensional vector space. Lie algebras arise from associative algebras in a ltrery simple way. Let lI be an associative algebra. lf x,yell, thqn we define the Li,e product or (additive) commutator of. x and y AS ( 9)
txyl: xy - lx .
One checks immediately that ln * rcz, lf : [x'yj * Ixryl, lr, !, * yrl : Ixyrl * [x!r] , alxyl: lax, yf : lr, dyl . Moreover, lxrl:tct-x':0,
llxylzl+ [yzlxl + llzxlyl : (xy- yr)z- z(xy- yr) + (yz - zy)r - r(yz - zt) * kx - rz)y tkx
-',t"z) - 0 .
fhus the product Ixyl satisfies all the conditions on tire product in a Lie algebra. The Lie algebra obtained in this way Iis called, the Lie algebra of the associative algebra lI. We shall defrote this Lie algebra as 2Is. In particular, we have the Lie algebrd @z obtained from G. Any subalgebra I of CIz is called a Li,e algebra of linear transformations. we shall see later that every Lie algebra is isomorphic to a subalgebra of a Lie algebra ?Ir, g[ associative. In view of the result just proved on associative algebras this is equivalent to showing that every Lie algebra is isomordric to a Lie algebra of linear transformations. we shall consider now some important instances od subalgebras of Lie algebras @r, @ the associative algebra of linehr transformations in a vector space llt over a field O. orthogonal Lie algebra. Let !)t be equipped with a nbn-degenerate symmetric bilinear form (x, y) and assume !)t finite-dimensional. Then any linear transformation A in !)l has an adjoint ..4* relative to (x,y); that is, Ax is linear and satisfies: (rd, y) : (r, tA*). The mapping A -. A* is an anti-automorphism in the algebra G: (A+ B ) *: A * + B* , (a A 1 * : a A * , (A B)* : B* A * . l l -et € denote the set of Ae @which are skew in the sense that A* * - A. Then is a subspace of @ and if. A* - - A, B* : - B, fhen lABl* :
7
I. BASIC CONCEPTS
-LABI. Hence (AB - BA)* -- B*A* - A*B* : BA - AB: lBAl: tABle 6 and 6 is a subalgebra of 0r. If @ is the field of real numbers, then the Lie algebta 6 is the Lie algebra of the orthogonal group of llt relative to (r' y). This is the group of linear transformations O in 9]t which are orthogonal r,! in !It. For this reason we in the sense that (xO,yO):(x,!), the orthogonal Li.e algebra relative to (r, y). shall call Syruplectic Lie algebra. Here we suppose (r, y) is a non-degenerate and again dim l}t < oo. We recall that alternate form: (r,x):0 these conditions imply that dim tn - 2l is even. Again let ,4* be the adjoint of ,4( e O) relative to (r, y). Then the set 6 of skew (A* : - A) linear transformations is a subalgebra of @2. This is related to the symplectic group and so we shall call it the symPlectic Lie algebra 6 of the alternate form (x, y). c!]l-*Dt Triangular linear transfarmations. Let 0c1]tr cl}tzc''' let ! be I and IJtt: dim 9Jt such that of of subspaces be a chain the set of linear transformations T such that lltrT G TJtt. It is clear that E is a subalgebra of the associative algebra @: hence E; is a 'We can choose a basis (xr, rr, "', x*) for lll so subalgebra of @2. Then if TeT, Tft;T S !]h that (trr,trr, "', x;) is a basis for fii. implies that the matrix of. T relative to (xr, Nr, "', x*) is of the form
(10)
,:[:: :'
:]
Such a matrix is called triangular and correspondingly we shall call any TeT a triangular linear transformation. Deriaation algebras. Let ?I be an arbitrary non-associative algebra. A deriaation D in ?I is a linear mapping of lI into ?I satisfying
(11)
(xy)D:
(xD)y * x(yD) .
Let D(?I) denote the set of derivations in !I. then
lf Dr,DzeS([),
(xy)(D, * D,) - (ry)D1 + @y)Dz- (rD,)t * r(!D) * (xD)y * x(yD,) : (r(Dr * Dr))y* x(y(h + Dr))
LIE ALGEBRAS Hence Dt * Dz e D(U;. a,e0. We have
Similarly,
one checks that
a,DrQS(U) if
(xy)D,Dz: (@Db * x(yD,))Dz : (xD,Dz)y * (xD,1lyD') + (xDr)(tD,) + x(vD,Dr) ' Interchange of l, 2 and subtraction gives (ry)lD,Dzl:
i
@lDD'l)Y + r(YlD'D,l) .
Hence lDrDzle E(lI) and so S(U) is a subalgebra of Gz, where € is the algebra of linear transformations in the vector space lI. We shalt call this the Lie algebra of deriaations or deriadtion algebra
of a. The Lie algebra lD(U) is the Lie algebra of the group of automorphisms of lI if lt is a finite-dimensional algebra over the field of real numbers. We shall not prove any of our asseftions on the relation between Lie groups and Lie algebras but refef the reader to the literature on Lie groups for this. However, in rthe present instance we shall indicate the link between the group;of automor' phisms and the Lie algebra of derivations. Let D be a derivation. Then induction on n gives the Leibniz rule:
(12)
(ry)D" :
D i)(vD'- t) V^("r)rx
If the characteristic of O is 0 we can divide bY n! and obtain (12')
,"O#:E(+.D)(hyD".).
If ll is finite-dimensionalover the field of reals, then it is easy to prove (cf. Jacobson[2], vol. II, p. 197)that the seriesl (13)
r+D*+++ +"'
converges for every linear mapping D in l[, and the linear mapping (12',), exp D defined by (13) is 1: 1. Also it is easy to see, using : (rGXyG). (ry)G : exP D satisfies that if D is a derivation, then G Hence G is an automorphism of ?I. A connection between automorphisms and derivations can be established in a purely algebraic setting which has lmportant ap' plications. Here we suppose the base field of U iS arbitrary of
I. BASICCONCEPTS characteristic 0. Let D be a nilpotent derivation, sY, Consider the mapping
DN :0.
G:expD:L+D++*...+d+
(14)
+(Dn-'l(N-1)!) Z:D+(D'zpD+... We write this as G:L*2, t h e i n v e r s eL - Z + HenceG:1*Zhas and notethat ZN:0. Z' + ... -f Z"-' anld so G is 1: 1 of lI onto lI. We have
: (E+X,!,#) (rG)(yG) zry_z/
r_/ yo"-' 11 :,FoEr "D'11 it 11n-tyt)) zJy_2
: E @ y,' , D" _ nl n=o .lv-l
D"
n=o
n!
(by r?t)
-- (xv)GHence G is an automorphism of lI. 3.
fnner deriiations
of aesoeiatiue and Lie algebras
If a is any element of a non-associative algebra lI, then a deter' mines two mappings az: x -+ ax and ani x -, xQ of lI into itself. These are called the left multiplication and right multiplication deter' mined by a. The defining conditions (1) and (2) for an algebra show that at and aa are linear mappings and the mappings a + (zL, an are linear of !t into the space 0 of linear transformations a ---> in !t. Now let ?t be associative and set D" : (zn - av Hence Do is the linear mapping x--+tut - &r. We have (15) rja - axy -- (ra - ax)t * x(ya - ay) ; hence D" is a derivation in the associative algebra ?I. We shall call this the i,nner deriaation determined by a. Next let I be a Lie algebra. Because of the way Lie algebras arise from associative ones it is customary to denote the product in 8 by [ry] and we shall do this from now on. Also, it is usual to denote the right multiptication an (: - az since lxal - - lax)) by ad a and to call this the adioint mapping determind by a. We have
IO
LIE ALGEBRAS
tlxyial + llyalx) + llaxJyl : 0, llxy)a) - - llyalrl - lLar) yl - lrlyall + llxa)y1 ; hence ad 6 v -,fxaf is a derivation. We call this also the i'nner deriaation determined by ae8,. A subset E of a non-associative algebra ?I is called i an ideal if (1) 8 is a subspace of the vector space ?I, (2) ab, ba€ E for any a in ll, 6 in E. Consider the set of elements of the forfn l,a;bt ai, Dt in ?I. We denote this set as lI2 and we can check ,that this is an ideal in lI. If ?I : 8 is a Lie algebra, then it is customary to write 8' for 8' and to call this the deriued algebra (or:ideal) of 8. If 8 is a Lie algebra, then the skew symmetry of the multiplication implies that a subspace E of I is an ideal if and only if labl (or [Da]) is in E for every aeSJ,DeE. It follows that the subset G of elements c such that [ac] - 0 for all a e I is an lideal. This is called the center of 8. 8 is called abelian if I ==6, which is equivalent to 8' : 0. Pnorosrrrox 2. If lI is associatiue or Lie, then the innQr deriuations l forrn an ideal 3(ll) iz the deriaation algebra D(?l). Proof: In any non'associative algebra we have (a + b)r,: ar, * bt, (aa)": dazt @ * b)n : aR * bn, (aa)*-- dan. Hence if ' D"-- an - at; then D,+ a: D, * Dt, Dro - aDo and the inner deriv]ations of an associative or of a Lie algebra form a subspace of S(U). Let D be a derivation in lI. Then (ax)D - (aD)x * a(xD), ot (ax)D - a(xD): (aD)x. In operator form this reads (xar)D - (xD)az',: x(aD)r' or atD - Da, - (aD)r Similarly, [a*Dl-@D)n and tconsequently lapl: also [D"D] : D,n. These formulas show that if ll is dssociative or Lie and .I is an inner derivation and D any derivatiori, then UDI is an inner derivation. Hence S0I) is an ideal in O(U). Erample. Let I be the algebra with basis (e,f) such that [efl - ffel and all other products of base elements are 0. Then e laa):O in Il and since dim I :2, 8 is a Lie algebra. The derived algebra g' : Ae. If D is a derivation in any algebra ?I, then ?IzDg ?Iz. Hence if D is a derivation in I then eD: 6e. Also a[(D/) has the property e(adD/) : le,6fl -- 6e. Hence if E: D - adp/, then E is a derivation and eE: A. Then e - lefl gives g - [e,fpl. It follows that fE -- Te. Now ad(- re) satisfies e ad(- re) :0' /ad(- Te) : re. Hence E-ad(- re) is inner and D: E*adDf lf, - yel-ylefl: is inner. Thus every derivation of [l : Ae + Af is infier.
I. BASICCONCEPTS
11
In group theory one defines a group to be complete if all of its automorphisms are inner and its center is the identity. If H is complete and invariant in G then ^Ff is a direct factor of G. By analogy we shall call a Lie algebta complete if its derivations are all inner and its center is 0. Pnorostuon 3. If R is complete and an ideal in t., then 8': S @ E where E is an ideal. Proof: We note first that if .f is an ideal in IJ, then the centralizer E of s, that is, the set of elements D such that [&D]: 0 for all &eS is an ideal. E is evidently a subspace and if DeE and a € g, then lhlbal) - - [alkb]l - lbtakll : 0 - [b, k'7, ft' : fak) c Ri Hence E is an ideal. hence tklball:0 for all k e 0 and [ba]e$. Now let ,R be complete. If c e S n E, then c is in the center of fr and so c : 0. Hence S n E - 0. Next let a € 8. Since S is an ideal in 8, ad a maps F into itself and hence it induces a deriva' tion D in S. This is inner and so we have a & e R such that E and a:b*k, r D - - l r a l - - l x k l f . o re v e r y r € S . T h e n b : a - k e r e q u i r e d. a s n @ E S *E beE, ft e n. Thus I is complete. example Af last Ae the of The algebra + Eram\le. 4,
Determination of the Lie algebras of low dimcnsionalities
We shall now determine all the Lie algebras IJ such that dim 8 S 3 . l f ( e r , € 2 , . . . , e , )i s a b a s i s f o r a L i e a l g e b r a S , t h e n l e ; e r ] : 0 and le;ei\:. - leie;\. Hence in giving the multiplication table for the basis, it suffices to give the products fe;ei f-or i < i. We shall use these abbreviated multiplication tables in our discussion. I. dim I : 1. Then 8 :Oe, leel - g. II. dimS-2. (a) 8' : 0, I is abelian. (b) 8' + 0. Since I -- Oe + Af,8' - a[ef] is one-dimensional. We may choose e so that 8/ : Oe. Then lefl - ae * 0 and replacement of.f by a-'f permits us to take lefl - e. Then I is the algebra of the example of $ 3. This can now be characterized as the non' abelian two-dimensional Lie algebra. III. dim 13- 3. (a) 8' :0, I abelian. (b) dim 8/ : 1, 8' S S the center. If 8' -- oe we write 8,: oe * Hence we may suppose lfgl - e. Thus Af + Ag. Then 8,' :0[fg7.
L2
LIE ALGEBRAS
8 has basis (e,f,g),
(16)
with
multiplication
lfsf- e,
lefl-0,
table
I e g -l 0 .
We have only one Lie algebra satisfying our conditions. (If we have (16), then the Jacobi condition is satisfied.) (c) dim 13': 1, !' E 0 the center. If 8' - O€,then,there is an / such that lef) + 0. Then [ef] - Fe + 0 and we may suppose lefj - e. Hence Oet Af is the non-abeliantwo'dimensionalalgebra n . S i n c e R = 8 ' , S i s a n i d e a ta n d s i n c e S i s c o m p l e t e8,- S O E , HenceIJ has basis (e,f,g) with multiplicatiorltable E:Og. (17)
lefl : e,
lesl- o,
[fsl -- o .
(d) dim 8' :2. 8' cannot be the non-abeliantwo-dimensional L i e a l g e b r a S . F o r t h e n8 : S @ E a n d 8 ' : S ' - A . B u t S / c S . Hencewe have 8' abelian. Let 8/ : Oe+ Af and I - Ae + af + Og. Then 8t:alegl+o17g1 and so adg inducesa 1:1 linear mapping in 8'. Hence we have basis (e,f, g) with (18)
lefl - 0,
[esl: ae * Ff ,
lfsl - re * 8f
: (; Convergely, in any t. a non-singular matrix. il " space I with basis (e,f, g) we can define a product [aD] so that [aa] - 0 and (18) holds. Then tteflsl + [tfg]el + llselfl = 0 and hence I is a Lie algebra. What changes can be made in the multiplica' tion table (18)? Our choice of basis amounts to thiS: We have chosen a basis (e,f) for It and supplemented this witt'. a 9 to get a basis for 8. A change of basis in 8' will change A to a similar matrix M-'AM. The type of change allowable f.or g [s to replace Thenle,pg*xl:plegl,'lf,pg *rl: it by pg+x, p+OinO,rinS/. plfgl so this changes A to pA. Hence the different matrices .4 which can be used in (18) are the non-zero multiples of ithe matrices similar to A. This means that we have a 1 : 1 correspondence be' tween the algebras I satisfying dim 8 : 3, dim 8' :2 and the con' jugacy classes in the two dimensional collineation grotrp. If the field is algebraically closed we can choose A in one of the following forms: where
(l )
a*0,
(; f) o+o
These give the multiplication tables
:
T3
I. BASIC CONCEPTS
lefl:0, lefl -0,
lfgl:af Ieg):e, lfgl- f ' legl: e + Bf ,
Difierent choices of a give different algebras unless ad' : \. Hence we get an infinite number of non-isomorphic algebras. Gj dim 8' : 3. Let (eb €2,€s) be a basis and set fere'l: f', Vrrrl -4 - fr, ferezl: fr. Then (fr, fr,/t) is a basis' Write f i [i='eiiet' -- (a;i) non-singular. The only Jacobi condition which has to be imposed is that lf 'erl * lfrerl * [/gea]: 0. This gives * asrlererl* aazlezesl * azrLe4zl* azslegezf 0 : arzlezer)* argleserf - - anfg * arrf, * dnfs - azsft - asrfz * atrft ' so A is a symmetric matrix. Let (7yEz,ds)_be Hence aii = a4; "nd, : a second basis where 4 : Lpitrt, M : (piil non'singular. Set /1 per' (i, cyclic any i, k) l,rdrl, fr: [d&rl, ir: [er,i- We have f'or mu ta ti on of .( L, 2, 3) 7, - ldprl - [l,rr &,,I,pr,eJ - Epi,ttn,f0,0,l : (pizltrs ' tttsltrr)f ,' * (pisttt , :}v;' f ' '
piillnsJfz* (pnp"' - ttizpt)fg
The matrix N: (vi.i) - adj M' : (M')-' det Mt . The matrix relat' ing the /'s to the a's is A and that relating the e's to the 7's is -M-1. Hence if A is the matrix (au) such that f t, \d.;id1, then (19)
{ - (det M'Y.M')-'AM-'
.
Two matrices A, Bare called multi.pticatiuelycogredient if' B:pN' AN where N is non-singular and p + 0 in O. In this case we may write B - po'(o-tN)t A@-tN), o : p det N and if the matrices are - oN-' and fi of three rows and columns, then we take llrf p(M-')'AM-t, 7c: poz: det M. _Thus we have the relation (19). Thus the conditions on A and A are that these symmetric matrices are multiplicatively cogredient. Hence with each I satisfying our conditions we can associate a unique class of non'singular multi' plicativety cogredient symmetric matrices. We have as many alge' bras as there are classes of such matrices. For the remainder of this section we assume the characteristic is not two. Then each cogredience class contains a diagonal matrix of the form diag {4, LL}, aF + 0. This implies that the basis can be chosen so that
(20)
fere2l: es ,
-legel) Pes.
T4
LIE ALGEBRAS
If the base field is the field of reals, then we have two different If the algebras obtained by taking a, : 9: 1 and a -- - l, p::1. field is algebraically closed we can take a: I : l. We shall now single out a particular algebra in thb family of algebras satisfying dim I : 3 : dim 8'. We impose the condition that I contains an element h such that ad lz has a characteristic root a #0 belonging b A. Then we have a vector e*0 suchthat and sincelhhl-U, e and h are linearly infehl-eadh:d€;e0 dependentand are part of a basis (?,,€2,0a)=(€,h,f). lf (f',fr,ft) are defined as before, the symmetric matrix (461)is now /dt
(21)
dp
du\
(o* dzz ol. \a00/
Then we have tehl - ae, lhhl:0, tfhl- -af - dt€ - dph, which implies that the characteristic roots of ad h are 0, a and - &. We may replace f by a characteristic vector belonging to the root - a of. ad h. This is linearly independent of (e, h) and may be used + df. If we f.or f. Hence we may suppose ttla;t Lehl: &€s lfhl: 2f. 2e, by Za-'h The form of h we obtain replace lfhl [eh] (21) now gives [ef] - Ph + 0. If we replace f by Fl'f, then we obtain the basis (e,f, h) such that
(22)
lehl:2n ,
lfhl:
- 2f ,
lefl -- 11. i
Thus we see that there is a unique algebra satisfying our condition. We shall see soon that any I such that dim I * dimS' = 3 is simple, that is, 8 has no ideals other than itself and 0 and B' + 0. The particular algebra we have singled out by the condition that it contains ft with ad ft having a non-zero characteristic foot in O is called thp sblit three-dimensional simple Lie algebra. It will play an important role in the sequel. 5.
Reprewntations
and modules
If lI is an associative algebra over a field O, then a representation of lI is a homomorphism of tr into an algebra 0 of linear transformations of a vector space llt over O. lf. a + A, b'+ B in the representation, then, by definition, a *b-+ A * B, qn-+aA, ae O, and, ab -+ AB. A right \-module for the associative al$ebra lI is a vector space l}t over @ together with a binary product of llt x U
I. BASTCCONCEPTS into IJt mapping (r,a), r CTft, a e \, into an element ra eIIt that 1. (rr * xz)d: rta * xza , 2. a(xa) -- (ax)a: x(aa) , 3. x(ab) - (xa)b .
15 such
x(a, * az): xar * xoz , d QO,
lf. a -- A is a representation of U, then the representation space llt can be made into a right ?I-moduie by defining tca: rA. Thus we will have (x, + xz)a: (xt + xr)A: xtA * xzA: xfl * *za x(ar * az): r(At * Ar): xA, * tr42: tch * frat a( r a) : a (x A): (a r)A - (&,fl a a(xA) - x(aA): x(aa) x(ab) - x(AB) - (xA)B - (ra)b . Conversely, if llt is any right ll-module, then for any 4 e lI we let A denote the mapping x -) tca. Then the first part of 1 and the first part o! 2 show that A is a linear transformation in IJt over o. The rest of the conditions in 1, 2, and 3 imply that a --+A is a representation. In the theory of representations and in other parts of the theory of associative algebras, algebras with an identity play a pre' ponderant role. In fact, for most considerations it is convenient to confine attention to these algebras and to consider only those homomorphisms which map the identity into the identity of the image algebra. In the sequel we shall find it useful at times to deal with We shall associative algebras which need not have identities. therefore adopt the following conventions on terminology: " Algebra" without any modifier witt fnean "associatiae algebra with an i'dentity." For these "subalgebra" will mean subalgebra in the usual sense containing the identity, and "homomorphism" will mean homomorphism in the usual sense mapping 1 into 1. In particular, this will be understood for representations. The corresponding notion of a module is defined by 1 through 3 above, together with the condition 4, t(I:x,
r;eDt.
If we wish to allow the possibility that t does not have an identity then we shalt speak of the "associative algebra" lI and if we wish
16
LIE ALGEBRAS
to drop 4, then we shall speak of a "module for associative alge' bra" rather than a module for "the algebra ?[." The algebra lI can be considered as a right ll-module by taking xa to be the product as defined in lI. Then 1, 2, and 3 hold as a consequence of the axioms for an algebra and 4. holds since 1 is the identity. The representation a -).4 where A is the linear transformation x -+ xa is called the regular Tepresentation; We have seen ($ 2) tl:ort the regular representation is faithful, that is, an isomorphism. Now let 8 bei a Lie algebra. Then we define a reprvsentation of. I3 to be a homomorphism I -- L of I into a Lie algebro Gz, @ the algebra of linear transforr-nations of a vector space Dt over O. The conditions here are that i| t, -. Lr, lr-+ lz, then (23)
l, * l, --' Lt * Lr, al1-+ a2, llrlrJ - [L'Lz) : LrLz - LrL' .
We now define xl f.or .r e !Dt, I e I by rl : xL. linearity of. L gives the following conditions: 1. (x, * xr)l : xrl * rzl , 2. a(xl) - (ax)l : x(al) , 3. xllJl - (rl)lz - @l)tr
x(1, * lr):
Then (23) and the
xl, * rlz ,
We shatl now use these conditions to define the concept of an 8-modute, [J a Lie algebra. This is a vector space I]t over O with a mapping of Ift x 8 to llt such that the result r/ satisfies 1, 2, and 3 above. As in the associative case, the concepts of module and representation are equivalent. Thus we have indicated that if I -+ L is a representation, then the representation space IJt can be considered as a module. On the other hand, if tlt is any module, then for any Then Z is linear in / e 8 we let L denote the mapping x -'fl. I)t over O and I --' L is a representation of I by lineaf transformations in TJI over O. We note next that 53itself can be considered as an F-module by taking rl to be the product [r/] in 8. Then 1 and 2 are consequences of the axioms for an algebra and 3. follows frotn the Jacobi identity and skew symmetry. We have denoted the representing transformation of. l:1s'-lxll by adl. The representation l-+ad'l determined by this module is called the adjoint reprdsentation of.
I. BASICCONCEPTS
t7
'We
recall that the mappings ad I are derivations in 8. If tJt is a module for the Lie algebra I then we can consider TJt Then the mappings as an abelian Lie algebra with product [xy]:0. of the triviality --+ because 9)t and in transformations linear xl are r of the multiplication in llt, these are derivations. More generally, we suppose now that fi is an !-module which is at the same time -t xl are a Lie algebra, and we assume that the module mappings x algeLie for a derivations in St. Thus, in addition to the axioms also bra in I and in l}t and 1, 2, and 3 above, we have 8.
4.
lxrxzll
-
fxrl, rrl * lxr, xzll .
Now let S be the direct sum of the two vector spaces I and !Ut' We introduce in R a multiplication luul by means of the formula
[r, + lr, x, * lrl: lrrxzl * rJz xzlt+ Utlr] . It is clear that this compositionis bilinear, so it makes the vector space S into a non'associativealgebra. Moreover, lx * l, x * I): 0 * xl - rla 0 : 0 . [[rr * lt, xz * lzlxs* 1'1- l[xaix'] * l[xJr]x'l
(24)
+ l?,xzlx'11 [[/'/'lra] * [[rrrz]/al + llrrlzllsl + [U'r,]ral+ [U'rrUJ - llx*t)xrl + [xrlr, rs] - lxzlr, xsl - xtllJrl + lrrxzlls * (x,l)h - (xzl)h + t[r'rr]r'l : llxfizlxsf * lxJz, xsf- lrzlb xal- (r"lr)lz * (xJil, a lrJa, rrl I fxr, xzhJ * (xrlr)lg- (xzl)I, + IUJrU'l . If we permute L, 2, and 3 cyclically and add, then the terms involving three ,'s or three /'s add up to 0 by the Jacobi identity in I and in 'lt. The terms involving two r's and one / are lxtlz, xsl - lx2lu rrl * txrlg, xz]l* lrt, xzlsl * lxzlg,xl - fxrlz, xr) * fxzlv xt) * lx2, xslt) - 0. * fxslb xrl - f,xtlu xzl * lxslz,rtl * lxa, xJzl The terms involving two /'s and one .r are - (xrlr)lz + (xslz)h t UJ)ls - @zlr)ls - (xrlr)I" * (xJilz * @zl")l'- (rslrV, - (rzl)1, * (xzl)h * (xalr)lz- (rrl')lz - $ .
18
LIE ALGEBRAS
This shows that ft - I @ l}t is a Lie algebra. It is imrhediate from (24) and, the fact that [r/] € !ft for r in IJt, / in 8, that I is a subalgebra of R and !)t is an ideal in A. We shall call S' the split extension of I by fi. An important special case of this is obtained by taking llt : I to be any Lie algebra and I - S, the derivation algebra. Since 0 is, by definition, a Lie algebra of linear transforrhations, the identity mapping is a representation of O in 8. I bepomes a 0' module by defining the module product lD : lD, / b 8, D e S. The split extension of E by 8, 6 : D @ 8 is called the holomorph of 8. This is the analogue of the holomorph of a group lwhich is an extension of the group of automorphisms of a group by the group. - Sr O 8 where 8 is any We can make the same construction S Lie algebra and S, is a subalgebra of the derivation algebra. In particular, itisoften useful to do this for Or: oD, the subalgebra of multiples of one derivation D of 8. Another important special case of a split extensiort is obtained by taking IJ and Dt to be arbitrary Lie algebras and considering l)t as a trivial module for I by defining ml :0, ffi e tJl, / e 8. The Lie algebra R - I @ llt is the direct sum of Il and lll. ,More generally, if lJ,, ,8r,' '', I' are Lie algebras then the direct sum 8 . . . tl, @ 8, @ @ 8" is the vector space direct sum of the 8r with the Lie product [Xi/;, ZTm,i : :,Tllimil. As in group theory, if I] is a Lie algebra and I contains ideals ,8c such that S:8'@8r@ ... O I J " , as v ec to r s p a c e , th e n L l tl tl € 8 a fl 8 1 :0 i f /t € 8t, /r € 8r Then I is isomorphic to the direct sum of the Lie and i + j. '8 is a direct sum bf the ideals algebras 8c and we shall say that IJr of 8. The kernel S' of a homomorphism 11of a Lie algebr4 I into a Lie algebra !ft is an ideat in I and the image 8Z is a subalsebra of IIt' The fundamental theorem of homomorphisms states that 8n " 8/S under the correspondence / + n - hl. We recall that 8/S is the vector space 8/ft' considered as an algebra relative to the multiplication U, + S, /, + Sl : ltJrl + A. This is a Lie algebra. The kernel : of the adjoint representation is the set of elements c suchlthat [rc] $ : is ! ad 8 just imag€, The of 8. o the center for all, x. This is : 0, = E, 6 If have 8/G we and derivations the algebra of inner ad I is a Lie algebra of linear transformations isomdrphic to 8' Thus in this case we obtain in an easy way a faithfulr representation of 8. We shall see later that every I has a faithfUl represent-
I. BASIC CONCEPTS
19
ation and that every finite'dimensional I has a faithful finitedimensional representation, that is, a faithful representation acting in a finite-dimensional space. Eramples. We shall now determine the matrices of the adjoint representations of two of our examples. ( a ) S t h e L i e a l g e b r a w i t h b a s i s( e , f ) , l e f l : e . W e h a v ee a d e : 0 , f ade - ei eadf : e, f adf - 0. Hence relative to the basis (e,f) the matrices are
,-'(-l3),'-(l 3) (b) 8 the Lie algebra with basis (er,er, ee) such that Iepzf : es, ---- - qst e, ad €t : €z', leresl er, les€tl €2. Here 0r ?d €t 0, 0r 3d e, 4"d,02:
€s; qzadcz:0,
€sades:0.
egad€z:
-
€i
€rader:
*
€zt €zvd€s:
€rt
Hence the representation by matrices is
0r /00 n,--(oo -l), \0 1 0/
€z-
-1 10 er-l I 0 \00
( rsil s)
Note that the matrices are skew symmetric and form a basis for the space of skew symmetric matrices. Hence we see that I is isomorphic to the Lie algebra of skew symmetric matrices in the matrix algebra Os. 6.
Some basic module operationa
The notion of a submodule It of a module 9Jl for a Lie or associative algebra is clear: It is a subspace of llt closed under the composition by elements of the algebra. If It is a submodule, then we obtain the factor module lll/ft which is the coset space tn/tt with the module compositions (r * Tt)a : xa + tt, a in the algebra. If YJtrand !)tz are two modules for an associative or a Lie algebra, then the spac€ Ut' O fi, is a module relative to the composition (xt * rz)a : xfi * rzar !c; € IJlr. This module is called the direct sum Tft, @ Ilt, of the two given modules. A similar construction can be given for any number of modules. The module concepts we have just indicated are applicable to
20
LIE ALGEBRAS
both associative and Lie algebras. We shall now cdnsider some notions which apply only in the Lie case. These are hnalogous to well-known concepts of the representation theory of $roups. The principal ones we shall consider are the tensor product of modules and the contragredient module. We Suppose first that I -, Lr and / -, Lz are two re$resentations of a Lie algebra I by linear transformations acting in the same vector Space Tft over O. We assume' moreover, that if Lt is any representing transformation from the first representatign and M" is any representing transformation from the second refresentation, -- 0. We shall now define a new mapthen [LrMzJ: LrM, - MrL, of linear transformationF in TJt by @, ping of 8 into the algebra
(25)
l-+Lt*Lr.
- L, it since this is the sum of the linear mappings l - L, and I -mapping new in the that so Mr, n't. Mt, is linear. Now let m we have m - M, * Mz. Then (26)
"v"[L,M') lL, * Lr, M, + Mzl - [L,M,J * lL,Mz\ + ILzM] : [L1M'l I [LrMzl
and since ltml--,lLrMrl a lLrMr\, the new mapping is ja representation. (Note: Nothing like this holds in the associativB case') We suppose next that fir and !]tz are any two modtlles for a Lie - utl 8llt, (= IJlr IrTJt') the tensor (0r Kronecker algebra 8. Let Ift oidirect) product of llt, and 9ltz. We recall that if 4; is a linear l l,xtA'E^ yiAz' transformation in IJli, then the mapping )ri I li Az lin ']tt I ffir. At I transformation linear xt € Tltr, yi € [!tz, is a We have the rules
(27)
(Ar + B')8 Az: e' A ' 8 ( A , + B r ): . 4 ' a(Ar6 B,) : aArS (,4' 8 Ar\,Br@Bz):
8 A 2+ B ' 8 A ' I Az* Ar@ B' B ' : A r @a B r , A'Br@ AzBz'
I a QO
I,, It is clear from these relations that the mapping Ar+4'8Lr, of O(!)t') algebfa the of homomorphism a is fir, in the identity linear transformations in fir into the algebra @(yJt'I tnd). similarly' Now Az + lr 8 A, is a homomorphism of €(m') into O(fi' @ IJtr). linear The !)ti' the by we have the representation R; determined yol. The resultant transformation /nd associated with t e I is'r1--+
2T
I. BASIC CONCEPTS
of the Lie homomorphism I -, l&i with the associative (hence Lie) homomorphism of @(lltr)into @(!]trI !lt') is a representation of I acting The two representations of I obtained in this way are in fi, I fir. (28)
l-alBr@lz
and
l-lt&l*'.
lf. l, m e 8, then (29)
(/or €) lrXl, I
m*r) : l"' I
m":
(1, 6 mBzylP'I
1r) .
Hence the commutativity condition of the last paragraph holds. It now follows that
(30)
I -. l * ' 8
1 , + 1 1@ l n z
is a representation of I acting in !lt' I TJlz. In this way I)l' I fit is an 8-module with the module composition defined by
(311
(X", &y)l - \xit Syr + I,,n&yl
The module Dl' S llt' obtained in this way is called the tensor product of the two modules fie. The same terminology is applied to the representation, which we denote as fr, I Rr. We consider next a Lie module llt and the dual space IJt* of linear functions on IJt (to the base field). We shall denote the value of a linear function y* at the vector r e IJt by (*, t*). Then (x, y*) e o and this product is bilinear: ( n * x r,!* ) : (x r,y * ) + (x r, y * )
(32)
(r, yl + fi> : (x,yl) + Q, yf) (ox, y*) : a(x,y*) : (x, o!*) .
Also the product is non-degenerate. Any linear transformation 24, in 9lt determines an adjoint (or transpose) transformation A* in tJt* such that (33)
(xA,J*) : (x, t* A*)
The mapping 4 -, A* is an associativeanti-homomorphismof 0(tn) into G(IJI*). Now consider the mapping A -, - A*. This is linear and [- A* [A*, B*l : A*B* - B*A* (34) (BA)* - (AB)* - IBAI* - lABl* Hence A -. - A* is homomorphism of @(fi)z into O(!]l*)2. If we
22
LIE ALGEBRAS
now take the resultant of the representation I -.lR determined by llt with A -, - A* we obtain a new representation I -+ r--(l8)* of 8. For the corresponding module llt* we have
(r'Y*t) =Y-':iifl.'
Hence the characteristic property relating the two modules is
(35)
(x,t*l) * (xl,y*):0.
We call the module TJt*definedin this way the contragredientrnodule !lt* of llt and denote the correspondingrepresentation by R*. We recall that if tJt is a finite-dimensional space, thdn there is a natural isomorphism of !|n S llt* onto the vector spadeC(!n). If Xiri I yt e tn S !lt*, then the correspondinglinear transformation in Dt is r -, 2;(r, yt)x;. If tlt is a module for 8, then fi* and !m S IJt* are modules. By definition, (\xa8Yf)t - l,xal SYi * X"o Qc^Yf I. is -(r")n. Then in the If we denotex-'rl by tu then r*-'r*/ to / sends mapping correspondin$ representationin IIt I Dtn the of O(!lt) The elemehts into 2xi8yi'(/')*. }xnlo @yf Err SyI associatedwith these two transformations are A: r__ !,(r, yf )x, . B: x -' )(r, y{)xil" - >il, yf (to)*)*n * \(x,yt)xnl* * Z(*l*,v{)n It is clear from these formulas that B : lA,ln). We can interpret this result as follows: Consideran arbitrary 8-modulefi and the algebra G(9n). If l8 is the representingtransformation,of l, then the mapping X--+lx,l"l is a representationof I actin$ in G. The result we have shown is that this representation is equivalent to the representation in fi 81]t*; that is, the module O is isomorphic to llt I tJt*. The result just indicated can be generalizedto a pair of vector spaces[l,,IJlr. We recall that the set of linear tranSformations O(!ltr,Dtr) of Dtz into lltr is a vector spaceunder usual compositions of addition and scalar multiplication. If the spaces are finite' dimensional, then there is a natural isomorphismsof !)t, I mf
I. BASIC CONCEPTS
23
.f,; € !Itr, Jf eDti' onto @(Iltr,I)lr) mapping the element LriSyf, of Dlz into Dtr. If tnr into the linear mapping y -+ >: fl or k > rt so that we have Cr "' C5WrDr "' Dt assumption to contrary Ex Hence Wro:0 and Wz*€ 2.
Nil weaklY closed seta
The main result we shall obtain on weakly closed systems is the following Tnnonnu 1. Iet 8 be a weakly closed subset of the associ'atiue algebra @of tinear transformations of a rtnfie-dimensional aector space [Il oaer A. Assume euery ]7e S is associatiue nilpotent, that is, Wk :0 for sorne positi.ueinteger k. Then the enaeloping associati,ae algebra tr\* o/ IB fs nilPotent. Proof: We shall prove the result by induction on dim TJt. The result is clear if dim fi - 0 or if [S : {0}. Hence we assume dim Dt > 0, tl$ + {0}. Let Q be the collection of subsystems E of B such that E* is nilpotent and let E be an element of l] such that dim E* is maximal (for the elements of J]). We shall show that Ex : S* and this will imply the theorem. We note first that Then by I, E* + 0. Thus let W be a non-zero element of S. is the : Since and a subsystem is Is n W}* {W}* $* {w}*. 3 { set of polynomials in W with constant term 0, {W}* is nilpotent. Hence g e 9. Since 3 + 0 it follows that E* + 0. This implies that the subspace[t spanned by all the vectors tcB*, r e Tlt, B* € E*, is not 0. Also It + !]t. For otherwise, any x : 2 xoBl, x; e VJt,
U
LIE ALGEBRAS
aI e Et. If we use similar expressionsfor the r;i we obtain r:X yfilCf , Bf , Cl in E*. A repetition of this pfocessgives r : x zrBtrtBtr-.. Bt,, Bfi in E*. since E* is nilpoteni tthis im"plies fi :0, and rlt : 0 contrary to assumption. Hencewg havdllt = ll > 0. Let 6 be the subsetof I of elementsc such that gtcrs tt. Then it is clear that 6 is a subsystemcontaining E. MoreoverG induces weakly closedsystemsof nilpotent linear transformationsin !t and in the factor spacerltlyt. Since dim ft, dim !lt/[ < dirn I]t we may assumethat the inducedsystemshave nilpotent envelodinqassociative algebras. The nilpotency of the algebra in rJtlrt limptiestnat there exists a positive integer p such that if r is an5feLment of Ilt and Cr,Cr, ,.., C, are any C; in 0 then xCtCz.. . Cr e It. Also there exists an integer 4 such that if Dr, Dr, . . ., Doe fu and y e tt then JDr ... Do: 0. This implies that if Cr, ..., Cp+a€6 then CtCr... Cr*o: 0. Thus O* is nilpotent and 6 e g. We can now concludethat 8* : E*. otherwise, by III., there exists w e w, E E* such that B x W e E* for all B in E. By - BII., f lve have for any r-oUl, B*eE*, xBxW-r(WBl+nl), eE*. Hence nWgn so We6,. Since WdE*, dimG*>dimEfr and sinbe E e ll this contradictsthe choiceof E. Hence E* : s*iis nilpotent. If [B is a set of nil triangular matrices, that is, triangular matrices with 0 diagonalelements,then g* is containdain the associativealgebraof nil triangular matrices. The latter is nilpotent; hence 8* is nilpotent. The following result is therefofeessentially just a slightly different formulation of Theorem l. Tsponnu 1'. Let un be as in Theorem1. Then there existsa basisfor ffi such that the matricesof ail the w e g haaenil triangular form relatiae to this basis. Proof: we may assumerlt + 0. Then the proof ot' Theorem 1 shows lhat tn r m8* where tjts* is the space spairned by the vectors fiw*, r € Dl, I,[z*e 8*. In general, if rt is a $ubspacland 6 is a set of linear transformations,then we shall write rt6 for the subspacespannedby all the vectors ys, y e rt, s e 6. Then it is immediate that tn(r8*)' : (DruB*)uBx.Also if tnr8* + 0 the argument used before shows that llt8* ) tn(!B*)t. Hdnce we have a chainIJt= UIIB*:Dt(8*;'=fi(S*), =...=Dt(IB*)x-'l!rJl(IB*)r: 0, if (8t)n : 0 and ([B*)r-t + 0. We now choosea basis (dr, . . ., en)for f i s u c ht h a t( e r ,. . . t n r )i s a b a s i sf o r l l t ( S * ) n - r ,( e r , . . . , 4 n 1 , . . . , € n ; n 2 ) i s a b a s i s f o r l l t 1 8 * ; r - 2 , . . . , ( € r , . . . , € n r + n z + . . .i+snl ap ) b a s i s f o r
N. SOLVABLE AND NILPOTENT LIE ALGEBRAS
35
tJt([B*)*-t Then the relations l]l(UB*)x-]USc tn(8*)nr-&+r imply that the matrix of any W € S has the form
.tu tl
| nz
hl0
ltz :
*
- [r" NN
Theorem 1 can be generalized to a theorem about ideals in weakly closed sets. For this purpose we shall derive another general proper' ty of enveloping associative algebras of weakly closed sets: IV. Izt W be a weahb closed subset of an associatiae algebra and let E be an i,deat in 8, E*, !B* the enueloping associatiae algebras of E and W, respectiuely. Then
(2) (3)
g (a*)*a* + (E*)o, uB*(E*)o (E*)oS*. 8*(E*)o+ (E*)*, g 8*(E*)t , (E*s*)t E (E*)e$* (uB*E*)*
Proof: By II. of $1, the conditionthat Bx W €E for BeE, w es&, implies that E*w s I,l4B*+ E*. It follows that if * E*. Hence W r , W r , . . . , W , e S , t h e n E * W r W z . . .W r Q 2 S * E + E*m*.[B*E* + E*. Inductionnow proves(E*)&IB*gIB*(E*)& + (E*)0. Similarly the condition that S is a left ideal implies that s*(E*)* g (E*)o[B*+ (E*)*. Hence (2) holds. Now (3) is clear for h: t and if it holds for some &, then ([B*E*;r*r : ([B*E*).[B*E* g m*(E*)rgr:8* = [s*(g*E*e + E*o)E* g g*(E*)k+' . Similarly, the secondpart of (3) holds. We can now prove Tnnonpn 2" Izt 8 bea weaklyclosed,set of li,neartransforma.tions acting in a finite-dimensionaluectorspaceand let E be an ideal in W such that euery element of E is nitpotent. Then E*(henceE) l's containedin the radical S o/ S*. Proof: S*E* + E* is an ideal since S*(!B*E* + E*) E S*E* and
LIE ALGEBRA
36
(m*E*+E*Xsxs [B*([8xE*+E*)+s*E*+E*-IB*E*+E*. Also (EB*E* + E*)o = [B*E* + (E*)0. By Theorem 1, E* is nilpotent. On the Hence h can be chosen so that (S*E* + E*)o g [B*$*. : 0. g (S*E*)r so that taken can be I (mn5*), [B*(E*)'and hand other Thus 8*E* + E* is a nilpotent ideal and so it is conitained in S' It follows that E* and E are contained in fi, 3.
Engel's theorem
Theorem 1 applies in particular to Lie algebras. In this case it is known as Engel's theorem on Lie algebras of linpar transfor' mations: If g is a Lie algebra of linear transformatiorls i'n a fi'nite' dimensional aector space and eaery L e g is nilpotenl, then 8* fs nilpotent. The conclusion can also be stated that a I basis exists for the unclerlying space so that all the matrices are nil triangular' These results can be applied via the adjoint reprdsentation to arbitrary finite-dimensional Lie algebras. Thus we have Engel's theorem on abstract Lie algebras. If t, is a,rtnite'dirnen' sionai Lie algebra, then g is nitbotent if and only if ad E is nilpotent fo r euer yae8, . Proof : A Lie algebra 8 is nilpotent if and only if I there exists This an integer N such that I...La&zlasl-..4^n,l:0 fot stet. a's. ^ A / l p r o d u c t contaihs if the impliesttut t...lxalal...al:0 and g. finite-dirt'rensional Conversely, let I be Hence (ad a)tr-t : of I ad set The 8. a e assume that ad.a is nilpotent for every nilof al[ebra Lie is a (acting 8) in linear transformations ad a potent linear transformations. Hence (ad 8)* is nilpotent. This means t hat t her e i s a n i n te g e r N s u c h th a t a d a.ad,as" ' adax:0 for cr e 8. Thus l"'[aarlar1"' an\:0 and so 8rn'=+ 0' we can apply Theorem 2 of the last section tp obtain two characterizations of the nil radical of a Lie algebra' ' Tsuonpu 3. Let g be a fi,nite-dimensional Lie algebfa. Then the nil rad.icat Tt of g can be characterized in the follow\ng two ways: (I) For eaery a e Tl, ad a (acti.ng in 8) es nilpotent and if E is any id,ealsuch that adb (in 8) is nilpotent for eaery beE, then Egtlt. (2) Tt is the set of elements 6 e I such that ad b e th'e radical ft of (ad 8)*. i Proof: for (1) If DeTt and aeg, then [ab]eIt and t"'Lablbil"'bl:0
U. SOLVABLEAND NILPOTENT LIE ALGEBRAS
37
enough ,'s. Hence ad b is nilpotent. Next let E be an ideal such that ad D is nilpotent for every b e E. Then the restriction ad56 of ad D to E is nilpotent and E is nilpotent by Engel's theorem. Hence E g tt. (2) The image ad R of It under the adjoint mapping is an ideal in ad I consisting of ni,lpotent transformations. Hence, by Theo' rem 2, ad It g S. It is clear that S n ad I is an ideal in ad 8; hence its inverse image !t, under the adjoint representation is an ideal in 8. But }tr is the set of elements b such that ad b e S. Every adylrb,b e Ttr, is nilpotent. Hence Tt, is nilpotent and Ttr g It. We saw before that ad Tt s ft so that Tt I Tl'. Hence It : Ttr. 4.
Primara
components.
Weight spaces
For a linear transformation Aina finite-dimensional vector space tjt the well-known Fitting's lemrna asserts that fi:fio.r@fir, where the lltra are invariant relative to A and the induced transformation in lJtoe is nilpotent and in IJt,r is an isomorphism. We shall call fio, and !Il1a, respectively, the Fitting null anq one component of l}t relative to A. The space lJtrr: flLrl)t,4' and IJtga: {zl zAr : 0 for some r}. The proof of the decomposition Hence there runs as follows. We have yIt2YItA2tIItAz 2 "'. tMA' ' : : : IJtre. Let 3t : " TItA'*t exists an r such that ' ', c ' so we have an s such that $, 0}. Then 3, tr 3, {z;l a,{ ' _ max (r, s). Then [Jtot: $6 and lJlra: &*r : " : Tftoe. Let t tlftAt. Let r e IIt. Then tcA' : yAzt for some y since tIftAt :TI\A". Thus x: (x - yA') + yA' and yAt € IJt,e, while (x - yA').4' : 0 so x - yA'€ lJtor. Hence lJt - TJto,r* tlltta' Let z € TJtoefl IJlt'' Then z: wA' and 0 : zA' : roA't. Since u)A" :0, u e Tltsa- $, and - 0; hence sIft- l]to, O Wtr* wAt - 0. Hence z :0 and Dto,ro lJtrr Since TJlsa: gr, A' : 0 in [Ilsa. Since [Itla: 'lnA' : TIIA'*' : TItlaA, 4 is surjective in lJtr,r. Hence / is an isomorphism of llt,e. We recall also another type of decomposition of llt relative to A, namely, the decomposition into pri,mary cornbonents. Here 9 m- Mor a$ I J t o2,e'r ' ' @ T [to ,e w h e re rc i: r;(tr), { rr' (i ),' ' ' , tr,(A )} are the irreducible factors with leading coefficients 1 of the minimum (or of the characteristic) polynomial of A, and if p(,i) is any polynomial, then (4) Tltp.t: {zlza(A)' : 0 for some r} (cf. Jacobson, [2], vol. II, p. 130).
LIE ALGEBRAS
38
is irreducible with leading coefficient one, then IIt",r :0 lf p(l) : 7q(1) : nri for some i. The space Tftpe is evidently invariant unless ft relative to A. The characteristic polynomial of the restliction of A to sto1,r has the form n;(f,)'i and dim Wtnaa: zir; where ril : deg zc(t). If zrr(,i): l, then the characteristic pqlynomial of A ih Dhe is i', so A' :.0 in llh,r and IJhe € lllor. If zdi) + l, the characteristic polynomial of the restriction of. A to Wtn4 is not divisible by I' so "', so this restriction is an isomorphism. Hence [Jtop,:\JtnilA: Since m - !Ito, O $Jtu : Thus Xna+rlltnae E IIts. Wtx&tE lltrr. it follows that IJtor : IJtr,r, Iltre : Ela+^lltoar. In lJh,, O Xoa#r$Jtn6e, particular, we see that the Fitting null component is lthe charac' teristic space of the characteristic root 0 of ,4 and dirh lltoe is the multiplicity of the root 0 in the characteristic polynornial of. A, We shall now extend these results to nilpotent Lie algebras of It would suffice to consider the primary linear transformations. decomposition since the result on the Fitting decompdsition could be deduced from it. However, the Fitting decomposit[on is appli' cable in other situations (e.g., vector spaces over division rings)' so it seems to be of interest to treat this separately. We shall need two important commutation formulas. Let lI be an associative algebra, a. an elertrent of lI and consider the innef derivation Dlx->tct= [rc] in lI. We write xtt-(vtk-rt)', xrc): iv Then one t proves directly by induction the following formulas:
(5)
xah: akr* (l)r-'-'
* (t)ao-'r" + "' + xtk'.
(6) 'We
apply this to linear transformations to prove
Lpuru.r,1. Let A and.B belinear transformationsin a'fi'nite'd'im'en' intpger N such sional aector sfage. Sufrbosethere existsa bosi.ti'ae fioe, fire that l. . .LlB'AlAl . . . al : O. Then the Fitting componefu.ls of 'm rebti'ue to A are i.naari.antunder B. -N*m-L P r o o f : S u p p o sxeA n : 0 . T h e n f . o rk
rBAh- ,(eon. (f )or'"'+ "' * (r1 ,)ro-"*'r'r'-") 0 ' HencexBelJhr.Nextletr€Ill,r.Iffistheintegerusedin
II. SOLVABLE AND NILPOTENT LIE ALGEBRAS the proof of Fitting's Then
lemma, then we can write
39
tt = yA'*n-t-
- r)B' . A'**-' xB -- r4t+nts - t(ae,*Ir-r - ( i
+ ...* (;1;
t ) r ' ' - " o ' ) e r I t A ' 0n: r e
Hence IftroB E[ftrt. Tsponpu 4. Izt 8, be a nilpotent Lie algebra of linear transfor' mations in a f,nite'dimensional aector space fll and let Tfto: fl ae8[loe, Then the tfti are inaariant under 8 (that is, under Wt1: OL'Ilt(8*)c. eoery B e 8) and Wt - mo CI9]tt. Moreouer; fi1 : XeeB!]tra. Proofz Suppose first that IJt : IJtoe for every A e 8. Then Tltla:0 for all A and XUlr,r : 0. Also by Engel's theorem (8*)t :0 for some N. Hence Dtr : O L'fi(8*)t : 0. Thus the result holds in this case. We shall now use induction on dim llt and we may assume Wtol * Ilt for some A. By Lemma 1, the B e I map lltor into itself; hence 8 induces a nilpotent Lie algebra of linear transformations in St : l}toe c lll. We can write gt : Ilo O ltr where lto : 9to and llr are invariant 11,6gTtor, Itr : fl tlt(l!*)d : IaeBTtra "rd It is clear from the definitions Ultr* tt' Tft tto 8. Then under O O that llo: Illo: naeglDtoa and Il, * Ilt,e S.Xaegllt,r g nrDt(8*)d. On the other hand, by Engel's theorem the atgebra induced by 8* in llto is nilpotent so that we have an N such that llt(8*)t :0. Then tn(B*)t:
Dto(8*)'+ Il,(8*)t + IJl,n(8*)' g !t,(8*)' * rll,r(8*) g Tt' * !)1,,r .
Hence Ilr * Dlrr: Xleegtlt,a: 6rDt(8*)i and the theorem is proved. Our discussion of the primary decomposition will be based on a criterion for multiple factors of polynomials. Let Ol]1 denote the polynomial ring in the indeterminate i with coefficients in O. \{e define a sequenceof linear mappings Do - l, Dt, Dz, .'. in O[i] as follows: D; is the linear mapping in @[,1]whose effect on the element ^i in the basis (1, l, f,',...) for 0[t] over @ is given by
(7) where we take (l) :0
f,tDr - ( "
if i
T h u s w e h a v e A i D r:i l i -t,
so
LIE ALGEBRAS
that D, is the usual formal derivation in All'l. Also ^i Do,j (j - 1) . . . ( j - i + l )i ' -d :i l (to )i l -n - i l A i D ;. H ence i f thecharacteristic is zero, then D.i : $lit)D6t If d(f) e O[i] we' shall write $e = 6Dr. Then we have the Leibniz rule:
(8)
k
@0* - E 6;Q*-r. I=O
It suffices to check this for 6 : Then
f,i, Q :
f,i' in the b a s i s( 1 , , 1^,' , " ' ) .
-o*' *, onp*-,:* (t,) (u'- ;)il-']o'
: (t (t)G'-o))^'."'-r ' Since >f=r?)G|n) : (t+t'\ the abovereducesto (r+l')fi|'-h : (li+t'rr. Hence (8) is valid. We can now prove 2. If q(l)o*'Ip(t), then T()')| p;(i), i : 0,t,2, 1" , k. Luurrr.e, -Proof: This is true f.ot k: 0 since ps(t) : p('i). \iVrite p(^) Then we mav assurriethat 41p1, d(f),r(i) where TklO and vl{. j : 0 , ! , 2 , . . . , k 1 , a n d I l * 0 . T h e nn l @ * ) * = p x b y ( 8 ) . Vlfu, and multiply (5) bV d'rcand sumlon &. This Let r/(i) I,;l,--sat,lft glves (9)
x Q @ ) - < P ( a ) r + Q , @ ) r+ ' Qr@)r" + "'
* dttr"' ,
which we shall use to establish Lpuun 3. Let A, B be li,nPar transformations in a tfi'nite'dimen'
E 0 for some sional aectorspacesatisfying B'Nt: t. .-1Bm:iJ N. Let AD be a fotlnomial and let T[tp,t: {ylyp(A)^',:0 for some mj. Then lltpe as inuariant under B. --0. Proof: Let y € ffi*, and suppose ys(A)^ p( ^ )* N . T h e n ,.b y (9 ), {(^ ) : qQ ) *: B * ( A ) - 4 , ( A ) B+ , P , ( A ) B ' + ' ' '
Setl z(i) : P(1)*'
* *n-,(A)Bq-l) .
By Lemma 2, u1Q)l{{1),i = t't - t- Henceyhi@) -P, i = Nand yBlt(A) = 0. HenceYB e Wno.
1,
Tnponpu 5. Iat 8, be a nilpotent Lie algebra of Iiloear transformations i,n a' fi.nite'dimensi.onalaector space tn' Then we ca.n
il. SOLVABLE AND NILPOTENT
LIE ALGEBRAS
4I
decompose[!t: ![t, o!]t, o ... o IIt, where TJ?iis inaariant under B and. the mini,mum potynomial of the restriction of eaery A e 8' to TIti i.s a prirne pou)er. Proof: If the minimum polynomial of every 4 e 8 is a prime power, then there is nothing to prove. Otherwise' w€ have an ,4 e 8 such that ![t - IJt"r,aO ''' O ffinre, tti,: r+(l) irreducible, Wtone * 0, s ) 1. By Lemma 3, IftnreB ?Wn# for every B e 8. This permits us to complete the proof by induction on dim fi. If the base field is algebraically closed, then the minimum polynomials in the subspaces IItr dr€ of the form (^ - a(A);tir, A € 8. Setting Z;(A) - A.- a.(A) for the sPac€ Dt; we see that Zi(A) is a nilpotent transformation in IJI;. Hence every A e 8, is a scalar plus a nilpotent in lJti. We therefore have the following Conor,urnv (Zassenhaus). If 8, is a nilpotent Lie algebra of linear transformati,ons in a fi,nite-dimensi.anal aector space oaer afi alge' braically closed field, then the space lJt can be decomPosed,ts fi, O tjt. O . . . O T!t, where the Ylt; are inaariant subspa.cessuch that the restriction of eaery A e 8 to IJti is a scalar plus a nilpotent linear transformation. Consider a decomposition of IJt as in Theorem 5. For each i and each A we may associate the prime polynomial ne(]) such that nin(f,)kiais the minimum polynomial of A restricted to lltr. Then the mapping tra:A -> nu(f,) is a primary function for 8 in the sense of the following DonNtnoN 3. Let 8 be a Lie algebra of linear transformations in IJt. Then a mapping n: A-+rt(A), A in 8, rt()), a prime polynomial with leading coefficient one, is called a primary functi,on of. !)l for 8 if there exists a non-zero vector r such that xn/A)^'"e' - O for every ,4 e 8. The set of vectors (zero included) satisfying this condition is a subspace called the primary component W" corre' sponding to tt. Using this terminology we may say that the space llti in Theorem 5 is contained in the primary component Dt,,6. By adding together certain of the IJtr, if necessary, we may suppose that if i + j, then tt;i A + z;,r(l) and n7:.A -->ntt(A) determined by fi; and fiy are distinct. We shall now show that in this case IJtr coincides with IJto,and the n are the only primary functions. Thus let r A+rla(l) be a primary function and let x € lll,,. Assume n * r;, so that
LIE ALGEBRAS
42
l
there exists A; e Ssuch that reo(]) + tt;,t{}). Write x: lr+ "' { rrt h e T)ft* Since x e t[Itn there exists a positive integer fe such that 0:
X r , t t ( A i ) ^ : X g c t r ( A ; )+^ . . .
1y,r1n(A;)^',
Since the decomposition Yt - Dt' 0 .'' O Un' is direct, we have It * tr;an(]) this implies ' rc;:0. ranar(Ai)*: Q and since TrAr(^) c o n t rat h i s a n d f o l l o w st h a t i f n * r i f o r i : 1 , 2 , , . - , f , t h e n r : 0 the zt are thei Thus primary function. a dicts the definition of only primary functions. The same argument shows that if tc etlll,,t, then x € lJtii hence IJti : SJt,,c. The argument just given was based on the following itwo properties of the decomposition: Ifti tr TJtn,wher€ rct iS a prim{ry function and ni * rt if. i + j. The existence of such a decomposition is a consequence of Theorem 5. Hence we have the following Tnponpu 6. If g is a nil4otent Lie algebra of linear:transforma' tions in a finite-d.imensi.onalaector space W, then B has Qnly a finite nurnber of primary functions. The corresponding primary compo' nents are subrnod.ulesand. Tft is a direct sum of these. Moreouer, if of ' masadi rect sum a ,n ! d e c o fn p o T i t i on !n - I J t , olJ t r @. . . @ D t,.i s (L) for' each i the that of subspac7sIftt + 0 inuariant under B such A e 8l to TIt; is a euery of m,inimuru fobnomi.al of the restriction an 'F such that eri'sts (2) there i power if + pri.me ru())^io and i, " r,r, are the i :L,2, A' n ;o ()), m a p p i n g s th e r;A) ) * r c ia( ) ) t, h e n primary functions and the TIt; are the corresponding prtimary cofn' ponents. It is easy to establish the relation between the primary decom' position and the Fitting decomposition: IJto is the primary component ln^ it A--+ t is a primary function and is 0 otherwise; mt is the sum of the primary components IJt,,, r * ),. We leaVe it to the ' reader to verify this. We shalt now assume that the nilpotent Lie algebra 8 of linear transformations has the property that the characteristic roots of every A e g are in the base field. A Lie algebra of linear transformations having this property will be called split. Evidently, if the base field @ is algebraically closed, then any Lie algebra of linear transformations in a finite-dimensional vector space ovdt A is split. Now let 8 be nilpotent and split. It is clear that the characteristic polynomial of the restriction of A to the primary component correis of the form ttn())'. Since thi$ is a factor sponding to A-rn(l)
il. SOLVABLEAND NILPOTENT LIE ALGEBRAS
43
of the characteristic polynomial of. A and zrr(t) is irreducible, our - ] - d.(A), a(A) e A. Under these hypothesis implies that no}) -+ z,r(,1)by ciriumstances it is natural to replace the mapping A following -+ the formulate to O and a(A) of B into the mapping A Dpr.rNruoN 4. Let 8 be a Lie algebra of linear transformations Then a mappin g a: A --+a(A) of 8 into the base field O is in fi. called a weight of IJt for 8 if there exists a non-zero vector r such (zero ttult r('A _ a(A}I)mx,A :0 for all .A e 8. The set of vectors included) satisfying this condition is a subspace fllt" called the wei.ght sface of llt corresponding to the weight a' Theorem 6 specializes to the following result on weights and weight spaces. Tnronpu 7. Let I be a s|li,t nilpotent Lie algebra of linear transformations in a rt,nite'd.imensional aector space. Then t has only a fi,ni.te number of distinct weights, the weight spaces are submodules, and. m l's a direct surn af these. Moreouer, let tm into subspaces m - !lt, o llt, o . . . o Tft, be any decomposition of llh ;e 0 inuariant under B such that (l) for each i, the restriction of any A e g has onl! one characteri.sticroot at(A) (with a certain multipilctty) in Tlti and (2) if i + i, then there erists an A e 8 such that Then the mabfings A -. a;(A) are the wei'ghts and in@) + a{A). the spaces Tft;,ara the weight spaces. 5.
Lie algebras with semi'simple enueloping assoeiatioe algebraa
Our main result in this section gives the structure of a Lie alge' bra 8 of linear transformations whose enveloping associative alge' bra 8* is semi-simple. In the next section the proof of Lie's theo' rems will also be based on this result. For all of this we shall have to assume that the characteristic is 0. we recall that the trace of. a linear transformation A in a finite'dimensional vector space, which is defined to be >ifau, for any matrix (a;) of .4, is 'i' the sum of the characteristic roots Pb : !, ' ' ' t fr, of A' Also p; 0, so tr Ab :0, : the all tr Ak Z?=rpt. lf. A is nitpotent "r€ .. converse holds: . . O is 0, the :1,2, of characteristic If the h w e have Thus : 1, 2, n i l p o te n t' : 0, A i s th e n k "', If tr Ak
The formulas (Newton's identitiesi cf. 1pt -0, k:L,2, p. 110) expressingXpi in terms of the elI, vol. Jacobson,fzl,
LIE ALGEBRAS
ementary symmetric functions of the p; show that [f the characteristic is 0 our conditions imply that all the elementdry symmetric functions of the p; 0, so all the p; are 0. flence A is "te nilpotent. We use this result in the following ] Lslrul 4. Let C e 8,, the algebra of linear transforrh.ati,ons in a rtnile'dimensi.onal uectar sfuce oaer a field of characteri.stic 0 and s u p p o s eC : X i [ A i B r , l , A r , B i e O a n d l C A i l = 0 , i * L , 2 , . . . , r . Then C is nilpotent. ( C o- 1 ) . i H e n c eC k : Proof: We have lCk-'A;l:0, k:L,2,... ZiCk-'(AtBn - B;A): Xr(A;(r*-tB;) - 1g*-rg)A): )ilAi, ,*-tBil. Since the trace of any commutator is 0, this gives tr Ce - 0 for h : L, 2, . . - . He n c e C i s n i l p o te n t. The key result for the present considerations is the following Tnponpu 8. Let 8, be a Lie algebra of linear transf$rmations in a finite-dimensional uector space oaer a fi.eld of char',acteristic 0. Assume that the enueloping associ.atiae algebra 8* is semi-simfle. Then 8 :8' @6 where 6, is the center of 8, and 8,, is an ideal of 8 which is semi-simple (as a Lie algebra). Proof: Let 6 be the radical of 8. We sho* first ithat 6 : 6 the center of 8. Otherwis€, 6, : [89] is a non-zero solvable ideal. Suppose 6{u) : 0, 6lo-t) + 0 and set 6z: glu-t', 6s F [g;r8]. If C € 6 ' , C : E l A i B ; . ] , A i € 6 2 , B i € 8 , a n d [ C A r ] : 0 s l n c e9 ' t r € z and 6, is abelian. Hence, by Lemma 4, C is nilpofent. Thus every element of the ideal €a of I is nilpotent; hencd, by Theorem 2,6g is in the radicat of 8*. Since 8n is semi-sinple, €s - Q. Since €s : [gr8] this shows that 6, tr 6. Since 6r E [86] g 8' every element C of. 6' has the form l,[A;B;1, Ai, Bi f 8, and we have [CA;l: 0, since 6s g 6. The argument just used limplies that 6z:0, which contradicts our original assumption that €[ : [89] + 0. Hence we have [89] - 0, and 6 : O. The argument wb have used twice can be applied to conclude also that 6 n 8' : 0. I Hence we can find a subspace 81 3 ,8' such that I : I, O 6. Slnce 8, contains 8', 8, is an ideal. Also 8r:8/6 : 8/6 is semi-sitnple. This concludes the proof. Remark: In the next chapter we shall show that 8r e 8', so we shall have I : 8'€) G, 8' semi-simple. Conorulnv 1. I-et I be as in the theorem, g* serni-s;npie. Then 8, i,s solaable if and only if 8 ds abelian. More generflly, if 8 r's
II. SOLVABLE AND NILPOTENT LIE ALGEBRAS
45
g*, then ll*lm ls a commutatiae soluable and ft is the rad.ical af algebra. 8* Proof: If I is abelian it is solvable, and if 13is solvable and the Since : semi'simple' is G O 8, where 8, is semi-simple, then I have only Lie atgebra which is solvable and semi-simple is 0 we it is statement prove second the : To O is abelian. 8r :0 and I as I consider and slightly point view of the convenient to change over algebra associative finite'dimensional U a ?[r, a subalgebra of a field of characteristic 0. since any such 1l can be considered as a subalgebra of an associative algebra @ of tinear transformations in a finite-dimensional vector space, Theorem 8 is applicable to I and lI. We now assume I solvable, so (S + m)m, which is a homomorphic image of 8, is solvable. Moreover, the enveloping associative algebra of this Lie algebra is th.e semi-simple associative Hence (S + lt)/lt is abelian. This implies that algebra 8*/n, (b + S) (a + S) for any a,b e 8,. Since the cosets ta1 ml (b + n) a * ft generate 8*/n, it follows that 3x/S is commutative. conou,.lny 2. Let g be a Li,e algebra of li,near transformati'ons i,n a fi,nite-d,imensional uector space oaer a field 9f characteristi'c 0, let 6 be the radical of g and, ft the radical of 8*. Then I n S is the totality of nit\otent elements of 6 and tgSl g ft' Proof:, Since S is associative nilpotent it is Lie pilpotent' Hence M o r e o v e r , t h e e l e m e n t s oS f arenilpotent so I nStr9o I nftg6. g. If So denotes the radical of of elements nilpotent of set the the enveloping associative algebra @* then, by Corollary L,6*/So is commutative. Now any nilpotent element of a commutative algebra generates a nilqotent ideal and so belongs to the radical. Since g,*/ffi0 is semi-simple, it has no non-zero nilpotent elements. It follows that fro is the set of nilpotent elements of 6* and so 6o : fro 0 6 is a subspace of 8. Next consider (8 + S)/S. The enveloping associative algebra of this Lie algebra is 8*/m, which is semi-simple- Hence the radical of (8 + m)& is contained in its center' Since (g + ft)ffi is a solvable ideat i. (8 * fr)/ft we must have
g l(6 + S)/m,(s + mynl : 0, which means that t68l S.
Hence
Since tgsl g I n S E 60. Then [908] S 6o and 6o is an ideal in 8. its elements are nilpotent, 6o S S by Theorem 2. Hence 6o I I n S and 6o : I n S, which completes the proof. There is a more useful formulation of Theorem 8 in which the hypothesis on the structure of 8* is replaced by one on the action
46
LIE ALGEBRAS
of I on Dl. In order to give this we need to recall sdme standard notions on sets of linear transformations. Let 2 be a set of linear transformations in a finite-dimensional vector space IJI over a field O. We recall that the collection L(t) of subspace$ which are invariant under 2 (nA=n,A eJ) is a sublattice of the lattice of subspaces of llt. We refer to the elements of.L(t) aE Z-subspaces of llt. If Il is a subspace then the collection of A e1E such that nA E Tt is a subalgebra of @. It follows that if 9t e L(2), then 9t is invariant under every element of the envelopingi associative algebra .E* of J and under every element of the enveloping algebra ^tt. Thus we see that L(E): L(2*) - L(lt). The set J is called an irreducible set of linear transformations and !]t is called J-irreducible it L(2) - {D[, 0] and IJt + 0. J is ind,ecomposableand lll is J-indecomposable if there exisits no decomposition tlt - Dt, O IJtz where the IJL are non-zero elemEnts of L(2), implies indecomposability. , J (and lyt Of course, irreducibility relative to J) is called completely red,ucible if !n - > m[, Dto € Ut) and llt" irreducible. We recall the following well-knolwn result.
Tnronnu 9. 2 is completelyreducibleif and only if iL(Z) is complemented,that i,s,for eaeryslte L(2) there existsanll'e L(E)such that Tft': ItOtt'. If the conditionholdsthenTft: IIlr OIJt, O . . . Ofi, where the Tfti,e L(2) and are ineducible. Proof: Assume m - ! l)la, Tft, irreducible in L(J) and let It e l,(J). If dim It : dim fi, Il : fi and IJf : It @'0. We now supposedim It < dim !)l and we may assumethe theoFemfor subspaces!t, such that dim It, ) dim It. Since It c Dl : [ !]la there is an llt" such that Tft,EIt. Consider the subspaceDd"n 9t. This is a J-subspace of the irreducible J-space Tllr. Hence either IlloflIl=Dt" or IJl"OIl:0. If m"OIl:!Ilr, ftSfi"contraryto We andltr:lt*IIfa:fte)Dt". a s s u m p t i o n .H e n c el l l , n I l : 0 can now apply the induction hypothesis to conclude that Un-ytOTtl whereTt': tt,Oytl, Ytie L(J). Then tJt -nel]t"Ottl completnented. Let is L(E) !D1"O ni e L(z). Conversely, assume Itrt, be a minimal element + 0 of. L(2). (Such elemen{sexist since dim l]t is finite.) Then we have m - Ilt' O It where ltle Z(t). We note now that the condition assumed for St carrie$ over to St. Thus let S be a J-subspaceof !t. Then we can writelllt : POP" Then, by Dedekind's modular [aw, $t : !]t rt 9t : $' e l(g). S + (S' n It) since lt = tS. If we set !p" S' n !t, thbn $" n S:
LIE ALGEBRAS il. soLvABLE AND NILPOTENTI
47
Hence!t:$OS" whereS"€ L(2)and$"9Tt' S nS'nIl-0. 'we can now repeat for It the step taken for Ilt; that is, we can writelt:firO$wherelJts,PeL(')andlJteisirreducible' of Continuing in this way we obtain, becauseof the finiteness ' ' ' (E irreducible' Dti fi", Dt, m, @ O dimensionality, that tn fir € L(r). This completesthe Proof' we supposenext that J - lI is a subalgebra of the associative algebra ri (possibtynot containing 1), and we obtain the following necessarycondition for complete reducibility' Tnponpu 10. If \I is an associatiaealgebra of linear transforma' tions i,na f.nite-dimensi,onalaectorspace,then lI completelyreducible that \' is semi'simfle. implies -Proof: - ) Dlta,'It" e L(U) Let S be the radical of ?I and supposetlt and irreducible. Considerthe subspaceDtdn spannedby the vectors contained of the form tN, t e IIlr, N e ffi. This is an ?I-subspace (cf' the cIIl' tltd'n have in !Jt". Since ft&: 0 for some ft we must that conclude can we irreducible SJt" is proof of Theorem t). Since 0, lltlt: that -implies this !n xut" 0 for every !Itr. since mrn :0. semi'simple' l[ is Hence that is, S Since ^I is completely reducible if and only if 2* and 2I are completelyreducible, we have the Conor,r,lny. If t i,scompletelyreducible,then Ex and 2t are semi' simPle. We shall say that a single linear transformation A is semi'simple prime if the minimum polynomial p(],) of.A is a product of distinct that condition the {A}l io polynomials. This conditionis equivalent : p(r) "'tr,(f,)" n(l)ettr(l)" if has no nilpotentelements+ 0. Thus } L' then Z: n{A)rr(A) "' 78,(A)is nilpotent and Z + 0 if someet Let L. 0t: all primes and distinct ?itfTt, conversely, supposethe : is divisible :0, and 0 d(,i)" Z' If : 6(4)' Z 0(A) Ue niipotent. We by pQ). Hencg d(,i) is divisible bv p(i) and Z A(A) 0' can now prove Tnponnu 11. Let I be a completely reducible Lie algebra of linear transformations in a fi.nite'dimensionaluector space ouer a : G O I' where 6' ,s the center field, of characteristic0. Then I g, and is a semi-sim\le ideal. Moreouer,the elementsof E' are semi' simflq. Proof:
If t3 is completely reducible, 8* is
Hence
48
LIE ALGEBRAS
the statement about 8 : G O 8, follows from TheorLm 8. Now suppose that 6 contains a C which is not semi-simple.J Then {C}t contains a non-zero nilpotent N. Moreover, {C}t i$ the set of polynomials in C, so that N is a polynomial in C. Hence N is in the center of 8t. Since this is the case, N8t : 8tN is an ideal in 81. Moreover, (N8t)* g N&8t : 0 if ft is sufficiently llarge. Since N e N81, N8t is a non-zero nilpotent ideal in 8t corltrary to the semi-simplicity of 81. We shall show later (Theorem 3.10) that the conYerse of this result holds, so the conditions given here are neces$ary and sufficient for complete reducibility of a Lie algebra of linpar transformations in the characteristic zero case. Nothing like this can hold for characteristic p + Q (cf.. S6.3). We remark also that the converse of Theorem 10 is valid too. For the associative case one therefore has a simple necessary and sufficient condition for com' plete reducibility, valid for any characteristic of the field. The converse of Theorem 10 is considerably deeper thani the theorem itself. This will not play an essential role in the sequel. 6.
I'ie's theorems
We need to recall the notion of a composition serie$ for a set J of linear transformations and its meaning in terms lof matrices. We recall that a chain ffi - 9Jt,) [ft2) "' r Dt, r fir*, :0 of Jsubspaces is a composition series for IJt relative to 2 if for every i there exists no llt' e L(t) such that Dti ) m' r lJtr+r. If It is a Jsubspace of IJt, then J induces a set E of linear tralnsformations in fi/It. As is well known (and easy to see) for groups with operators the i-subspaces of !]t - Wtll,t have the form S/![ where S is a J-subspace of tlt containing Yt. It follows that lthere exists no P e Z(.x) with IJI > p = It if and only if llt/It is f-irreducible. Dt, eL(t), is a composition H e n c et J t - I l t r ) T f t z ) . . . - ) I J l e a l : 0 , series if and only if every IJln/IJtr*,is i;-irreducible, 5; the set of induced transformations in lJtrllJti+r determined by the A e 2. The finiteness of dimensionality of TJt assures the efistence of a composition series. Thus let llts be a maximal J'subsface properly contained in IJtr : IJt. Then IJt, ) IJlz and fi'/TJtz i$ irreducible. Next let IJt, be a maximal invariant subspace of fi2, * I)ts, €tc. This leads to a composition series IJt, :IJt, r IJtsr "' = F' : TJtr*':0 for St.
II.SOLVABLEANDNILPOTENTLIEALGEBRAS49 b e a d e s c e n d i n gc h a i n o f Let lll-Uft, I1ltrrI]ts3 "'::Ilt'*r:0 . .,0r) . TJtsuch that (et, " ' , en') for basis a be (er, J-subspacesand let Then it is a basis for )ltr, (er, . . .,0n1+n2)is a basis for Tjte-r,etC. (e;) of the is to relative A of matrix the 2, is clear that if A e form
M, M_
(10)
(11)
t
I
,Mz It
Thus
0-l *
I
MrJ
we have €n1+...+n1-peA:+uf),enr*...*n1-1+t*x, k,l:L,"',tli,
Hence we have (10) with Mt: r is a vector in fir-ia2. k: L, "',?li Q t f l l . T h e c o s e t s 7 n 1 + . . . + n J - r +=r z n t + . . - + * J - 1 +*r l l t s - r + s r form a basis for the factor space 1]tr-r+,/Ilts-r'+r zlod (11) gives
where
(12) which shows that the matrix of the linear transformation 24.induced in Ijts-r+r IAU-i+z is Mi. We shall now see what can be said about solvable Lie algebras of linear transformations in a finite'dimensional vector space over an algebraically closed field of characteristic zero. For this we need the following Lpuu.r, 5. Let g be an abelian Lie algebra of linear transformations in the fi,nite-dimensional space tJt oaer O, O algebraically closed' Suppose !)t ls g-irreducible. Then IJt ds one-dimensional. Prod: If A e g, A has a non-zero characteristic vectot x. Thus xA: a.x, d e O. Now let lft" be the set of vectors y satisfying this equation. Then if Be8 and y€!It', (yB)A: yAB: eyB. Hence yBe\!tr. This shows that ')to is invariant under eYery Be 8' -sitr." fi is irreducible, fi : Dt", which means that A: aL in fi. Now this holds for every A e 8. It follows that any subspace of Dt is an 8-subspace. Since TJt is 8-irreducible, l)t has no subspaces other than itself and 0. Hence lJt is one-dimensional.
50
LIE ALGEBRAS
We can now prove Lie's theorem. If 8 is a solaable Lie algebra of lintfor transformations in a finite-dimensional uector space Tft ouer an filgebraically closed feld of characteristic 0, then the matrices of 8, dan be taken in simultaneous triangular form. Proof: Let llt - tlt' I IJtz3 . -. I l)t,+r :0 be a composition series for Dt relative to 8. Let E; denote the set of induced liinear transformations in the irreducible space TItJ)!l;*r. Then Er iS a solvable Lie algebra of linear transformations since it is a hpmomorphic image of 8. Also Er is irreducible, hence completely reducible in IJt,/IJln*,. Hence Er is abelian by Theorem 11. It follo*s from the lemma that dimlJtr/Ij?;*r - 1. This means that if we use a basis corresponding to the composition series then the matrices -1lr in (10) are one-rowed. Hence every M corresponding to the A e 8 is triangular. If 8 is nilpotent, it is solvable, so Lie's theorem is applicable. We observe also that Lemma 5 and, consequently, Lie's theorem are valid for any field of characteristic 0, provided the cl,aracteristic roots of the linear transformations belong to the field. We have called such Lie algebras of linear transformations split. If we combine this extension of Lie's theorem with Theorem )7 we obtain Tsponpu 12. Let L be a split nilpotent Lie algebra of [,inear trans' formations in a finite-dimensional uector space IIt oue* a rteM of characteristic 0. Then Wt is a direct sum of its weight spaces Wla and the matrices in the weight space Dlln can be taken shnultaneously in the form a(A) (13)
A-:['.'
:l
Proof: The fact that Dt is a direct sum of the weight spaces Wt, has been proved before. We have x'(A - a(A)l)* : g for every ra in llt", which implies that the only characteriistic root of A in IJl, is a(A). Since the diagonal elements in (10) are charac' teristic roots it follows that the Mi in (10) for A are al(A). Hence I we have the form (13) for the matrix of A in Tlt".
II. SOLVABLEAND NILPOTENT LIE ALGEBRAS
51
The proof of this result (or the form of (13)) shows that in Dlo a(A)x, A e 8,- If there exists a non'zero vector .r such that rA: (a(A) a(A a(B))x: * B)x, x(pA) * A, B € 8 we have x(A * B): (a(A)a(B) : a(B)a(A))x p A, xlABl pu(A)x a(p(A))x, e and a is a linear that imply of equations first two these a(lAB))x. The 8' is a algebra derived of the element Since every 8. function on linear combination of elements [AB], the linearity and the last condition imply that a(C): g, C e 8'. We therefore have the following important consequenceof Theorem 12. Conorr,nnv. Under the same assumpti'ons as in Theorent' L2, the weights a: A -, a(A) are linear functions on 8, which uanish on 8' 7. Applications to abstraet Lie algebras. Some eounter examples As usual, the results we have derived for Lie algebras of linear transformations apply to abstract Lie algebras on considering the set of linear transformations ad 8. We state two of the results which can be obtained in this way. Tnponsu 13. Let I be a finite-dimensional Lie algebra auer a field of characteristi,c0,6 the radical, and Tt the nil radical. Then [8s] g Yt. Proof: By Corollary 2 to Theorem 8, [ad I, ad 9] is contained in the radical of (ad 8)*. This implies that there exists an integer .A/ such that for any N transformations of the form fad a;, ad srl, a i e 8 , s ; € 6 , w e h a v e l a d a r , a d s t ] L a d a r ,a d s z l. . . l a d a 1 y , ? d s r ]: 0 . Hence ad [ars,]ad lazsz].''ad [assn'] : 6. Thus for any r € 8 we have 1...llxlars,ll[azszll. . . ,Iars*]J : 0. This implies that [68]rv+r0, so [68] is nilpotent. Since this is an ideal, [98] g tt. Conoruny 1. The deriued algebra of any finite-dimensional soluable Lie algebra of characteri,stic 0 is nilpotent. Proof: Take 8 : 6 in the theorem. Conoruo.ny 2. Let 6 be solaable finite-dimensional of characteristic 0, It the nil radical of 6. Then 6D = rft for any deriaation D of TE\
Proof: Let I - gO O D the splitextension of.OD byG (cf. $1.5). Here g is an ideal and [s, DJ : sD for s e 6. Since 8/6 is one-
52
LIE ALGEBRAS
dimensional, it is solvable. Hence 8' is a nilpoten{ ideal in 8. On the other hand, g' : @' + 6D, so g' + @D is a nllpotent ideal in I and €/ + eD g tl. Hence gD = ,t. Tnponpu L4 (Lie). Let I be a fi,nite.d,imensionalisolaable Li,e algebra oaer an algebraically closed fi.eld of characterktic 0. Then such that t h e r e e x i s t sa c h a i n o f i d e a l s 8 : , 8 , ) 8 o - r r " ' : , 8 t ] 0 dim 8; : i. Proof: Since I is solvable, ad I is a solvable Lle algebra of linear transformations acting in the vector space 8. Let I g, =,82 r . . . r,8r+r : 0 tle a composition series of I rel6tive to ad 8. Then 8; ad 8 s 8i is equivalent to the statement that $; is an ideal. By the proof of Lie's theorem we have that dim 8r/8r+l : 1. Hence the composition series provides a chain of ideals of the type required. We shall now show that the assumption that the bharacteristic is 0 is essential in the results of $$ 5 and 6. We begin our construction of counter examples in the characteristic p + 0 case with a P-dimensional vector space lJt ovel O of. characteristic 1. Let E and F be the linear transformations in Dt whose effdct on a basis (er,er, "'r op) is given bY (1 4)
i S, which is what we wished to prove PnoposnroN2. Izt 6 be a nilpotent subalgebra of the finitedimensionalLie algebra 8 and let 8: ,80O 8, be the Fltting decomposition of 8, relatiue to ad 6. Then ,80is a subalgebrabnd [8180]g a
,.;1. k
ProofzLeth e Oanda€ 80. Thent...noffilJ h.
*0forsome
Hence [ . . - [ a da a d h ] a d h l . . - a d h l 0 .
This relation and Lemma 1 of g 2.4 implies that the Fitting spaces 8o o and l]r a 0. We remark also that I is of rank 0 if and only if every ad h is nilpotent. By Engel's theorem this holds if and only if 8 is a nilpotent Lie algebra. Regular elements can be used to construct Cartan subalgebras, for we have the following Tnnonnru 1. If g is a finite-dimensionat Lie algebra ot)er an in' Fitting null fi,nite field 0 and, a is a regular element of 8, then the g subalgebra. is a cartan a to ad relatiae cornbonent 6 of Praof: Let I - 6 O S be the Fitting decomposition of I relative g S. to ad.a. Then, by Proposition 2, 6 is a subalgebra and t6nl We assert that every ad6b, b e 6, is nilpotent. Otherwise let 6 be We choose a an element of 6 such that ad5} is not nilpotent. basis for I which consists of a basis for 6 and a basis for S. Then the matrix of any ad h, h e S, relative to this basis has the form
(1)
,9) $) \ o (p)l
where (p,) is a matrix
(2)
of. ad,6h and (pz) is a matrix of adgh. Let
' , : (t''J(-,3) , B:(u'; .u,3)
respectively be the matrices lor ad a and ad D. Then we know that (az) is non-singular; hence det(ar) + 0. Also, by assumption' (F,) is not nilpotent. Hence if n - / is the rank then dim 6 - I and the characteristic polynomial of (F,) is not divisible by,lz. Now let f,, il, I be (algebraically independent) indeterminates and let F(1, p,v) - pA - vB). We be the characteristic polynomial F(f,, pt, y) : det(,ll where u) have F(1, pt,u) F1(i, 1t,v)Fz(],p, Fi(f,,P, v) : det(,il - P(e) - v(Fil ' We have seen that Fz(l,L,0) : det (r1 - (as)) is not divisible by A and R(,t,g, 1) - det(tl - (0,)) is not divisible by 'lt' Hence the highest power of t dividing F(1, p, r,) is l.t', l' < l. Since dl is infinite we can choose Fo, uo in @such that F(t, po, vo)is not divisible Set c : poa + vab. Then the characteristic polynomial by i,'*t. det (il - ad c) : det(,il - thA - voB) F(A, Po,vo) is not divisible by S,r'+t. Hence the multiplicity of the characteristic root 0 of ad-c is l' < l. This contradicts the regularity of. a. We have therefore proved that for every b e 6, adgb is nilpotent. Consequently, by Engel's theorem, O is a nilpoteni Lie algebra. Let 8o be the Fitting
LIE ALGEBRAS
null component of I relative to ad 6. Then 8o S 6 sirilcethe latter is the Fitting null component of ad a and a e6. Qn the other hand, we always have that 8o 2 S for a nilpotent subalgebra. Hence 8o : 6, and O is a Cartan subalgebra by Propdsition 1. Another useful remark about regular elements and Cartan subalgebras is that if a Cartan subalgebra b contains a regular element a, then 6 is uniquely determined by a as the Fitting null component of ad a. Thus if ft is this component then it is clear that tt = 6 since 6 is nilpotent. On the other hand, we have jrfst seen that Tt is nilpotent so that if !t : O, then It contains an ellement z 6 6 such that [e0] g 6 (cf. Exercise 1). This contradicts the assumption that S is a Cartan subalgebra. An immediate consequenceof our result is that if two Cartan subalgebras have a regular element in common then they coincide. We shall see later [Chapter IX) that if @ is algebraically closed of characteristic zerQ, then every Cartan subalgebra contains a regular element. We ndw indicate a fairly concrete way of determining the regular elemedts assuming again that O is infinite. For this purpose we need to introduce the notion of a generic element and the characteristic polynomial of a Lie algebra. Let 8 be a Lie algebra with basis (er, er, ' ' ' , 0o) over the field O. Let. tr, Er, " ' , En be indeterminates and let P : CI(Er, Ez " ', fr), the field of rational expressions in the tr. We form the extension 8.p: Pe, * Pe, + . -. * Pen. The element x: LTE;e; o[ 8r is called a generic element of 8 and the characteristic polyndmial f,(l) of. ad r (in 8") is called the characteristic polynomi.al of the Lie algebra 8. If we use the basis (er,er, "',0*) for 8r', then welcan write (3)
Le;x7:i.poiei,
where the p;l are homo;";""". It follows that (4)
i:L,2,"',/t,
expressions of degree one in the fr.
I f,(l):det(,11 -(P)) : f,n* r,(€)1"-' * +(E)]"-z - ... + (- L)rr*1,(€)]r,
where rr is a homogeneous polynomial of degree I in i the f's and Since xadx:A and r*4, r*-t(€)+0 but tn4+n:Qr if. k>0. : det(p) 0 and , > 0. The characteristic polynomial I of any (t : ,n, in(4). I-le.iere I is obtained by specializing Er: a; i:1,2,t' the char' 0 root of the for Hence it is clear that the multiplicity
III.CARTAN'SCRITERIONANDITSCONSEQUENCES6I acteristic polynomial of ad a is at least /. On the other hand, if @ is an infinite field then, since the polynomial r"-r(E) + 0 in the p o l yn o m ial algebr a O lE r,Er," ' ,€ o l , w e c a n c h o o s e E t: et so that - \aih has exactly / characteristic ,*-i@) + O. Then ad a for a roots 0, and so a is regular. Thus we see that for an infinite field, a is regular if and only if r,-r(a) + 0 .
(5)
In this sense "almost all" the elements of 8 are regular. (In the sense of algebraic geometry the regular elements form an open set') It is also clear that n - / is the rank of 8. All of this depends on the choice of the basis (e). However, it is easy to see what happens if we change to another basis T h u s i f , 1 r , 4 2",' r Q n a r g i n d e t e r ' ( f r , f r , . . . , f * ) w h e r ef r . : L k f l i . minates, then y : );tl+f+: Zrlil-tti€i. Hence the characteristic poly' nomial /"(,1) is obtained from /"(r) bv the substitutions Ei->,;tlitt;t in its coefficients (of the powers of ,i). If I is any extension field of O, then (e) is a basis for 8o ov€r P. Hence x - 2E& csll be considered also as a generic element of ,8p and the characteristic polynomial f.()') is unchanged on ex' tending the base field O to 9. It is clear from this also that if @ is infinite, a € 8 is regular in 8 if and only if a is regular as an element in 8o. (In either case, (5) is the condition for regularity') We have seen that the Fitting null component 6 of ada, aregular is a Cartan subalgebra. {he dimensionality of O is /, which is the multiplicity of the characteristic root 0. It follows that the Cartan subalgebra determined by a in 8a is 6r. 2.
Prodacts
of weight sPaces
It is convenient to carry over the notion of weights and weight spaces for a Lie algebra of linear transformations to an abstract Lie algebra ,8 and a representation rR of 8. Let IJt be the module for 8. A mapping a--a(a) of I into @ is called a weight of. IIt if' there exists a non'zero vector x in TIt such that
(6)
x(a* - a(a)L)k -g
for a suitable &. The set of vectors satisfying this condition to' gether with 0 is a subspace IJt, called the weight space corresponding to the weight a. If t3 is nilpotent, then Lemma 2.1 shows that !Jl, is a submodule. If tlt : Ift,, then we shall say that lJt is a
LIE ALGEBRAS
weight module for 8 corresponding to the weight a. Let 8 be a Lie algebra and let tn be a finite-dimenslonal weight module for I relative to the weight d. Then for ,2ny x e Tft, x(a* - a(a)L)h: 0 if ft is sufficiently large. Moreover, if dim T!t: ?t, then (,1- a(a))" is the characteristic polynomial of an.i Hence we have x(a* - a(a)l)" - g for all r e tDt. We consider I the contragredient module Dt* which carries the representation rt* satisfying (7) r e Tft, J* e Dt*.
1*ao,y*)+(x,ya*'):O, 'We have
(S)
-(a(a)x,y*)+ (x,a(a)y*):0,
l
which we can add to (7) to obtain
(e)
(x(ao - a(a)L),y*) + (x, y*(ao* + a(a)I)):
Iteration of this gives (10)
(r(a* - a(a)l)h,J*) + (r,y*(a*'+(-1)'-'
0 .l
j o(a)l)*) - 0 .
If h,: ,rt, r(ao - a(a)I)" - g for all r and consequently, by (10), (x, !*(a*' * a(a)l)") : 0. Hence J*(a"' * a(a)l)":0 for all y* e llt*. This shows that lltx is a weight module utith the weight - a. Pnorosruor.r 3. If Wt is a fi,nite-di,mensional weight mbdule for 8 with the weight a, then the contragredient module llt*iis a uteight l module wi,th the wei.ght -a. We consider next what happens if we take the tensor product be weight mddules of I of two weight spaces. Thus let fi,It relative to the weights a and 9. Let .R and S denote the repre' sentations in llt and Tl, respectively. Then any rc e Iftr satisfies (11)
r(a* - a(a)I)h - g
forsomepositiveintegerk,and'everyJeytsatisfies (rz)
t(as - g@)l)k' :0
for some positive integer &'. Let !p.: lDt8It and denote the repre' Then we have sentation of I in S by ". (13) (x@y)a':xa*Qy+ x&yau or a* : aR81 + 1 I a". Hence
III. CARTAN'S CRITERION AND ITS CONSEQUENCES
(14)
63
ar -@@)+ 9@DL : ( a * 8 1 - a ( a ) L €1)) + ( 1 8 a u - 1 8 9 @ ) I ) .
Since the two transformations in the parentheses commute can apply the binomial theorem to obtain (15)
(a' - (a(a) + p@)lL)*
r - a(a)€) 1)t(1@au :E(T)(a"8
18 9@))^-t
If we apply this to rOY, we obtain (16)
(x89@'-(a(a)+ F@DL)* *-.(T\r[848-"] implies that every element of 6 is a sum of terms of the {orm lere-"1. The restriction of. hI to )Jto has the single characteristic root p(h,). Hence the restriction of, (hI)' has the single characteristic root p(hr)', and if np is the dimensionality of fi', then we have l 0 : tr (hI), : \npp(h,)' . By the lemma, p(h,): rpa(hr), ro rational. Hence a(h)4,(2nori) - 0. Since the n, are positive integers, this implies that a(h') : 0 and hence p(h,) - 0. Since the p are linear functions and Every h e 0 is a sum of elements of the form ltr, hF, ''', etc., We see that p(h) -- o. Thus 0 is the only weight for !lt; that ils, we have $Jt: TJto. If a is a root then (20) now implies that Plt8" :0 for every a * 0. This means the kernel S of R containsi all the 8", d + 0. Hence 8* : 8/S is a homomorphic image of &. Thus 88 l is nilpotent and 8 is solvable contrary to 8' : 8. llet J? be its then If the base field @ is not algebraically closed, algebraic closure. Then fip is a module for 8o and So iis the kernel of the corresponding representation. Since n is solyable, So is : solvable' Next we note that the condition tr(ao)'= 0 and tr aRbB tr bRaRimply that
III. CARTAN'S CRITERIONAND IT5 CONSEQUENCES tr anbn:
+ff(aLbn
69
+ bR an)
: + ftr@R + bo)'2-
tr(aR)z- tr(bn)'z]- o .
- Zrn,itt a?af -0. Hence if the ai e g and arr e g, then tr(}o'iaf)' Hence the condition (2) holds also in 8p. The first part of the proof therefore implies that ,8o is solvable. Hence I is solvable and the proof is comPlete. Conor,ur,ny. If 0 is'of characteristic 0 then I is soluable if and anly if tr (ad a)' : 0 for eaerY a e 8,t. Proof: The sufficiency of the condition is a consequence of Cartan's criterion since the kernel of the adjoint representation is the center. Conversely] assume I solvable. Then, by Corollaty 2 to Theorem 2.8, applied to ad 8, the elements adga, a e 8' , are in the radical of (ad8)*. Hence adga is nilpotent and tr(adga)z Q. Let R be a representation of a Lie algebra in a finite-dimensional space llt. Then the function f(a, b) = tr aRbn is evidently a symmetric bilinear form on Dt with values in O. Such a form will be catted a trace forrn for 13. In particular, if I is finite-dimensional, then we have the trace form tr(ad a)(adb), which we shall call the KiUing form of 8. If / is the trace form determined by the representation R then
(22)
f(lacl, b) + f(a, [Dc]): tr(laclRbR+ a*Lbcl*) : tr (laBc*lbo+ a*lb*c*11 t r L a n b R , c:"0l . A bilinear form f(a, b) on I which satisfiesthis condition (23)
f(tacl, b) * f(a, [Dc])- g
is called an inuaria,nt form on 8. Hence we have verified that We note next that if. f(a,&) is any trace forms are invariant. symmetric invariant form on 8, then the radical 8a of the form; that is, the set of elements e such that f(a, z) : 0 for all a e 8, is an ideal. This is clear since f(a,lzbl) - - f([ab], e) : 0. We can now derive Cartan's criteri.on for semi,-sirnplicity. If 8, is a finite'dimensional
70
LIE ALGEBRAS
semi'simple Lie algebra ouer a fi.etd of characteristic 0, then the trace forrn of any l:l representationof g is non-d.egenerate. If the Kitting form is non-degenerate, then g is semi-simple. Proof: Let R be a 1: 1 representation of g in a finite,dimensional space llt and let f(a,D) be the associated trace form. Then gr is an ideal of I and f(a, a) : tr(a')': 0 for every a € ga, Hence gr is, solvable by the first cartan criterion. since u is semi-simple, 8a : 0 and f(a, b) is non-degenerate. Next suppose that g is not semi'simple. Then I has an abelian ideal t + b. If #e choose a basis for I such that the first vectors form a basis, for E, then the matrices of ad a, a e 8, and adb, b € E, are, respectively, of the forms
(l ),
(:3)
This implies that tr(ad DXada) : 0. Hence E g ga an{ the Killing form is degenerate. lf. f(a, b) is a symmetric bilinear form in a finitefdimensional space and (er, e", ' ' ' , €n) is a basis for the space, then it is well known that f is non-degenerate if and only if det(f(e;,e)) +0. If 8 is a finite'dimensional Lie algebra of characteristic zero with basis (er,er,...,e*) and we set 0u: trad erxdrt, then g is semisimple if and only if det (B;r.)+ 0. This is the determinant form of Cartan's criterion which we have just proved. If J? is an ex. tension of the base field of 8 then (er, er, . . ., €n) is a lbasis for go over g. Hence it is clear that we have the following iconsequence of our criterion. conou,rnv. A finite-dirnensional Lie algebra g ouer a fietd a of characteristic zero is semi-simple if and only if go is setni-simple for euer! extension rteM I of A. 5.
Structure
of serni-simple algebras
we are now in a position to obtain the main structure theorem on semi-simple Lie algebras. The proof of this resullt which we shall give is a simplification, due to Dieudonnd, of Cartan's original proof. The argument is actually applicable to arbitraryinon-associative algebras and we shall give it in this generar forrd. Let lI be a non-associative algebra over a field @.t A bilinear
III. CARTAN'S CRITERION AND ITS CONSEQUENCES
7I
form f(a, b) on ?l (to O) is called associ'atiue if
(24)
f(ac,b) : f(a, cb).
gIf / is an invariant form in a Lie algebra then f(tacl, b) + f(a, [Dc])Hence f(lac|,b) f(a, [c&]) - 0 and / is associative. Now let f(a,b) be a symmetric associative bilinear form on l[ and let E be an ideal in lI. Let a e Ea so that f(a, b) : 0 for all b e s' Then for any c in 21, f(ac,b) -f(a,cD) - g since cD e E. Also f(ca,b) f(b, ca) f(bc, a) :0 since bc e E. Hence Er is an ideal. The importance of associative forms is indicated in the follow' ing result. THponsM 3. Let \ be a finite-dimensional non'a,ssociatiuealgebra aaer a fi.eld A such that (L) V, has a non'degenerate syrnmetric as' sociatiue form f and (2) lI has no ideals E with Ez : A. Tlwn lI is a direct sum of i.deals which are simple algebras. (We recall that lI simple means that lt has no ideals + 0, ?I, and ?12+ 0.) Proof: Let E be a minimal ideal ( + 0) in lI. Then E n Er is an ideal contained in E. Hence either E n Ea : E or E n Er :0. Suppose the first case holds and let bs bz e E, a e 2I. Then g. ' Since f is non"degenerate b'b, - g and f(brbr, q) - f(bv b2a): It is well E2 : 0 contrary to hypothesis. Hence E n Er :0. This ideal. is an Ef lI and E El implies that this that known O decomposition implies that EEr : 0 : EtE; hence every E'ideal is an ideal. Consequently, E is simple. Moreover, St satisfies the same conditions as ?I since the restriction of, f to Er is non' degenerate and any Er-ideal is an ideal. Hence, induction on dim ?I implies that Ea : ?I, (E '.' @ U" where the ?Ie are ideals and are we have 2[:?Ir@?trO "'OU", s i m p t e a l g e b r a s .T h e n f o r l l r : E lli simple and ideals. This result and the non-degeneracy of the Killing form for a semi-simple Lie algebra of characteristic zero imply the difficult half of the fundamental Structure theorem. A finite-di.mensional Lie algebra oaer a rteld "'@8' of characteristic 0 is semi'simgleif and only if 8:8'@8rO where the 8i are ideals which are simple algebras. Proof: If I is semi-simple, then I has the structure indicated. 8 i i d eal s and si mpl e. Co n ve rs ely , s uppos e8 = 8 ' O8 rO" ' @ 8 ' , 'l{e consider the set of linear transformations ad I : {ad a I a e 8}
72
LIE ALGEBRAS
acting in 8. The invariant subspaces relative to this set are the i d e a l so f 8 . S i n c e8 : 8 r O 8 r O . . . 08, where the g;lareirreducible, we see that the set ad I is completely reducible. jHence if E is any ideal + 0 in 8, then 8: E@o where o is an idehl (Theorem 2.9). Moreover, the proof of the theorem referred to shows that wecantakeEtohave the form D:IirO8,rO...g;* for a subset {8r,} of the 8i. Then E=8/(ll;rO... @8re)=.8rr68ile...68r,r where the 81oare the remaining 8;. Since 8r is simple, 8? : ,8r. Hence (8a O 8ir@ ... @ 8r,)' : 8r.rO 8r.2O . .. O 8i, and|consequently E' - A. Thus E is not solvable. We have iherefore proved that 8 has no non-zero solvable ideals; so 8 is semi-sirnple. T h e a r g u m e n tj u s t g i v e n t h a t * - . 8 r r @ 8 r r @ . . . 8 8 i , has the following consequence. Conor,r,.r,ny1. Any ideal in a semi-simple Li.e algebra of characteristic 0 is semi-si,mple. If 8r is simple then the derived algebra 8{ : 8;i hence the structure theorem implies the following Conor,ulny 2. If 8, is semi-simple of characteristic 0,ithen 8' : 8. Remark. We have proved in Chapter II that if I isla completely reducible Lie algebra of linear transformations in a ]finite-dimensional vector space over a field of characteristic 0, thdn 8 : 6 O,8, where G is the center and I, is a semi-simple ideal.I Then 8/ 8i : 8,. Hence 8 : 6 e 8', 8' semi-simple. We prove next the following general uniqueness thborem. THponpu 4. If 2I is a
algebra and
l I : ? [ rO l t r O . . . O U , : E r O E , O . . . O $ , where the \Ii and Et are tdeals and are simple, then s and the Eis coincide with the W";'s(excebtfor order). Proof: Consider 2Ir 0 Ei, j : L,2, . .., s. This is An ideal contained in llr and 81. Hence if lI' fi Er' + 0, then lI, :illr n Ei - E; since lI' and E; are simple. It follows that Ur o Ej' * 0 for at most one Er'. On the other hand, if ?I' n E, :0 for all j then llr8r S 2[' n Ei:0 this implies for all j. Sincel[:E'OE'O...OE,n that lIr?I : 0 contrary to the assumption that lli + 0. r Hence there is a i such that ?Ir - fl51. Similarly, we have that every lI; coincides with one of the El and every E1 coincides with cjne of the ?Ii.
III. CARTAN'S CRITERION AND ITS CONSEQUENCES
The result follows from this. It is easy to see also that if ?I is as in Theorem 4, then lI has just 2" ideals, namely, the ideals 2Ir,O ' . ' O \r*, {ir, "', ir} a subset of {1, 2, . - -, r}. We omit the proof . The main structure theorem fails if the characteristic is y' + 0. To obtain a counter example we consider the Lie algebra Et of linear transformations in a vector space lJt whose dimensionality z is divisible by f. It is easy to prove (Exercise 1.20)that the only ideals in Or are @L and oL, the set of multiples of 1. Since @i is AL g VL. the set of linear transformations of trace 0 and tr 1 : n:0, g,ilO| has only one ideal, namely, @ilAt, and the latter Hence I : is simple. This implies that 62101 is semi'simple, but since ELIOL is the only ideal in Ezl0l, E"IOL is not a direct sum of simple ideals. This and Theorem 3 imply that Erl01 possesses no nondegenerate symmetric associative bilinear form. We conclude this section with the following characterization of the radical in the characteristic 0 case. THponpu 5. If 8 is a fi,nite-dimensional Lie algebra ouer a field of characteristic 0, then the radical 6 of 8, is the orthogonal com' plement g't of gt relatiue to the Kitting form f(a,b). :f(b,D) :0. Proof: E : 8'a is an ideal and if 6 € E', then tr(ad gb)'z The kernel of the representation a + adga, a e E, is abelian. Hence E is solvable, by Cartan's criterion, and E s @. Next let s € 6, a,b e 8,. Then f(s,[ab]) - f([sa],b). We have seen (Corollary 2 to Theorem 2.8) that adtsal is contained in the radical of the envelop' ing associative algebra (ad 8)*. Consequently, ad [sa] ad D is nilpotent for every b andhencef(lsal,b):0. and s € 8'r. Thus f(s,labl):0 ThusOgS'randso6:8'4. 6.
Deriuations
We recall that ad a is a derivation called innzr and the set ad 8 of these derivations is an ideal in the derivation algebra O(8). In fact, we have the formula fad a, Dl : ad aD f.or D a derivation. Hence ladaad'b, Dl: which implies that
ad aad(bD) * ad (aD)adb
LIE ALGEBRAS
74
0 : tr [ad a ad b, D) - tr ad a ad (bD) * tr ad (aD)tadb Thus, for the Killing form f(a,b): trad aadb we have f(a, bD) + f(aD, b) : 0 ;
(25)
that is, every derivation is a skew-symmetric transformaltion relative to the Killing form. We prove next the following theorem which is due toiZassenhaus. Tnponnu 6. If 8, i,s a finite-di,mensional Lie algebra which has a non-degenerateKilti.ng form, then euery deriuation D of'8 is inner. is a linear mapping of I into Proof: The mapping r--tr(adr)D space 8* of 8. Since coniugate of the A; that is, it is an element exists i an element there that it follows non-degenerate J@, b) is
d e g such that f(d', x): tr(ad x)D for all .r e 8' Let E be the derivation D - ad d- Then tr (ad x)E : tr (ad x)D - tr (ad r)(ad d) : tr (ad r)D - f(d, r) : 0 ' Thus t r ( a dx ) E - 0 . (26) Now consider f(xE''y) : : : : -
tr (ad xE)adY tr lad x, E)ad'Y t r ( ( a dx ) E a d Y- E a d r a d Y ) , t r ( E a d y a dx - E a d x a d Y ) i tr E[ad y, ad x) tr E adlYxl: 0 ,
this implies that.0 - 0. Hence by (26). Since/ is non-degenerate, : is inner. d ad D This result implies that the derivations of any finitb dimensional semi-simpleLie algebra over a field of characteristicrzero are all inner. We recall also that if 6 is solvable, finite'dimensionalof characteristic 0, then 6 is mapped into the nil radical Il by every derivation of 6 (Corollary2 to Theorem 2.1-3). We cdn now proYe Tnuonpu 7. (l) Int 8 be a finite-dimensi.onalLie Algebraouer a g the radical, It the ni,l radics'|. Then any field of characteri,stic0, deriuation D of g maps6 into Tt. (2) Let 8 be an ideal in a finite' d.imensionalalgebra gr, @r, 9tr the radical and nil ladical of 8r
III. CARTAN'S CRITERION AND ITS CONSEQUENCES
Then6:8 O 6r, It:8 O llr. Proof: We first prove (2) f,or the radical. Thus it is clear that 8 n g, is a solvable ideal in l], hence I n 6r s g. Then g(8 n gr) is a solvable ideal in 81(8 n 6r). On the other hand, 8(8 n @') = (,8 * 6')/6,, which is an ideal in 8'/€'. Hence (8 + gr)/gr and fl 6r. and €:8 8/(8 n €,) are semi-simple. Hence 6(8 n 6,):0 Now let 8r be the holomorph of 8, €,, 9t, the radical and nil radical of 8r. Then we know that [8rgr] 5 ltr (Theorem 2.13). Since 6 s 6' by the first part of the argument, [8'g] I (!t' n 8) E It. This implies that every derivation of I maps 6 into It, which proves (1). Now let 8r be any finite'dimensional Lie algebra containing I as an ideal. If ar € 8r then ad ar induces a derivation in 8. Hence Ttad,a, gIt by (1). This means that 9t is an ideal in ll so that It S Tlr fl 8, 9tr the nil radical of ,8'. Since the reverse inequality is clear, lt: Itr fl ,8. This result fails for characteristic p + 0. To construct a counter example we consider first the commutative associative algebra 3 T h e r adi cal ft of 3 wi th th e bas is ( L, 2, 22,...,2 o -' ) w i th z p :A . has the basis (2, z', . . . , z'-') and 3/ft : AL. It is easy to prove that if ar is any element of 3, then there exists a derivation of 3 mapping z into w. In particular, there is a derivation D such that zD: !. Now let E be any simple Lie algebra and let I be the The elements of this algebra have the form Lie algebra E 13. g, and [(D&z)(b'&z')l -lbbt]@zz'. Then I E, bi zi Q zi, € )}i8 is a Lie algebra (Exercise 1.23) and E I ft is a nilpotent ideal in 8. i s s i mp l e . H e n c e E On i s the Mo re o ver , S ( E 8n) = E 8O I:E radical and the nil radical of 8. lt. D is any derivation in the as' sociative algebra S, then the mapping )Dc I z; + 2h I eiD is a derivation in 8. If we take D so that zD :1 and let b * 0 in E H e n c e w e h a v e a d eri vati on w hi ch t h e n Dfi -z - t $lE lS 8S . does not leave the radical invariant. 7.
Complete red,ucibilitA of the representations of semi-aimple algebras
In this section we shall prove the main structure theorem for modules of a semi-simple Lie algebra of characteristic zero and we shall obtain its most important consequences. The main theorem is due to Weyl and was proved by him by transcendental methods based on the connection between Lie algebras and compact groups.
LIE ALGEBRAS The first algebraic proof of the result was given by Casimir and van der Waerden. The proof we shall give is in es$ence due to Whitehead. It should be mentioned that Whitehead'$ proof was one of the stepping stones to the cohomology theory of Lie algebras which we shall consider in $ 10. We note also that in the characteristic p case there appears to be little connection hetween the structure of a Lie algebra and the structure of its modt{les since, as wilt be shown later, every finite'dimensional Lie algdbra of characteristic P +0 has faithful representations which are ndt completely reducible and also faithful representations which ard completely reducible We obtain first a criterion that a set ^5 of linear trafisformations in a finite-dimensional vector space l}t be completely reducible. We have seen (Theorem 2.9) that J is completety reducibld if and only if every invariant subspace Tt of IJt has a complement S which is invariant relative to J. Now let Tt' be any complerhentary subspace to Il: Un - n e n'. Such a decomposition is as{ociated with a projection .E of lJt onto It. Thus if. r e T!1, then v'1e can write r in one and only one way as .r : y + !', ! € It, J' le ttt, and -E is the linear mapping x-+y. Conversely, if ,E is any idempotent linear mapping such that Tt - !It.E then It/ : IJt(1 - E)'is a comple' ment of Tt in IJt. Now let A e 2 and consider the llinear transformation LAE| = AE - EA. If r e [It, xAE e It and xEA e Tt; Hence hence tAEl maps llt into Tt. If y e Tt then yAE = yA. tranformalineAr of the set I denotes Then if 0. into tAEl maps It It is tions of Dt which map IJI into It and It into 0, [AE] e f. clear that t is a subspace of the space G of linear trNnsformations in !Jt. We now prove the following Lnuu.r, 2. Tt has a complement S which is inuarianl if and only - lADl for all A e t. Here if there existsa D e X. such that lAEl [t. onto E is any froiection Proof: Let $ be a complement of Tt which is invafiant relative to J and let F be the projection of St onto Tt deterrhined by the decomposition tn - n CI S. Since $ is invariant, F cQmmutes with every Ae 2, that is, [AF] -0. Hence LADI-lAEl fbr D: E- F' Also, since ^E and F are projections on It, E - F maps !ft into It Conver5ely, suppose and !l into 0. Hence D e f" as required. Thdn F:E-D there exists a DeX, such that tAEl-lADltc(E- D) e It xF: llt then If r e t. Ae commutes with every
III. CARTAN'S CRITERIONAND ITS CONSEQUENCES
77
Then H e n c eF " : F a n d l t :Il tF. an d i f y e t t t hen y F - y E :y . J under is invariant and !t of is a complement !lt(l F") $ S: since F commutes with every A e 2. Suppose now that I is a Lie algebra and !)t is an !-module, It a submodule. We can apply our considerations to the set 8i of re' presenting transformations a' determined by !]t. Let E be a projection of.Ift onto !t. If a e 8, set f(a): la*, El. Then a-+f(a) is a linear mapping of I into the space fr of linear transformations llt into !t and !t into 0. If Xe f and a € 8 then of fiwhichmap if r e tlt then x[Xa"] e sIt and if. y e It then Thus X,. e IXaRJ by oi. yXa*: 0 and ya*X: O. We denote the mapping y-rlXanl -, a* is a representation of 8 whose associatIt is immediate that a ed module is the space l. We have f(labl) - flablB, El : IIaRb")Ol
- l[anElb"l+ la"[b"Ell - If@),6"1+ Iao,f(b)l - f(a)bfr- f(b)ai . We are now led to consider the following situation: We have a of I into f such module f for 8 and a linear mapping a-.f(a) that
(27)
f(tabl) - f(a)b - f(b)a .
example of such a mapping is obtained by taking A "trivial" where d is an element of f. For, we have da, f(a): f(abl) - dtabl - (da)b - (db)a : f(a)b f(b)a . The key result for the proof of complete reducibility of the modules for a semi-simple Lie algebra of characteristic zero is the following Lnuu.e, 3 (Whitehead). Let 8, be fini'te-dimensional semi'simple of characteristic zero and let X, be a fini,te-dimensional module for 8, and a-+f(a) a linear mafbing af 8. into X, satisfying (27). Then there exi.stsa d e *" such that f(a) - da. Proof: The proof will be based on the important notion of a Casi,mir operatm. First, suppose that 8 is a Lie algebra and Er and Ez are ideals in I such that the representations of I in E' and Ez Zlr€ contragredient. Thus we are assuming that the spaces Er andEgareconnected by a bilinear form (bt,br), D; € Ei, (bubz)e A, which is non-degenerate and that for any a e 8, we have
78 (28)
LIE ALGEBRAS (lb,al, b") + (b,,lbral) - 0 .
lf (ur, . . . , il^) is a basis for Er then we can choosea co{nplementary or dual basis (It', u' , . ., u^) for Es satisfying (ur, ui'\ - d;.r. Let -lu;al : !,ia,;iui and fuka): },r}*ilt. Then (Lu;a),uk) (Eatiut, uk): dik and (u;,luoal) : (?h,I,,lr,pt) : B*;. Hencel (28) implies 2au6i*: that air : - 9u, that is, the matrices (a) and (p) determined by dual bases satisfy (F) : - (a)' ((a)/ the transpose of (a)). Now let
R be a representationof 8. Then the element
7 - iufu'" t:L is called a Casimir operator oI R.
We have
LI-,anl: \lufa"fu'*
+ \utlu''anl
: \aa1ufu'* + Zl;iufuiR r ,J t, J
- \aipfu,'o t,,
-
\a1sf;ui* t,J
: \a;iufrai, - \ariufu'n -0. Hence we have the important property that /' commuJtes with all the representing transformations aa. Now let I satisfy the hypotheses of the lemma. ILet S be the kernel of the representation R determined by I . Then we can write I : F O 8, where I' is an ideal. Then the restriction of R to 8r is 1 : I and ,8, is semi'simple. Hence the trace fOrm (br, br) : trblbf , b; in 8r, is non'degenerate on ,8r. Also we knbw that the trace form of. a representation is invariant. Hence lthe equation (28) holds for Di e 8r and a e 8,. Thus the representation of 8 in 8r coincides with its contragredient and if (uu ''', u^)i (ut, " ', il^) - I,,Lruf;u'R is a are bases for 8, satisfying (ur ui) - dr., then /' 't|Ve note also a*. Casimir operator which commutes with every that tr l- : X; tr ufuio - 2 (ui, u') : re : dim 8r. We now decompose I into its Fitting components Ii and fr relative to l- so that I induces a nilpotent linear transformation in Io and a non-singular one in f,. Since TaR --aRl, friani9I1 so that the f; are submodules. we can write .f(a): fo@) * f'(a) where is a lineaf mappingof fi@) e I; and it is immediate tbat a-fi@)
III. CARTAN'S CRITERIONAND ITS CONSEQUENCES
79
8 into 17 satisfying (27). Now if both spaces f, are * 0, then H e n c e w e c a n u s e i n d u cti on on di mf di ml i < dim l f or j: 0, t . to conclude that there is a d1 € X,i such that f/a) : dP. Then d.: do * d, satisfies f(a) * da as required. Thus it remains to consider the following two cases: X : Xo and I : Xr. X,: fro: In this case /' is nilpotent. Hence m : tt f : 0. This means that the kernel of R is the whole of 8, that is, ao :0 for al| a. Then the condition (27) is that f(labl) : 0, a, b e 8,. Thus th i s i m p l i e s that.f(a):0 so f (a ') - 0 f or all a' € 8' . Si n c e 8 ' :8 , that d: 0 satisfies the condition. where the (ai) and (ud) are dual bases f : Ir: Set y :}1=rf(u;)ui for It as before. Then ya:
ZU@t)ut)a = \U@t)a)u" * |,f(u)lu"al : \(f(u;)a)ur
+ \giif@r)ui i. J
= \(f(ut)a)u'
-
- \(f(u)a)u'
-
= \(f(u;)a)u'
\a6f@)u' Z(f[uia))ui ZU@;)a)u' + l(f(a)u)u'
- f(a)r Since /- is non-singular, d: yT-' satisfies the required condition f(a) - da. This completes the proof of Whitehead's lemma. We can now prove the following fundamental theorem: Tnponnu 8. If 8, is fi,nite-dimensional semi-simple of characteristic 0, then eaery finite-dimensional module for 8, is completely redacible. Proof: Let llt be a finite'dimensional 8-module, !t a submodule. Let t be the space of linear transformations of llt which map llt into lt, !t into 0, and consider f as 8'module relative to the composition Xa = IX, a*1, R the representation of St. Let E be any projection of !t onto It and set f(a) - la*, El. Then /(a) satisfies the conditions of Whitehead's lemma. Hence there exists a D e X, such that f(a) : Da - lD, a*1. As we saw before, this implies that It has a complementary subspace which is invariant under 8. Since this applies to every submodule It, llt is completely reducible. If 8 is a subalgebra of a Lie algebra E, then a deriuation D of
80
LIE ALGEBRAS
l
g into E is a linear mapping of I into E such that
(29)
'
lt,l,lD- u,D,l,l+ u,,l,D\
for every lr,lz e g. It is immediate that the set O(8jl8) of deriva' tions of I into E is a subspace of the space of lineaf transforma' tions of 8 into E. Whitehead's lemma to the theorerh on complete reducibility has the following important consequenceon derivations: Tuponpru 9. Let E be a fi.nite'dim,ensional Lie algebra of char' acteri,stic A and. let g be a semi'simple suhalgebra of E, Then eaery sA' deriuati.on of g into E can be extended to an inner deniuation of Proaf: Consider E as 8-module relative to the multflplication [6/]' T h e n a d e r i v a t i o n D o f -8 i n t o E d e f i n e s , f ( ' ) : l D /e8,6e13. condition the satisfying f(llrlrl) - U,lrlD : fl r, lrDl + [l'D,li - lf(t), t,] - [f(tz), t,] of Whitehead's lemma. Hence there exists a d e E such that lD: deter' f(t) : ld,IJ. Then D can be extended to the inner derivation d. mined by the element We recalt that we have shown in Chapter II (Theqrem 2.11) that if 8 is a completely reducibte Lie algebra of linear trlansformations in a finite-dimensional vector space llt over a field of ] characteristic 0, then 8 : 8, O 6 where [!' is a semi'simple ideal and 6 is the center. Moreover, the elements C e G are semi'simplb in the sense that their minimum polynomials are products of distipct irreducible potynomiats. we are now in a position to establish the converse of this result. Our proof will be based on a field extension argument of the following type: Suppose we have a sdt J of linear transformations in lJt ovet o. If g is an extension fileld of o, every A e Z has a unique extension to a linear transformhtion, denoted again by A, in Dto. In this way we get a set J - {A} bf linear transformations in llto ovet 9. We shall now prove the ifollowing Lpuu.o, 4. I-et 2 be a set of linear transformatioths in a finite' dimensional aector space Ift ouer O and let 2 be the set of extensions of these transformations to Wto ouer !), Q an extertsion field of A' Su\fose the'set 2 i.n \fto is completely reducible. Then the original X is comPletely ,reducible in Tft. Proof: Let Tt be a subspace of SJt which is invdria nt under J l
III. CARTAN'S CRITERIONAND ITS CONSEQUENCES
81
and let E be a projection on It. Then our criterion for complementation (Lemma 2) shows that Tt will have a complement which is invariant relative to -I if and only if there exists a linear transformation D of !)t mapping tn into It, It into 0 such that IAE) tAD) for all A e Z. lf Ar, Az, "', A* is a maximal set of linearly independent elements in .X, and we set Bi: lArEl, then it suffices Br i : t,2, "', k. This is a system to find a D such that IAD\: of k linear equations f.or D in the finite'dimensional space f of linear transformations of TJt mapping Dt into yt, [t into 0. Thus if we have the basis (Ur, Ur, " ', U,) for X, we can write Bi: * li|lUi, then our equations are Ztr=r\rrtl,, lAr(Iuf :ZrTnrU,, D system: ordinary the DnTu"rdn: Ftu, i : L,2, '", k, equivalent to s : 1, 2, . ", r, for the d1 in O. Hence It has a J'invariant complement if and only if this system has a solution. We now pass to !)to and the invariant subspace lfo relative to the set J of extensions of the A e 2. Then our hypothesis is that lto has a J-invariant complement in llto. Now the extension E of. E is a projection of l]to onto !tp. Hence we have a linear mapping D of. IJto mapping l}to into lto, Tlointo 0, such that IA;DI : Bu = lA&), i : !,2, ' '', hThe extensions Ur, Ur, ''', U, form a basis for the space of linear transformations of fio mapping IJto into \lr, Ilo into 0' Hence if D : Xid i.Jt then the 8 satisfy the system Xarie,be : Fa. Since the rinc ?1d Fa belong to O, it follows that this system has a solution (d,, ..., d"), d's in O. Hence there exists a D e f such that[AD]: [AE], A e 2, and so It has a J'invariant complement in lft. We can now prove the following Trrponpu 10. Let 8, be a Lie algebra of linear transformatians in a fi.nite-dimensional uector space Wl oaer a field of characteristic zero. Then 8 fs completely reducible in Wt i,f and only if the following condi.tions hold: (1) I : Ir e 6, 8, a serni,-simbleideal and E the center and (2) the elements of 6 are semi'simple. Proof: The necessity has been proved before. Now assume (l) and (2) and let g be the algebraic closure of the base field. Then the lemma shows that it suffices to prove that the set I of extensions of the elements of 8 is completely reducible in llto. The set of. !}-linear combinations of the elements of I can be identified with 8c and similar statements hold for I' and O. Now let Ce6. Since the minimum polynomial of C in llt has distinct irreducible factors and since the field is of characteristic 0, the minimum
82
LIE ALGEBRAS
polynomial of C in lllo has distinct linear factors in A. Consequently, we can decomposefis as !J1",O ''' O IJt"* where Tltrn: {x; I x; € fio, xiC : aix;} and,ar, dzt . . ., ar are the different characteristic rootsi of' C. Since AC : CA for A e g, ySl,tA c Y)ldi We can apply the same procedure to the Tft,, relative to any other D e G. This leads to a i nto f i ri vari ant subd e c om pos it ionof IJ to :IJ I,O IJ I,O ...e )tn ' spaces such that the transformation induced in the'IJtr by every C e O is a scalar multiplication. To prove I completbly reducible in TJtoit suffices to show that the sets of induced transformations in the TJtrare completely reducible and since the elenlents of G are scalars in IJtr it suffices to show that 8r is completely reducible in every !]t;. The invariant subspaces of IJtr relative to lJ' are invariant relative to J?8r, the set of g-linear combinatiofirs of the elements of .8,. Now J?8,is a homomorphic (actually isomprphic) image of the extension aigebra 8ro, which is semi-simple. flence 9.8.-is semi-simple and consequently this Lie algebta of linegr transformations is completely reducible by Theorem 8. Thus wd have proved that 8 is completely reducible in IJls and hence in 'lt; We now shift our point of view and consider a finite dimensional Lie algebra 8 of characteristic 0 and two finite-dimdnsional completely reducible modules lJt and yt for 8. We shhll show that lft O Tl is completely reducible. Now the space p + :Ut@ Tt is a module relative to the product (x + y)l: xl * !1, i e TJt, y e tt. Evidently IS is completely reducible and 9Jt@ It is a Fubmodule of S I $. Hence it suffices to prove that S I S is conlpletely reducible. If we replace ti by 8/S where R is the kernelrof the representation in $, then we may assume that the associalted representwhere f, is ation R in S is 1:1. Then we know that 8:8,@0 a semi-simple ideal and 6 is the center. Moreover, the elements C*, C e O, are semi-simple. Now, in general, if ,R is a faithful representation of a Lie algebra lJ, then the represerftation l? I R is als o fa i th fu l . T h u s , i f a e I and a' i F not a scal ar i n s ap mUltiplication, then, since the algebra of linear tran$formations in S A S is the tensor product of the algebras of linqar transforma t ions in S , a o &a * , c ft8 1 , I@ a R a n d 181 arel l i nearl y i nde' pendent, so d* I 1 + | & aR + 0. Hence if a@o :0, a* must be a ia : 0. Since scalar, say a' : a. Then a@* :2a (in S I S) and We can now conclude that R is 1 : 1 this implies that a :0.
III. CARTAN'S CRITERIONAND ITS CONSEQUENCES
83
I]san - Sfos * 6nsn where 8i'8" - [3, is semi-simple and Onso is the center. Our result will therefore follow from the criterion of Th. 10 provided that we can prove that every gnsn:ce81+18c", C e O, is semi-simple. Let I be the algebraic closure of the base field and let dr, dz, . . . r dk be the different characteristic roots of C*. Then the proof of Theorem 10 shows that lpo : Srr 0 Sr, O . .. O !$"* where xr.QR : aixo,r,for rr, e E r. Hence (S @r$)p : To Sot$o : .Xt[tr; I Sr1 and It follows that the !C^@o : (a; + ar)l for every -/ € S"r 8 S"j. minimum polynomial of 6aan has distinct ioots in (S I ![')o. Since this is also the minimum polynomial of 6nan in S I $, it follows that this polynomial is a product of distinct irreducible factors. Thus C@* is semi-simple and we have proved THoonpu 11. Izt 8, be a fi.nite-dimensional Lie algebra ouer a field of characteristic zero and let Tft and Tt be finite-dimensianal completely reducible rnodules for 8. Then IX I tt ls completely reducible. 8.
Representations of the aplit three-dimetuional simple Lie algebra
In $ 1.4 we called a three-dimensional simple Lie algebra S split if ft' contains an element h such that ad /a has a non-zero characteristic root p belonging to the base field. We showed that any such algebra has a basis (e,f ,h) with the multiplication table
(30)
[ehl - 2u,
[fh]:
lefl - h
The representation theory of this algebra is the key for unlocking the deeper parts of the structure and representation theory of semisimple Lie algebras (Chapters IV, VII, and VIID. We consider this now for the case of a field CI of. characteristic 0. We suppose first that @ is algebraically closed and that IJt is a finite-dimensional module for S. The representation in Dt is determined by the images E,F, H of. the base elements e,.f,h and we have
(31)
[E, H]
[F, HI _
IE,FJ-H.
any three linear transformations
E, F, H satisfying
84
LIE ALGEBRAS
S'' these relations determine a representation of S and hence a module. Let a be a characteristic root of. HandraOorresponding characteristic vector: r + O, xH : ax. Then (32)
@E)H - x(HE + 2E) - (xD)(a + 2) '
for lL xE + 0 then (32) shows that a * 2 is a characteristic root t replace We can vector. characteristic corresponding xE a H and nonof by xE and repeat the process. This leads to a seqtrence zero vectors t, xE, JcEz,"', belonging to the charaqteristic roots a, a * 2, a * 4, . .., respectively, f.or H. Now f/ ha$ only a finite number of distinct characteristic roots; hence, our se{uence breaks off and this means that we obtain a k such thatt: xEk * 0 and xEk*' :0. If we replace r by xEh we may suppose at the start that r + 0 and (33)
xH:
Now set ro : r and let ri: obtain 34)
ax , x;-tF.
rE:O
'
Then, analogous to (32)' we
xtH : (a - 2i)x; ,
and the argument used for the vectors rEd shows that there exists '''I Xvlvtu + 0 but Xrn+r: a non-negative integer zz suChthat ro, )r,t, 0. Thus xF^+r - 0, xF^ + 0. to Then ri, o = i = m, is a characteristic vector oL H belonging Z i . d , d Z , d ' 4 " " ' a 2 m Since t h e c h a r a c t e r i s t i cr o o t a Let are all different it follows that the n; ?fE linearly indEpendent. 'ltof subspace (m l)-dimensional is an * Tt : xto oxi so that Tt to relative irreduclble and invariant is It that show We shall now formula the establish n. We first
(35)
x;E: (- ia + i(i - 1))rr-'
- t' as given in (35)' Assumei (35) for i Thus we have )hE:0 Then x; E -- r;-rFE - r;-t(EF - H) - (- (i - L)a + (i - lxt - 2))ri-zF - (a - 2Q - 1))rr-r - (- ia + i(i - l))ri-r
III. CARTAN'S CRITERION AND ITS CONSEQUENCES
85
as required. It is now clear from (34), (35), and xtF: )ci+rthat 9t i s a S-subspace of llt. Since 11 - [EF] we must have tryTH : 0. This, using (34), gives (m * l)a - m(m * 1) : 9. Hence we obtain t the result that a,: rn. Our formulas now read i :0, "', rtt !c;H - (m - 2i')n , (36) i--0,...,rn-1, x^F:0 hF:ri+t, i - 1, . ..,rn x;E -- (- mi + i(i- l))r,i-r, xsE: 0, and we note that in the last equation
-mi+t(l-1)+0. Now let llr be a non-zero invariant subspace of !t and let ! : F;x; * 0i+rri+r+ "'
* F^x* ,
Bi.* 0, be in Tlr. Then xrn: Fr'lF*-o € ltr. Hence by the last equation of (36) every h € \\ and Itr : ll. Hence if ![l is S-irre' ducible to begin with, then lll - tl. In general, the theorem on complete reducibility shows that IIt is a direct sum of irreducible invariant subspaces which are like the space !t. We can now drop the hypothesis that A is algebraically closed, assuming only that @ is of characteristic 0. We note first the following LsMuA. 5. Let R be the spli,t three-dimensional simple Lie algebra ouer a fi.eH A of characteristiczero and let e-sE, f -F, h-H defi,ne a fi,nite-dimensional representation of fi. Then the charac' teristic roots of H are integers. Proof: If tJt is the module of the representation and .? is the algebraic closure of O then llto is a module for Ro which satisfies the same conditions over Q as S over @. Then !]to is a direct sum of irreducible subspacesIt with bases (No,Nr, ..., x*) satisfying (36). Hence if we choose a suitable basis for Tfto then the matrix of H relative to this is a diagonal matrix with integral entries. Hence the characteristic roots of H in lltp are integers. These are also the characteristic roots of. H in Dt. 'We can now prove the following THsonpM 12. Let fr be the split three-dimensional simfle Lie algebra oaer a fi,eld of characteristic 0. Then for each integer m: 0,1,2,..- there exists one and, in the sense of isomorphi,sm,only
LIE ALGEBRAS
86
one irreducible fi-module \Il of dimensi,on m * l- TIt: has a basis (xr, xr, ..., r,o) such that the representing transformatibns (E, F, H) corresponding to the canonical basis (e,f,h) are giuen 4y (36). proofi Let llt be a finite-dimensional irreducible rriodule for tr. Then the characteristic roots of H are integers. Hence we can : dtc- As find an integer a and.a vector x * 0 in Sl such that rH m and a: that obtain Then we before we may suppose xE:o. .,1c*) for. . (36) These holdl. that (tcr, such Nr, that 9lt has a basis !)t' | of m * dimensionalityt the by mulas are completely determined are $ for modules irreducible (rn l)-dimensional * Hence any two isomorphic. It remains to show that there is an irredupible (m + 1)' ' '. ' To see this dimensional module for S for every rn :0,1, ' ', r-) and we define y/e let lJt be a space with the basis (xo, trr,' the linear transformations E, F,If by (36). Then we have x{EH-HE)
: (- mi * i(i - I))(m - 2(i' - l))ri-r - (m - 2i,)(- mi * i(i - t))x+, - 2(- rni * i,(i - 1))x*'
:2xrE , : (m - 2(i * 1))16,',- (m - Zi)x*' HF) x;(FH-
*{EF-FE)
-
l
Zxt+t
:-Zx;F, : (- mi.* i(i - r))n + @Q + 1) - $ + r)i)xt : (m - 2i)r; :X;H.
Hence E, F, and H satisfy the required commutation rblations and so they define a representation of S. As befor€, ffi is, S-irreducible' The theorem of complete reducibility applies herp also and together with the foregoing result gives the structure of any finitedimensional S-module. g.
The theorema of Levi and Maleev'Hariehlchand'ra
The .,radical splitting" theorem of Levi asserts that if E is a finite-dimensional Lie algebra of characteristic 0 rwith solvable radical Gi, then E contains a semi-simple subalgebfa 13such that and so that S:8OG + 6. It will follow that 8 n 6:0 E:8
III. CARTAN'S CRITERIONAND ITS CONSEQUENCES
87
g =El@. Thus the subalgebra 8 is isomorphic to the difference algebra of E modulo its radical. Conversely, if E contains a sub' algebra 13isomorphic to E/9, then 5Jis semi'simple. Hence I n I :0 an d si n ce dim 8: dim G + d i m E/€ : d i mG * d i m 8 , E - I + 6. We note next that it suffices to prove the theorem for the case 9' :0, that is, g is abelian. Thus suppose 62 + 0. Then if E : E/6t, dim E < dim E. . Hence if we use induction -@on the dimen' :61Q2 is the sionality we may assume the result for E. Now Hence E contains a subalgebra radical of E and Etd =E/9. E : E/@. As subalgebra of E, E has the form Erl6' where E, is a subalgebra of E containing @'. Now 6t is the radical of E' and ErlQ'=r E/g so that dim Er < dim E. The induction hypothesis can therefore be used to conclude that Er contains a subalgebra I = E/6, and this completes the proof for E. We now assume that @2:0 and for the moment we drop the assumption that E : Ei6 is semi-simple. Now 6 is a submodule of E for E (adjoint representation). Since @' :0, 6 is in the kernel of the representation of E determined by_the module 6. Hence we have an induced representation for E : E/@. For the corresponding module we have sD: [s, r]' s 9 6, D e E. We can find a 1:1 linear mapping o:b-+b' of. A into E such thatF - b. Such a mapping is obtained by writing E : 6 CI CI where 6 is a subspace. Then we have a projection of E onto S defined by this decomposition. Since I is the kernel, we have an -induced linear isomorphism a of E onto 6; hence into E. It b t h at s o D g a n d b " : g then by definition s*g, s€6,ge6, -' linear F : A: 5 as required. Conversely, le-t D b' be any 1 : I' Then 6 : Eo is a commapping of E into E such that b' :6. p t e m e n i o f6 i n E . I f s e @ a n d b e E , t h e n s b : s d : [ s 9 ] - [ s & " ] holds for the module multiplications in 6. Let br,5r e E and consider the element
(37)
€E. lbi,b{l - tb,b,1"
ot' E onto E and make If we apply the algebra homomorphism b:! : we obtain lb9-bil- tE,El _ lb,b,)and use of the property 6" b Ibr6rl" :'l6Er1: lb'bzl. Hence we see that -blzl g(br, br) = 1b'r, - fbrbrl" € g . one verifies immediately that (D,,br)- g(br, br)is a bilinear mapping ofExEintog.
88
LIE ALGEBRAS
Now suppose CI : E' is a subalgebra of E. T!"q tai, Sfl e 6; :0 for all 6 n 6 : 0 so that we must have-g(D,,bu) hence -g$i,6r)e T he g (br,br):0ri mpl i es that c onv e rs e i s a l s o c l e a r s i n c e b ,, br . if the only if and i a subalgebra [brbr)" € 6. Hence 6 is l6'r,5{t bilinear mapping I is 0. If E' is not a subalgebra, then -we seek to modify o to obtain a second mapping r of E so that E' is a subalgebra. Suppose this is possible. Then we have a 1:1 linear mapping r of E into E : such that F? - b and, t|i|tl - lb,br7' = 0 for all b,, br. Now let p that E such E into of o r. Then p is a linear mapping
F:F -T :5 -5: Q. Hence bo e g, and we can consider p aS a linear mapping of E into 9. Also we have
e(b;,br)- lb{,El - lb,b,l" -ull - [b,6'l' - [b,b,]o : [ti + bi,6g+ : L6i5il- lbg,til - lb,b,l' If sb is defined as before, we have t6-: [sD'I. Thr1s, if we can somehow choose a comp]eryent of I which is a subalgebra then the bilinear mapping 9(5,, &r) of E x E into 6 can be expressed in terms of the linear mapping p of E into 6 by the formula
s(6,,b,)- blb,- bg6' lb,6,lo Conversely, suppose we have a linear mapping p of E into 6 -p is another 1:1 linear satisfying this iondition. Then r:6 =br : b- and one can re-trace the ,.nupping-of 6 into € such that steps to show that [tl, Uil : lbrbrl' so that E" is subalgebra. Our results can be stated in the following way: Criterion. Let E be a Lie algebra, 6 an ideat il n such that 6, : 0 and set E: E/6. Then 6 is a E'module relatiae to the compositionsb - [sD]. Also there exist 1:1 linear rnappings6 oJ E into E such that F: D, b e E. If o is such a mapping then q s(br,br)-lbibll-lb,b,)" e ' Moreoaer,6 has a complementaryspace which is a subalgebraif and only if there erists a linear mabbing p of E into @ such that g(5r,6r): blb,- 596,- fb,brlo. (38)
III. CARTAN'S CRITERIONAND ITS CONSEQUENCES
89
We observe next that the bilinear mapping 9, which we shall call a E factor set in 6, satisfies certain conditions which are consequences of the speciat properties of the multiplication in a Lie algebra. Thus it is clear that
g(b,E)- o
(39)
which implies g(5r,8) - - g(br,br). We next write
Ib{,6{l: Ib,brl"a sQ,,b,) and calculate
F.,)oi"l u",l+ Ig(b,, Itft, 6{lb{l: [[6,&,]" - llb,brl6rl"* s(lb,br),Fl + Ig(b,,6),t{l . If we permute 6r,6r,Dgcyctically, add, and make use of the Jacobi identities in E and E, we obtain
(40)
b_,,) + [s(b,,6j, 5j1+ g(16_,b_"1, s{b,6,1,a11 b,),bil- 0 . 6r)+ [g(D', + ls(br,b),5i11 e([5'b'),
Our proof of Levi's theorem will be completed by proving the following lemma, which is due to Whitehead. Lsuur 6. Iat 8, be a finite-dimensional semi-simple Lie algebra of characteristic 0, Ilt a finite-dimensional S,-moduleand (lylr)--+g(lulz) a bilinear mapping of 8, x 8, into TJt such that (i ) (ii)
s(l,l):0 , g(fl,tzl,lr) + g(lr,lz)ls * g(Urlrl,l,) i g(lr, lr)l' * g(llsl'), lr) + g(ls, l)12 0 .
Then there erists a linear mafling (iii)
g(lr, lr) :
I --'f of 8 into Tlt such that
lllz - lgl' - l|'lrlo .
Proof: Let S, gr,2i, t4i,T be as in the proof of Whitehead's first lemma: 0 is the kernel of the representation, 8r is an ideal such that 8 : R O,8r, (ur), and (ai), l : 1, .. ., ffi, are dual bases of It relative to the trace form of the given representation, and I'is the Casimir operator determined by the u; and. ud. We recall that l" Set /s : ui in (ii) and take is the mapping r - I,l!,( xnr)u' in fi. the module product with z'. Add for e. This gives
90
LIE ALGEBRAS 0 - Lg(ULlzf, ui)ui * g(lr l)T ,, + xg(tlzu,f,lr)uo * 2@(lr, u;)I,)yd t,t + :.g([zrlrf,l2)ui + Z@@;,lr)l)ui tL
: 9(1,,l)r + I,;q{lJ,l, u;)ur t + \t(lzuil,lr)u' tt
* \g(tr, u\[lruil
* T,(g(/',u;)ui)l,+ >ig((tui,lrf ,lr)ui tt* \t(ur tt
lr)ltrui) + 2(g@tlr)ui)1, .
If we make use of luulj- Eauilt, luill _ Zgnfii, and recall that F;t : - ajt (cf. (28))we can verify that r (41) zg?r, ut)upil : \g(tr, [u;l,])ui (42) Zg(ur, tr)Urui): Zg(u;lr),l,)ui . These and the skew symmetry of g permit the cancellation of four terms in the foregoing equations. Hence we obtain
(43)
- 9(lr,l)r : l,,g(Ur,lrl,u;)u" * \@(1,, ui)u')ht Z@@r,l)ui)I" .
If I' is non-singular we define (M)
,L
lo - Zg(1, ui)u'T-' {:r
Then (43) gives the required relation (iii). If /. is nilpotent, then, as in the proof of Whitehead's first lemm?, ffi -0, S-8, so that the representation is a zero representation. Then (ii) reduces to (ii')
s(UJzl,lr)+ g(tlrlrl,l,)* g(UslJ,lr):0 .
Now let r denote the vector space of linear mappifrgs of g into tjt. we make this into an 8-module by defining f.or A€x., r,/eg, x(Al): -[xl]A, that is, Al: -(adl)A. It is east to see that this satisfies the module conditions (cf. S 1.0). For each / e g we define an element Ar e I as the mapping r + 9(x,/) e gn. Then l - Ar is a linear mapping of I into I and
III. CARTAN'S CRTTERION AND ITS CONSEQUENCES
91
xApgrt: g(x,UJ"l), xA4lz: - g(filrl,lr), xA4lr - - g(fxlr),lr). Hence the skew symmetry of I and (ii/) imply that (45)
Agp2t:
Arrl, -
Atrlt
Thus the hypothesisof Whitehead'sfirst lemma holds. The conclusion states that there exists a p e X" such that A1: pl. This meansthat we have a linear mapping p of I into IJt such that (46)
g(x,I)- -lr,l7o .
By definition of f; as module, this gives (iii). This proves the result for the case /- nilpotent. If f is neither non-singular nor nilpotent' then we have the decompositiorr of llt as Ut. @ Ilt' where the llti are the Fitting components of SJtrelative to f and these are + 0. These spaces are submodules and we can rilrite 9(lt,lr) : 9o(lulr) + gr(lr,lr), gt c TIt6. Then the g; satisfy the conditions imposed oD g, so we can represent these in the form (iii), by virtue of an induc' tion hypothesis on the dimensionality of !lt. This gives the result for fi by adding the linear transformations for the lltr. As we have noted before, the lemma completes our proof of Leui's theorem. If A is a finite-dimensional Lie algebra of charac' teristic zero with radical6 then there exists a semi'sirnple subalgebra 8,of E suchthat E:809. A subalgebra I satisfying these conditions is called,a Leai factor of E. A first consequenceof Levi's theorem is the following result: Conoru,nv 1. Let E, 6, and I be as in the theorem. Then t8el:E'n6. Since so that E' : t881 1 t8gl. Proof: We have E : 8@9 : ([88] n 6) + t8gl - [Eg]. t!$el tr 6 have E' n 6 We have seen that tg8l S It the nil radical of E (Theorem 2.13), so we can now state that E' n g g yt. We know also that the radical of an ideal is the intersection of the ideal with the radical of the containing algebra. Hence E' n 6 is the radical of E'. We therefore have the following Conorl,.onv 2. The radical of the deriaed algebra of a finite' dimensional Lie algebra of characteristic 0 is ni'lbotent.
92
LIE ALGEBRAS
We take up next the question of uniqueness of the I Levi factors. It will turn out that these are not usually unique; hpwever, they are conjugate in a rather strong sense which we shall now define. We recalt that if. zelt, the nil radical of E, then ad a is nilpotent. Since ad e is a derivation we know also that A : exP (ad a) is an automorphism. Let lI denote the group of automorphisms generat' ed by the elements exp (ad e), z e !t. Then we have the following conJugacy
Theoremof Malcec'-Harish-Chandra.Let E: 6 @ I where 6 is a solaableideal and 8, is a semi'simple subalgebraand let 8' be a semi-simplesubalgebraof E. Assume E finite-dilnensional and of characteristic 0. Then there exists an automorphism A e 2l such that 8f q 8. Proof: Any /r € 8r can be written in one and onl$ one way as l, : ll + l{, where /l e S and /i e 6 so that we have the linear mappings I and d of ,8, into 8 and 6, respectively. Since8r is semi-simple,8, o 6 : 0; hence ,l is 1 : 1. lf- lz e 8r lhen
(u)
llrlrl-ll,Irl^ +IIJ.it"
: t/i/|l + uitn + ltitil + ltitil . '
Hence (48)
It,t,)x- ullll , lt,t,)"- vitn - lflil + ll",l{1.
The second of these equations shows that UrlrT' € [EF] I It, the nil radical of E. Since 8{ : 8, this implies that l{ etn fdr every /' €,8r and so 13,g 8 O tt. We shall prove bi induction thdt there exists an'automorphism Ai € ?I (A, - 1) such that 8fi g I + !t'i) where ltit is the ith derived algebra of [t. Since It is solvable this will prove the result. Since we have proved that 8r g I + Tt it suffices to prove the inductive step and we may simplify the notation and assume that 8r g I + Tl'&'. Then we shall show that there exists A e lI such that 8f g I + 1{rc+t). If we use the notation introduced before, 5J,g U + tt(&) implies that fi € tt'&), /, € 8,. The first equation in (a8) implies that if we set z/r --lz,tll, z € nf&), h€ 8,, then this rnakes Il'o' into an 8r-module. Now 1{ft+t) is a, submodule so where e e It(*) that 9t(h/!|t&+r1is an 8r-module relative to Zlr:litij of the a nd t : z *91{ t c + r ). We n o w ta k e th e c o s e t srel ati vei tott(& + r) (e+'), (a8). we have Since € 9t of equation terms in the second tl{lil
III. CARTAN'S CRITERION AND ITS CONSEQUENCES
93
: vrtil- wTl -Tih -Et, . Jt,t,l" Now set /(/,) - ti then I, -,f(t,) is a linear mapping of 8r into the and the foregoing equation can be re'written as lr-module 91t'tr191t&+t'
f(U,t,D- fQ)lz - f(I,)l' .
(4e)
g1ttryltrc+t) Hence by Whitehead's first lemma there exists a 2 C that means which such that f(l'):21t, (50)
n :
V7ll ot l{ = lz,/ll (mod ltt+rr; '
Let A - exp (ad e). Then
tf : t, + Ual+ Uzllll,zlzl+ . . . (51)
= h * Urzl (mod !t(&+r))
= /f + l{ + ltizl + lt{zl(mod9t(&+r)) = /l (mod tl'u*") .
Now 8f = 8, and (51) shows that 8l g I + Tt'&*". We can therefore prove the result bY induction on &. Conor,urny 1. Any semi'simpte subalgebra of a finite'dimensional Lie algebra of characteristic zero can be imbedded in a l*ai' factor. Proof: If. A is as in the theorem, then 8r is contained in the Levi factor 8'-t. @6 where 8r and 8z are serni'' Conor,uny 2. If E- 8,06:,lJz simple subalgebras then there exists an automorphi'sm A e 2I such that 8f : 8r. This is an immediate consequence of the theorem. 70. Cohomology groups of a Lie algebra The two lemmas of \Mhiteheail can be formulated as theorems in the cohomology theory of Lie algebras. Historically, these con' stituted one of the clues which led to the discovery of this theory. Another impetus to the theory came from the study of the topology of Lie groups which was initiated by Cartan. In this section we give the definition of the cohomology groups which is concrete and we indicate an extension of the "l- non-singular" case of White' head's lemmas to a general cohomology theorem. Later (Chapter
LIE ALGEBRAS
V) we shall give the definition of the cohomology groups which follows the general pattern of derived functors of Cartan'Eilenberg. Let 8 be a Lie algebra, fi an 8-module. lt i 2 t, an'd-dimensional Nt-cochainfor 8 is a skew symmetric f-linear mapping of 18x8 x'' . x8 (i times) into !]t. Such a mapping/sends an i'tuple (lr,lr, "',li), /o e 8, into f(lu - - -,li) e !]t in such a rvay that for fixed values of lr, . . . ,lo-r, lq+rr..., /; the mapping ln-+f(lr, '' ',1;) is a lirrear mapping of 8 into St. The skew symmetry means that f is changed to -/ if any two of the /i are interchanged (the remaining ones unchanged). lt i: 0 one defines a $-dimensional\I\-cochain for 8 as a "constant" function from I to llt, that is, a mappinE I n u, u a fixed element of lft. If / is an i-dimensional cochain (or simply "an a-cochain"), i )- 0, / determines an (f + l)-dimensiondl cochain /d, called the coboundary of f, defined by the formula t+r
(52)
fd(l',' ' ', /t*,) - X(q=r
l)'*'-of(lr, ' ' ' , tn, ' ' ' ,l r* r)l n
t+l
A
. .., I.n,-.. , tr, "'r li*rrUolrl) + E t- L)'+qf(l', 41r=l
Here the ^ over an argument means that this argument is omitted this is to be interpreted as (e.g., f(11,fr,7r1-f(lr,t)). For r:0 (/d'Xr) - il|, if / is the mapping r -> u c $!t. The set Co(8, fi) of f-cochains for llt is a vector space relative to the usual definitions of addition and scalar multiplication of is a linear mapping, the coboundary functions. Moreover /-'ld operator, of Ci(8,IIt) into Ct*'(8, tn), i > 0. Besides the case
fil(l) - sa1,
(53)
if. f: x -+ tr. ,
we have (54) (55)
fd(lb l,) : - .f(1,)1,+ f(1,)1" - f(llJi) , (.fd)(l',lr,lr) : f(lr,l")l'- f(l.r,lr)1,a f(11,l)lt - f(lr,ll'lrD + f(lr, U'l'l) - f(l',[/r/'l)
.
An a-cochain / is called a cocycle it f6: 0 and a'coboundary if g8 for some (f - l)-cochain g. The set Zi(8,0n) df i-cocycles is f the kernel of the homomorphism d of Ci into Ci+', sb Zi is a subspace of Ci. Similarly, the set B{(8, Dt) of e'coboundariesis a subspace of.Ci since it is the image under d of Ci-t. It can be proved fairly directly that Bi g Zi, that is, coboundaries lare cocycles.
III. CARTAN'S CRITERION AND ITS CONSEQUENCES
95
the coboundary This amounts to the fundamental property: d2:0 of general case at the in operator. we shall not give the verification later point abstract 9f viery this point since it will follow from the :0 f.or f a verify to fd' content be shall on. At this poirit we + r,t x mapping the is is, that u, if. ,f f 0- or a l-cochain. Thus - - ulzlt * ulrlz - utlrlzl'1 by the then fy(t) - yg and f62Qt,lz) by given /z) is definition of a module' If / is a l-cochain' /d(/" (54). Hence, bY (55)' - f (Ll'l f (t)I rt, * f (t')l'l' .fB'(Ir, l r, l r) :' "1)l' - f(l,)l'l' * f(t)t,tz - f(1,)1,t, + f(l,l,l)l, - f(l')UJ'l 1 fQ1)lzh - fU.l'zl)I, '1 f(ll'lzl)l'
- f$J,l)l, * f(l')ll'ts7 + f(ltslt,t,11) - f(l,)url'J - f(l,tl,l,ll) + /(Jt,t'l)/' + JW'lUsI)'
:0 for any L'cochain f' one checks that this sum is 0; hence f6' :0 has been made, one can define the Once the verification D2 (sbace) of P olyti'ae to the module wt group i-dimensi.onal cohomologt tJt). lf i :0 we agree = tJt)/Bd(8, Zi(g, lft) as the factor space ff1t, l)'cochains. Hence in this since there are no (i B,:0 ;; ;;" be identificase it is understood that I/o(8,lJt) : Zo(g,Wt)- This can ul: 0 for ed with the subspace l(ilt) of elements z e !)t such tl1p;t lIt' module all t. such elements are called intariants of the l-cocycle every H'(g,fi) : 0 means that Zi(g,Ift) : Bd(8, TJt),that is, this states that if l->f(l) is a linear is a coboundary. For i:1 - f(tr)t, + f(t,)I, - f(ll|tzl): 0, then mapping of 8 into !ft such that This is just the typeof there exists a u in lft such that f(t):ul' lemma' Similarly' first statement which appears in Whitehead's the second coho' about statement is a Whitehead's second lemma now be stated in can results two these fact, groups. In mology waY. the following Tnponnu 13. If g is f.nite-dimensional semi-simpleof characteristic Q, then I/l(8, sJ|): 0 and, H,(g,fi) : 0 for eaery finite-dimensional mod,ule Ult of 8,. - $Jt,O S, where the l]ti are subIt is easy to see that if tX and modules of Dt, then F/t(8, fi) : F/i(8, tJt,) o Frd(8'rJt')' This the of the theorem of complete reducibility permits the reduction and irreducible 1lt F/r(8, tlt) for finite-dimensional fi to the case
LIE ALGEBRAS
96
In here one distinguishes two cases: (1) In8 + 0 and (2) Ut8 :0. implies dim IIt : 1, so; Ilt can be the second case irreducibility identified with the field @. Then an r'cochain is.a skew symmetric a-linear function of (/t, ...,1t) with values in0, and sirtce the rep' resentation is a zero representation, the coboundary fofmula reduces to (56)
f d ( l r ,- . . , l i * r ):
H
t - l ) ' r o . f ( l r.,. . , f o ,. . . , t , , . - . 1 , , 1 ; a 1 , l l o l , l )
It turns out that Lie algebras the cohomology -groups with values "r'r-*r-simple O are the really interesting ones, since in l}t these correspond to cohomology groups of Lie groups. On the other hand, the case lltS + 0 is not very interesting (for sertri'simple 5J, finite-dimensional irreducible llt) except for its applications to the theorem of complete reducibility and the Levi theoredl, since one has the following general result. Tnnonru 14 (Whitehead). Izt I be a finite-dimensional serni' siruple Lie algebra ouer a fi.eld of characteri,sti,c 0 and let TIt be a Then finite-dimensional irreducible module such that SJIS* 0. a l l i Z 0 . //'(8,!)t):0 for holds lf. i:0 the irreducibility and IJIS + 0 imply that u8:0 This means that F/o(8,fi) : 0. The proof for only for u :0. i > 0 is similar to the proof of the case: I- non-singular, in the 'We leave the details to the r€ader. two Whitehead lemmas. 1.L. More on eomplete reducibilitA
l
For our further study of this question we require a notion of a type of closure for Lie algebras of linear transformations and an imbedding theorem for nilpotent elements in three'dimensional split simple algebras. The first of these is based on a spedial case of a property of associative algebras (the so-calledWedderburn principal theorem), which is the analogue of Levi's theorem on L[e algebras. The result is the following THnonpu 15. I-et \ - Olrl be a finite-dimensional algebra (associa' tiae with identity l) generated by a single elernent x ouer 0 of charac' teristi,c zero and let ft be the radical of 11. Then \l contains a serni' simPle subalgebra\\ such that \I: ?I' O ft. Proof: Let /(i) be the minimum polynomial of r andi let
III.CARTAN'SCRITERIoNANDITSCoNSEQUENCES9T (57)
fQ) :
be the factorization of the leading coefficients deg zi(,i) > 0" We note zero nilpotent elements is nothing to Prove. In (58)
rr(J)"tnz(),)"'"'n,(f,)" f( ) into irreducible polynomials with one such that nQ) + r.t[) if i + i and first that if all the ei = 1: ?I -has nS lon(cf.. p. 47), so lI is semi-simple and there anY case, set
f'(l) : r'Q)rc2(l)"'n,(l)
T hen if e :ma x (e i ), z " :(fr(x ))" :rc r(x ) e" ' r,(x)c:$' an d z :fr(r ) . (e) geneso that z is nilpotent. Since lI is commutative, the ideal g (e) on theotherhand, s. ft@)=o hence rated by z is nilpotent; : tr + (z) i coset the polynomial of minimum (e)). the Hencs (mod generates in ?I/(e) is a product of distinct prime factors. Since .f It a/Ui), this means that ll(e) is semi-simple. Hence (z) ft. : is fr r * follows also easily that the minimum polynomial of r y whose an element fr(l). Hence it suffices to prove that lt contains minimum polynomial is fr(A). \4Ie shall obtain such an element by a method of "successive approximations" beginning with Jr': I' (mod S) and r= nr(modS)' Suppose To begin with we have f{r):0 gry&)and we have already determin ed, xr such that fr@r) = 0 (mod in determined : be is r,o to eu where I* * r = rr(mod S). Set rr+, theorem Taylor's by have, (mod we no*'). fte so that fr(rt +r): 0 for polynomials, ltlr-t'
- f,(xr* w) -f{x) + f!(x*)w*t#w' f,(xr+r)
+ "' '
Since the base field is of characteristic 0, ,f,(i) has distinct roots in the algebraic closure oI O. Hence /'(i) is prime to the derivatrQ). It follows that V -ffi : f 'r(r*)* S has an inverse D five f Then w = 0 (mod S&), so that in ?I/fi. Set a,o- - f,(x)a. ''(x)w (mod Yte*') f,(xx+) = ft@) + f = fr@i - f'r(r*)afr(rr) (mod no*') = fr(xi - f{tck) (mod S**') = 0 (mod nt*'). Thus we have determin€d .f,r+rsuch that f{n*r) = 0 (mod s&*t) and (mod fr). Since S is nilpotent this process leads to a y I z frrc+r suchthatf{y):0andy:r(modfr).HenceU'-oly)satisfies 2[ : lI, o a. since the minimum polynomial of .7 is .f'(A), it follows
98
LIE ALGEBRAS
that the minimum polynomial of y is /,(i) We can now prove
also.
Tnsoneu 16. Izt X be a linear transformation in a ,fi,nite-di,mensional aector space oaer a rteH of characteristic zero. Then we can write X: Y * Z where Y and Z are polynornials in $ such that Y is semi-simple and Zis nilpotent. Moreoaer, if X - 11 * Zr where Yr is semi-simple and Zt i.s nilpotent and Yt and Zt cdmmute wi,th X, then Yr : Y, Zt: Z. Proof: The existence of the decomposition X: Y *,Z is obtained by applying Theorem 15 to the algebra @fXl. Now suppose X : Yt a Z', where I.r and Z, have the properties stated in the theorem. Since IZ and Z are polynomials in X they commute with Y, and Zt. We have Y - Yt: Z, - Z. Since Z andl,Zt are nil' potent and commute, Z - Zt is nilpotent. Since Y land Yt a;re semi-simple and the base field is of characteristic zerb, the proof of Theorem 11 shows that Y - Yt is semi-simple. Since the only transformation which is both semi-simple and nilpoten{ is 0, Y-Yt:0:
Z-Zt.
Hence Y : Yr, Z: Zy We call the uniquely determined linear transformatidns Y and Z of Theorem 16 the semi-simple and the nilpotent compodents of X. Dprrr.rrrrou 3. A Lie algebra I of linear transformptions of a finite-dimensional vector space over a field of characteristic 0 is called almost algebraic* if it contains the nilpotent and semi-simple components of every X e 8. To prove our imbedding theorem we require the following two lemmas. Lpuui 7 (Morozov). Let 8 be a fi.nite-dimensional Lie algebra of characteristic 0 and suppose8, contains elementsf , h such rthat [fn1 - 2f and h e tS/1. Then there exists an element e € 8, buch that * This conceptis due to Malcev, who used the term spli,ttablic. We have changedthe term to "almost algebraic" since this is somewhatweakerthan Chevalley'snotion of an algebraic Lie algebra of linear trarjrsformations. Moreover,we have preferredto use the term "split Lie algebra" in a connection which is totally unrelatedto Malcev's notion.
III. CARTAN'S CRITERION AND ITS CONSEQUENCES
99
(lfhl: -Zf) . lefl - h lehl:2r, Proof: Thereexistsa ze9, such that h-[zf]. Set F:adf,
(59)
H : ad,h, Z:
ad z so that we have
(60)
H:lZFi. lFHl:-2F, The first of these relations implies that F is nilpotent (Lemma 2.4). Also tlzhl - 22,fl - llzf lhl + Lzlhfll Z[zf I :0*2h-2h:0. Hence [zhl - 2z * h where fr € 0, the subalgebra of elements r such that lxfl - Q. Since tFHl -- - 2F, if. b e S, then bHF -- b(FH + 2F): Q . Hence bH e S and so SH s S. Also we have [ Z F ' ] : l z F l F r - ' + F I Z F I F ; - s+ . . . + F t - ' I z F l : f{Pi-r + FHF|-'+ ..- + Fi-tH and since HFk : FkH +%kFk, we have (61) [zF'l : Fi-'(H + 2(i- 1) + H + 2(i - 2) + "' + H) - 'rt-t(H + (t - 1)) . Let b e A n SF'i-t. Then b - aFi-' and bF : aFn : 0. Hence iaPi-'(H + (t - 1)): a(zFt - Fnz): (az)Fi e 8Fi . Hence b(H + (t - 1)) e R n !Fi. It fotlows from this relation and the nilpotency of.F that if D is any element of S then (62)
bH(H+ L)(H+ 2) ... (H * m) - s
for some positive integer m. Thus the characteristic roots of the Hence H - 2 restriction of H to R are non-positive integers. and consequently in S non-singular linear transformation induces a there exists a yr e S such that yt(H - 2) : rr where rr is the element such that lzh):22 * rr. Then lyrh):Zlt * xr. Hence if we set z : z - Jr we have lehl - 2u. Also lefl: lzfl - h. Hence (59) holds. Lpuu^n 8. Let t, be a Li,e algebra of linear transformations in a fi.nite-dimensional aector space oaer a field of characteristi,c 0. Sufifose euery nilbotent elem.ent F + 0 of 8, can be imbedded in a subalgebra
lOO
LIE ALGEBRAS
- 28, l"FHl - - 2F, tEFl - H. with basi.s(E, F, H) such that tEHl Let ft be any subalgebra of ,J which has a complementary space Il in g inaariant under multiplication Dy s: I : 'QG) Yt' ltts]i g tt' Then fi has the ProPerty stated for 8. Proof: Let F be a non -zero nilpotent of R. Then we can choose E and // in 13 so that the indicated relations hold. Write f/: H r * Hr , H, € A , I/2 e n , E - E t* Ez , Er€ . S , E e Tl . Then w e have -2F - IFH) -- [FH') + [FH,] and'IFH,I e fi, [FH2]:e ft. Hence -zF - [FH,l. Also 17 -- lEFl: lE,F1 + lE,Fl. This lmplies that Ht : IE,FI e tAFl. Thus f/, satisfies for F, S the conditions on 'I{ in Lemma 7. Hence there exists E', H' in n such that IFH'I - 2F, IE'H'l -- 2E', [E'F] -- H' . The subalgebra generated by F, 8,, H, is a homomorphic image of the split three'dimensional simple algebra. Since F + 0 we have an isomorphism, so that F, E' , H' arelinearly independent and satisfy the required conditions We can now establish our secondcriterion for complete reducibilityTssonpru 17. Let g be a l.ie algebra of linear transformations in a fi.nite-dimensional aector space Yt ouer a field of chayacteristic 0' (1) Assume g completely reducible. Then euery non-zbro nilpotent elernent of g can be imbedd,ed in a three'dimensional sflit sirnfle subalgebra of g and 8, is almost algebraic. (2) Assume that eaery non-zero nilfotent elernent of t can be imbedded in a three'dimensional simpte subalgebra of g and that the center 6. of 8 is almbst algebraic. Then 5Jis comPletelY reduci'ble. Proof: (1) Assume ll is completely reducible and lbt 6 denote the complete algebra of linear transformations in 9Jt- Let F be a - tjtr e ilftf O ' ' ' G) 9Jt' nilpoteni linear transformation and let st be a decomposition of IJt into cyclic invariant subspaces relative ' to F. Thus in 9lt; we have a basis (tr', Ji,,.,", r^;) such that r1F: linear transformations the ri+r, fi^rF : O. We define f/ and E to be : (mt - ZJ)xi, xnE:0' xiH leaving'"u.ty !Jl; invariant and satisfying -1 ))r;-,, j > 0 (3 6 ). (c f. as i n $8, [E /{ ]: Then + iU ri E : q- m , i 2E,|FH}_-2F,tEr]:H.ThisshowsthatFcanbeimbedded in a subalgebra AE + OF + OH of. the type indicated. We shall show next that we can write @t :8 O tt where S is a sub' space such that [!t8] g tt. It will then follow from Lemma 8 that u.r"ry nilpotent element + 0 of I can be imbedded in a split three' dimensional simple subalgebra of [3. We recall that d^uas module relative to Il (adioint representation) is equivalent to Dt I tll*, tn*
III. CARTAN'S CRITERION AND ITS CONSEQUENCES
TO1
the contragredient module. It is also easy to see that rlt* is completely reducible. Hence, by Theorem 11, tn S rlt*, and consequently oz, is completely reducible relative to 8. Since I is a submodule of @z relative to 13,there exists a complement !t such that Gz: I O tt, [9t8] s tl. This completes the proof of the first assertion in (1). Now let x be any element of I and let y and z be the semi-simple and nilpotent components of X. Then ad6X:
adGY * adsZ ,
[ad6lzad6Zl:0 and ad.gZis nilpotent (Z^:0 implies (adgZ)2*-':0). Also the identification of G with !x I Dl* and the proof of Theorem 11 show that ad6lz is semi-simple. Hence ad6Y and ad,6z are the semi-simple and nilpotent components of ad6X and so these are polynomials in adgX. Since 8ad6X:c.8 and aci6lr, ad6Z are polynomials in ad6X, 8ad6Y s 8, I ad6Z g t. Thus L-+ILyl, L-\LZ) are derivations in Il. we can write ,8 : .8'O 6 where lJ' is semisimple and o is the center. Since the derivations of t' are all inner it follows that any derivation of I which maps G into 0 is an inner derivation determined by an element of 8'. since z is a polvnomial in x, IXCI - 0 implies [zc] -e. This impries that the derivation L-rVZl maps 6 into 0. Hence there exists a Z, e g, such that lLZl - [LZrl, L e 8. Since Z is nilpotent, ads,Zr: adg,z is nilpotent. since ll' is semi-simple, the result just proved (applied to ad8') implies that there exists an element u e fJ' such that lad,g,Z',adg,U] : 2 adg,Zr. Then lzrul : 2Zr, which implies g, lxzrl: that Zr is nilpotent. Since lXZl: 0 and since Z is a polynomial in X, lZZ,l: Q. It now follows that Z - Z, is nilpotent. Since [L, Z - Zr]:0, L € 8, and Z - Z, is in the enveloping -g*associative algebra 8*, z - z, is in the center of [J*. since i. completely reducible and z - z, is nilpotent, this implies Z - 21: e so Z:Zre8,. Hencealso Y:X-Zeg. T h i s c o m p l e t e st h e proof that I is almost algebraic. (z) Assume g has an almost algebraic center and has the property stated for nilpotent elements. Let € be the radical of I and let F e tsgl. Then we know that F is nilpotent (corollary 2 to Theorem z.g). rf. F is not zero it can be imbedded in a three-dimensional simple subalgebra.R. Since s n @ + 0 and s is simple,,R s 6 which is impossible becauseof the solvability of. @. Hence F: 0 and [86] * Q. This implies that 6 : O the center. By Levi's theorem ,8 : O @,8, where gr is a semi-simple subalgebra. Since 6 is the center this implies that
LOz
LIE ALGEBRAS
8, is an ideal. We can now invoke Theorem 10 to pfove that 8 is completely reducible, provided that we can show 1that every C e 6 is s em i- s im p l e . N o w w e a re a s s u mi n g th at C - D + E w here D is semi-simple, E nilpotent, and D and E are in 6. It' E + 0 we can imbed this in a three'dimensional simple subalgebra. Clearly this is impossible since E is in the center. This coinpletes the proof of (2). It is immediate that if 8 is almost algebraic, then the center 6 of I is almost algebraic. Hence we can replace the as$umption in (2) that 6 is almost algebraic by the assumption that I is almost algebraic. We recall that the centralizer 6s(S) of a subset S is the set of elements y e I such that [sy] :0 for alt s e S'j This is a subalgebra of 8. We shalt now use the foregoing criterion to prove Tsnonpn 18. Let g be a comptetely redacible Lie algebta of linear transformations in a finite-dimensional aector space Tlt',of charac' teristic zero and, let & be a completely reducible subalgebra of 8' Then the centralizer 8r: 6e(8') is completely reducible. i Proof: Let X € 82. Then since I is almost algebraid, the semisimple and nilpotent parts Y, Z of X are in 8. Since these are for C €,8r implies lCYl F 0: tCZl' polynomials in X, tCX):0 algebraic. We shalll show next is almost Hence Y, Z e.8, and 8, of I such that [8a8r] tr '8r' : subspace a is 8, ,8, where that I O 8a Lemma 8 that every niland, L7 Theorem from foltow It will then potent element of 8, can be imbedded in a three'dimensional split iimpte algebra. Then 8z will be completely reducible by Theorem L7. Now we know that adgSr is completely reducible (proof of Theorem 17). Since 8 is a submodule of 6 relative to 8r1,I is completely reducible relative to adg8r. Thus we may writd
8:fi,Ollt,O"'G)Dt* where [!]ti[,l g TJI;, i : !, . . ., k, and tJJti is irreducible relative to :0, i adg8,. we assume the !)ti are ordered so that Plt;8,J 1, 1..,h, and [fir8,] +0 if j > h. Since the subset & of elements g]t,i, it is immediate that zi such that [zt8r] :0 is a submodule of 8z : lltr + "'
* !]tr -
Set,8s: IJlp+r+ ... * Iltr. fhen I : 8rO8'. If. i > h, the\[!]tr'8'l +0 and [1]t;8,1+ [[!]tr$.l8,l + ... is an lr-submodule + 0 of [ti. Hence glti : pltr.8'l a ttlltr8rl,8'l * ' ' '. This implies that
III. CARTAN'S CRITERION AND ITS CONSEQUENCES
g, - [.8r8,]+ [[g3g'Jg'] + ... s [gg,] . On the other hand, I : ,8,@,Iai hence,[88,] - [8r8,Js ga. Hence 8, : [88,Jand [9,8,] - [[gg,lgz]tr [g[9,9,]l+ tlsg,lg,l tr [gg,] : g, . This shows that 8c is a complementof .8,in 8 suchthat [fJs.8e] s 8a, which is what we neededto prove. Exereiees In all these exercises the characteristic of the base field will be zero, and unless the number is indicated with an asterisk the dimensionalities of the spaces will be finite. 1. Show that if S is a cartan subalgebra of 8 then 0 is a maximal nil. potent subalgebra of 8. show that the converse is false for onz@ > 2). 2- Let 6 be a nilpotent Lie algebra of linear transformations in ![l and let sl - !n0 @ il?t be the Fitting decomposition relative to 0. Show that if @ is infinite, then there exists an A e 0 such that fio = Wto;., xlh : fllta, ffic* the Fitting components relative to A. 3. Show that the diagonal matrices of trace 0 form a Cartan subalgebra in the Lie algebra 8 of triangular matrices of trace 0. Show that g is complete 1. Let I be the subalgebra of e2s of matrices A satisfying s-rAts = - | where
s: (; ,J) (This is isomorphic to an orthogonal Lie algebra.) matrices d i a g { l t , . . . , f , r ,form
5.
a Cartan
subalgebra
)r,
Show that the diagonal
- i rl
of B.
Same as Exercise 4 but with S reptaced by
e:( r-
1:). 11
0)'
6. Generalize Exercise 2.9 to the following: Let B be a Lie algebra, 0 a nilpotent subalgebra of the derivation algebra of g. Suppose the only e l e m e n tl € 8 s u c h t h a t l D = 0 f o r a l l D € S i s l : 0 . Then provethatgis nilpotent. 7. Show that if 8r is a semi-simpleideal in 8 then 8=8r@gz where gr is a second ideal.
LIE ALGEBRAS
104
8. Let S be an ideal in 8 such that 8/S is semi-simple. Show that there exists a subalgebra8r of I such that 8: SO8r. 9. Let 8 be simple over an algebraically closed field @ and let /(4, b) be an invariant symmetric bilinear form on 8. Show that / is a rhultiple of the Kitling form. Generalize this to semi-simple 8. 10. Let Slr be the n-dimensional irreducible module for d split threedimensional simple algebra S. Obtain a decomposition of. llllns lll' into irreducible submodules. 1l*. Let e, h be elements of an associative algebra such thdt llehlhl: 0. Show that if h is algebra,in, that is, there exists a non-zero polynomial d(f) such that 6(h) :0, then lelal is nilpotent. l2*. Let ?I be an associative algebra with an identity element I and suppose ?I contains elements e, f, h such that lehl -- %, Ifhl - - 2f , lelJ : h. Show that if d(h) e?I is a polynomial in h then €id(h):6Qt,+2i)ei, i:0,t,2, 6(lift : fi6(ll + 2i). Also prove that if r and n are posifive integers, I r 5 n, then r
'lrlzj r I' ' 'Ie"flfl' ' '"fl = f c " ,i fI @ + n - t)si -t+ r-zJ
J= 0
t=l
. /n\t n -r\. . wnere cmr|= \ I l\ r _ zj)r l3*. 21,h, e, f as in Exercise 12. Show that if e- : 0, then zm-l
g(h*m-i):o 14. Prove that if e is an element of a semi'simple Lie algehra I of characteristic zero such that ad e is nilpotent, then eR is nilpotGnt for every representation ,B of 8. 15. Prove that if I is semi-simple over an algebraically closod field then 8 contains an e * 0 with ad e nilpotent. 16. Prove that every finite-dimensional Lie algebra * 0 over tan algebraic' ally closed field has indecomposablemodules of arbitrarily highlfinite dimen' sionalities. (Hint: Show that there exists an e e I and a reprbsentation .B such that eE is nilpotent * 0. If $t is the corresponding module, then the '" dimensionalities of the indecomposablecomponents of ff|, D?8tIt, $8m8![t, are not bounded.) 17. Prove that any semi-simple algebra has irreducible modules of arbitrarily high dimensionalities. 18. Show that the derivation algebra of any Lie algebra [s algebraic. (Hint: Use Exercise 2.8.) 19. A Lie algebra I is called reiluctiae if ad I is completely reducible. Show that I is reductive if and only if I has a 1 :1 completOly reducible representation.
III. CARTAN'S CRITERION AND ITS CONSEQUENCES
105
20. A subalgebra s of I is called reductiae in I if. adgs is completely reducible. Prove that if 8 is a completely reducible Lie algebra of linear transformations and s is reductive in I then s is completely reducible. 21. Show that any reductive commutative subalgebraof a semi.simpleLie algebra can be imbedded in a Cartan subalgebra. 22. Show that any semi-simple Lie algebra contains commutative Cartan subalgebras. 23. Let A be an automorphism of a semi-simpleLie algebra 8. show that the subalgebra of elements grsuch that U(A - L)^ = 0 for some ?c is a reductive subalgebra. (Hint: Use Exercise 2.5.) 24. (Mostow-Taft). Let G be a finite group of automorphisms in a Lie algebra. show that I has a Levi factor which is invariant under G. 25. Let /.(i) = det (tl - ad a), the characteristic polynomial of ad a in a Lie algebra 8, and let D be a derivation of U. Show that if t is an indeterminate, then /"+nn()t) = fo(A) qmodt2) . (Hint: use the f.act f,,1()') : .fo?) if A is an automorphism and the fact that exptD: l * tD+(tzDzpD+... i s a w e l l - d e f i n e da u t o m o r p h i s mi n g p , p the field of power series in t with coefdcients in @.) 26. Write f"(l)= l" -q(a)lo-t +n(a)lo-? +...-r-1- l),rr(a)p-t... and let rt(ar, . . . , ar) be the linearized form of rt defined by ct{ar,.:.,at)
:f
* ".*or) ["(o'
-!r1(ar* ...*d1*. ..*ad* )
)
r a ( o r *. . . . + 6 t * . . . * 6 r * .
J 0 o r then o*tlp*t t < 0. we shall refer to the ordering of sf just defined as the leri.cographic ordering determined by the ordered set of roots (ar, ar; . .., a,t)(which form a basis for 6i). Lprrru.r,1. Let pr, pz, "' , p,,c $t . Supposethe pr > A and (pu,pr')s 0 if i + j. Then the p's are li,nearly inde\endent oaer Q.
120
LIE ALGEBRAS
Proof: Supposepr: Ef-t,l,pr: 21'opn+ L ll'pr wherp 1S4, s 5 h - l ^ , t i > 0 , t l ' S 0 . S e t X i i p q : ( 1 , 2 1 ! r ' p r : " . S i n c ed r t ) 0 , o * 0 . We have (o, r) : >, ),'q^'r'(pq, o.) 2 0. Hence (p * ,o ) : (o , o ) * (o ,r) ) 0 . On the other hand, (pr, o) - 2l[@*, p) 3 0 which is a contradiction. Hence the p's are Q-independent. DprrNrnoN 1. Relative to the ordering defined in 6i we call a root a simple if. a> 0 and a cannot be written in the form F + r where I and r are positive roots. XV. Iat tr be the collection of simple roots relatiad to a fired lexicographic orderins of bt. Then: (i ) If o,0er, a+ P, thena- 0 is not a root. ( i i ) I f a , F e n a n d a + g , t h e n( a , 9 ) < 0 . (iii) The set tr is a basisfor bi oaer Q. If B isany hositiue root, then 0 -- Z"enkra where the k" are non-negatiue integers. (iv) If A is a positiae root and B 6 o, then there erists a,n d, e rE such that B - a is a positiae root. i s a p o si ti ve root, then a: Pr oof : ( i) I f . a ,l e tr a n d a -P c ont r a ry to th e d e fi n i ti o n o f r. l I a- F i + a negati ve 9 * @ - 9) root we again obtain a contradiction on writing B : (9 - a) + a. (ii) Let I - /&, I -(r -I)a, "', F + qa be the a-strinFcontaining P. Then 2(a, ill(a, d) : r - q. Since r :0 by (i) and (4, 4) > 0' (a, F) < 0. (iii) The linear independence of the roots tontained in Tc is clear from (ii) and the lemma. Let F be a positive root and suppose we already know that every root T such that F > r > 0 is of the form Zrenkr&, k' a non-negative integeri We may s u p p o s e a l s o t h a9t d t r , s o t h a t 9 : 0 ' . * 9 2 , f u > 0 . T h e n 9 > 9 + pr:.\htiu, h'", k'r: non'negative intisers; hence and pr:lh',a, p'J)a : is required form for p., If p is a which the 9 ZtkL + negative root then -F is a positive root. Hence F : X kra where the ho ?te integers 5 0. The first statement of (iii) fpllows from this and the linear independenceof the elements of r. (iv) Let F be a positive root not in z, The lemma and (iii) imply that there is (r,q a n a € z rs u c ht h a t ( 0 , 4 ) > 0 . T h e n Z ( F , a ) l @ , & ) : r * q > 0 p a < 0 , t hen is a root. If and 9-a as before). Hencer)0 a € r . H e r l c e c o n t r a r y t o : a : a n d a,- F > 0 9-a>O B +(a- F) and F - (P - a) * a where a € r. We now write 7t: (&t, &zt ..., cs) and we call this the simble
ry. SPLIT SEMI.SIMPLELIE ALGEBRAS
I2T
slsteln of roots for I relative to O and the given ordering in 6J. - Zk;at where the k; are integers We have seen that every root F and either all, ki ) 0 or all &r < 0. This property is characteristic of simple systems. Thus let fr : (dr, dzt ''', er') where / : dim O where the kt and the dt Ata roots such that every root B:}k;aa our Evidently too). (rationals do would are integers of like sign the introduce We for ft basis is a 6f. hypothesis implies that a;: or the based of Oi lexicographic ordering h0 Then the positive roots g : X k&; ?te those such that the k, > 0 and some h, > 0. It is clear that fio d; is a sum of positive roots. Hence the a; are simple. Since any simple system consists of / roots the set E : (&r, dzt . .., dt) is the simple system defined by the ordering. lf n : (&r, dr, . , ,, dt) is a simple system of roots' the matrix is called aCartanmatrix for 8 (relative (Aii), Aii:2(a;,,a)l(e;,d;) and the of this matrix are Ai+:2 entries diagonal to o). The -1, -2 or -3. (XII and XV (ii). off diagonal entries are A;t:0, If i + i the d+ zfid, e5 1tE linearly independent so that if dir is This gives the angle between ar and d; then 0 3 cos' 0;5 11. < 4; hence 0 5 A;tAii 0 , and4 r > 0 q a n d r i n t e g e r s , a r e isa root +0) wherethe a , s tri n g c o n ta i n ing th e 9= d\r* ..' * d;r,. a ndr ar e det er m i n e db y This string is known from (A;i). A similar argument shows that lorr, "' oi1r,fi€if :
slerr,"' eih,)
: where s is a non'zero integer which can be determined from the Ari. It follows that s [ et r ,' " e i y ,l : l e i r," ' e i * ,f$ 1 7 ' ' ' 0+ n' €i' 1 - -q (r * 1 )[e ;r," ' € 4 ,e i s + 2 )' tl ^ i r, " ' € ;r,e i rr+ 2"), ' e i k ' e i \ w here /1i s a rati onal Th us le; r ," ' e4, l: number which can be determined from the Aii. Thi$ reduces the discussion to the first case. The /'s can be treated Similarly. We can now prove the following basic theorem' , of roots : sYstern (dr, dzt "', az) be a si'mhle Tnponnu 2. Let r Cartan splltting for a split semi-simple Lie algebra 8, relatiae to a
W. SPLIT SEMI.SIMPLELIE ALGEBRAS
125
'll, generators of 8' subalgebraQ. I*t the et, f+, h'i,.i:L,2, -be ' ' be aefinea in Q7) and let the basisi,r, lerr'' 0;*1, lf'r'' {'-1 {:f I basis this ai specirt,ed,in XVII . Then the multi\lication table for matrir has rational coefficients which are determined by the Cartan
(Ai. '
€irc7, Proof: We have [hthi: 0 and le;r''' e+*hil: 2h'='Aii*le;r"' bv (28i. SimilarlY , lfr, "' fr*h51 to consider products of e terms and f terms. Since ferr''' -eitrl: ' l a d e a r a d e a r"l ' a d e a * l .and 1 . . . l e ; r e r r l" ' e i * \ , a d { e 6 , " e , ; n l : [ " ' ' ei*l' This can be obtained ' i -'xl" ad' irr! eaad " eo*ll lad i",lior" polynomial in 6y op"tutin! on r with a certain (non-commutative) It follows from this and a similar argument for adeir, ...,adei*. the i's that ii suffices to show that [[e;' "' e;*leil, llf1r^"' f'*lei' of oUr base fle;r " ' eiklf il, llfrr''' fr*\f i are rational combinations Atl. The the by can be determined ilements where the "oeih.i"r,ts so we two the first for as two argument is the same for the last ascertain we evaluate To lfeir"'eircleA consider the first two only. first whether or not 6 * ei, F: ai'r+ "' + aib is a root' If not' : root then the product is 0. on the other hand, if T P + 4' is a e'base the of multiple a rational is eiftleil then, by XVIII, l!err... element associated with r, the multiplier determined by the Cartan If k : ! the product is 0 matrix. Next consider llfrr"'fr*1ei. -ht. lf' k>:2 we shall show j, case in which lfareil: unless il: linear combination of rational product ir the that induction by " :2 the product is 0 unless / - i' of ,fjbu." elements. Thus, if k -+ llf'reilf il: j-- i,. For llfr,f ileiJ we have llf;rf ileil: lfirlf&il :0' The relation just lf trh5l: Airrfry since i, + i and so If;reil - Ai;rf;r' Now assume k>2' derivld impliel atso that Llfftrleilj, product : is 0. otherwise, let e'+r be the last then the If no i, equals 1. Then index in [,f,, "'fo*j which llfor''' fo,fi''' f4leil : llf;r''' f,,f,ieif',*r''' ft*l
: - [ ' ' ' 1 . f , , ' 'f'i , l h i ] ' ' 'f , * l + l l [ f ; r ' ' 'f i , ) e ] f o ' ' 'f ' * f '
The first term is a rational multipte determined by the A;i of an the same /-base element. The induction hypothesis establishes proof. the completes This element. claim for the second Before continuing our analysis, it will be well to Suuulnv. results which we have obtained. For any split the summarize semi-simple Lie algebra I with splitting Cartan subalgebra O we
tz6
LIE ALGEBRAS
have obtained a canonical set of generators e;,fr hr, i F t,2, "',1, These were obtained by satisfying the defining relations'(28). z The characteristic property of roots. choosing a simple *y.ie. ai ft, where tfie &r are all a: e, z every root of is that \lkna;, either non-negative or non-positive integers. Such systems are obtained by introducing lexicographic orderings in the lspace 6J of rational linear combinations of the roots and selecting lthe positive roots which are not of the form a I 9, d, P > 0, in such an ordering. A canonical set of generators associated with rT is hi: Zh*nl(ai,a1), €i: €o,t,ft: 2e-r6l(ai, a) where zai is 4nY non'zeto element of 9rn and e*r6 is chosen in ,8-"a so that lepne-r) - hat,. We observe ihat hr is uniquely determined by da while er c?D be replaced by any $tei, $r * A, in O. Then /r will betreplaced by pt,'fr. The elements er,fn, ho constitute a canonical basis for a split three-dimensional simple Lie algebra 8;. Moreover, [t is easy to check that if (eL,fl,h') is any canonical basis for $o such that h't e 8a n O, then h't: h;, e'r: pren,fl - pttfr. lf e;,frhi zre canonical generators we obtain a cahonical basis for I in the following manner: the basis /l; for 6 ahd for each acot [f.r" ' ft* l n o n- z er o r oot P a b a s e e l e me n t z y :fz \" ' e ;* l cording as p > 0or F < 0 where (ir, " ',ir) is a sequenpesuch that a n d e v e ry p a rti a l S U IrI4r, + " 1' * d+ n i s a d ;r * " ' + dir r : t F root. The multiplication table for the canonical basis is rational and is determined by the Cartan matrix (Au), Ari :2(ai, a)l(di, ai). -- 0 : Aii The 4;i are integers, A;r: 2 and if. i + i, then either 4r, or one of the numbers Ait, Aio is -1 and the other iF -L, -2 ot -3. We observe also that the group of orthogonal linpar transformations generated by the reflections
q:.'lr, S;= S'r: F* t- - ?9' (ai, a;) which is a subgroup of the Weyl group W is finiter (Later we shall see that this subgroup coincides with I7.) TheLreflection S,i can be described by the Cartan matrix. Thus if we t4ke the basis (dr, dr, ''', dr) for Oi, then S; is completely describedlby
,,jst: ai -ffi
dt : dj - A;iat'
The conditions we have noted on the A6i
"te
redundanlt. However,
ry. SPLIT SEMI.SIMPLELIE ALGEBRAS sometimes one subset of these conditions will other times another will be used. 4.
The isomorphism theorem.
127
be used while at
SimplicitA
Theorem 2 of the last section makes the following isomorphism theorem almost obvious. Tnsonnu 3. I-et 8,, 8' be sflit semi-simple L|e algebras oaer a. fietd O of characteristi,c 0 with sbtitti,ng Cartan subalgebra.s O, S' of the sarne dimensionatity l. I*t (ar, dzt ''', dr), (al, aL, "', a!) be simble systems of roots for I and 8' respecti,uely. suffose the Cartan matrices (2(a;, a)l(a;, a6)), (Z(a!t,d1)l@!t,a!)) are identical' Let e;,f;,h;, e't,fl,hla, i -1,2, "',1, be canonical generators for I and g, as in (27). Then there exists a unique isomorphism of 8 onto g' mab\ing 0; on eL,f; on fl.,ht on hL. Proof: By XVI we know that x h&; is a root for 6 if and only if >kp,t is a root f.or b'. If F is a positive root for s we write Then i s . a r o o t ,m 3 h . *dir so that d\ * "'+a;B:a\+..can We a root' d!i* is d!, everY + and + "' ailr* ilr+ "' + P' choose as base eliments in the tanonical basis (36) for I and 8' Then th e e l e m ent s lenr ' " €h ), l frr' " fr* 7 , l u | r' " e ' n ), tfl ' " ' f{ * )' Theorem 2 shows that the coefficients in the multiplication table for the basis for I and 8/ are identical. Hence the linear mapping which matches these base elements is the required isomorphism. The uniqueness is clear since the el,f;, hi are generators. The result we have just proved is basic for the problem of determining the simple Lie algebras. It is useful also for the study of automorphisms (which we shall consider later) of a single Lie algebra. We note here that the result implies that there exists -h;. This an automorphism of I mapping e;-+f;, f;+ €i, hi--+ -ai, ' '', /, is a simple system i :1, follows by observing that a!, : and e't: fr, fl : e,t, h't - *h,;. is a corresponding set of generators. We shall need this in $ 7. DsrrxrtloN 2. A simple system of roots E : (dv &zt "', dt) is called i.ndecomposableif it is impossibte to partition zr into nonvacuous non-overlapping sets z',2r" such that Ati-- 0 for every dt€nt,
a1 €ntt.
Tsponpu 4.
I
e's simfle
if and only if
the associated simple
LIE ALGEBRAS
L28
systern r of roots is tndecomposable Proof; supposefirst that rc: (dt,.'., dt) U (ar*r, "', a,r),L = k < l, i < k, i > h. Choose canonical gener4tors e;,fi,h;, so that A;i:0, and let 8r denote the subalgebra generated by the €t,ft, h, i I k. It is readily seen that 8r:0, * X'8r where Ot is the subspaceof 6 spanned by the h and the summation is taken overt the roots r which are linearly dependent on the ai, Hence 0 c 8r ct8. If. r) k, j < k, then Ar'" :0 and since di - dr is not a root, this implies that a5* d, is not a root. Hence le7e,1:0 as well as lft,erl:0. Also lhie,l - g and consequently ar is in the normblizer of 8r. Similarly, /" is in the normalizer of 8r. Since 8, is contained in its own normalizer it follows that I is the normalizer df 8r. Hence ,8, is an ideal and 8 is not simple. Conversely, supfose 8 is not simple. Then 8 : 8r O 8z where the ,8; are non-zero ideals. Let a Then 0r:0lr\ l,et', eS\ e8; be a non'zero root and let er€$r. and, le,hl - a(h)e, for h e O implies that lettnl - o(h)el'. Since 8, is one dimensional this implies that either 8" g 8r or 8o S 82. Since [8r8r] - 0 and [8"8-,] + 0 we have either 8, * 8-' S I' or 8"*,8-" I ,8s. In particular, we may order the canonical generators S i nce the za, f i s o t hat ey f r L " ' , e r, ft € 8 r, € t+ t,f* + u? " tl ttfr€8r. g-[e{e,f,11 -left,lA,Pi if. < and / 8i are non-zeroideals, L k. Hence Ai,: and the simple system of A,i:0 roots is decomposable. 5.
The determination
of the Cartan mqtrlees
The results of the last section reduce the problem of determining the simple Lie algebras to the following two problems: (1) determination of the Cartan matrices (Ai corresponding to indecomposable simple systems of roots and (2) determination of simple algebras associated with the Cartan matrices. We consider (1)l here and (2) in the next section. We observe first that the indefomposability condition amounts to saying that it is impossible to order the indices (or the at) so that the matrix has the block fbrm (B
\0
0\
c)
where B, C are not vacuous. \il'e now associatea diagram-the Dynkin diagratn-with the Cartan matrix A;;. We chooseI points drt &zt '", dr. a4d we connect
129
IV. SPLIT SEMI.SIMPLE LIE ALGEBRAS
a; to et, i + j, by AuAt; lines. Also we attach to each point ai the weight (a;, a;). If. Ati: 0 : Air, a, and' ai are not connected and if A;i *0, An* 0, then AtrlAu : (d* ai)l(at, a)- Hence AnlAtt and AuAr.; c€lr be determined from the diagram. Since Arr is non' positive this information determin€s ,4ii and An. Thus the matrix can be reconstructed from the diagram of points, lines and the weights. We consider two examples:
310-6
r^n\
(.t//
Gz
' 4z
d'1
111
o-o-o d,2 d1
. &g
11A .
O-o &t-t
,
.ctl
.
dt
Fo r G, w e hav e A r r lA rr: (d t,a )l (a z ,a z ):3 , ArrA r r:3 w hi ch i m' Hence the Cartan matrix is the plies that Arz - -1, A,-: -3. matrix in (29). For /r we have Au: 2, Ap: Azt : Aza: Asz: . i . _ Ar- r , , , : A r , , l- t- -1 a n d a l l th e o th e r A ;t:0 . H ence the matrix is
-2 -1
-1 2 -1 -1 2
(38) . -1
-1 2
In determining the Dynkin diagrams we at first drop the weights (ai, ai) on the points and consider only the collection of points and 'We have / points dt, &2, '", dt., ?nd ai the lines joining these. and are connected ?1d ax', i.+j, are not connected if A;tAi;:0 T h e a r are l i nearl y i nb y Au Ai; : 1, 2, or 3 l i n e s i f..A u A,r+ 0 . dependent vectors in a Euclidian space Eo over the rationals. This can be imbedded in the Euclidian space E: Eon over the field R of real numbers. If. 0;t denotes the angle between ai and c; then AriAn: 4 cost 0i; and cos dir ( 0. Any finite S€t 41, dz, . . ., &t of linearly independent vectors in a Euclidian space (over the reals) will be called an allowable confi,guration (a.c.) if 4 cosz0;i : 4(a;, a)z l(a;, a)(ai, a;) : 0,1,2 or 3 and coSd;i < 0 for every i, j, i + j . Ihus cos d;; :0, -L, -tv/T and accordingly 0u:90o, 120",135o, or 150o. We may or -]2/X
130
LIE ALGEBRAS
replace a; by the unit vector z; which is a positive mrlltiple of. aa. Then the conditions are (gg)
(utu;): 1,
4(ui,u)2:0,1,2 or 3, i+j,
(u;,u)150,
i,j:!,2,..-,1.
The Dynkin diagram (without weights) of an a.c. z isla collection of points %;, i :1, . - . ,1, and lines connecting these dccording to the rule given before: il,; and ui aty not connected if (u* u) :0 andu;andu5 are connectedby 4(ui,ut)' :1,2 or 3 lines otherwise. An a.c. is indecomposableif. it is impossible to partition z into non-overlapping non-vacuous subsets z/, z" such that (t't;,u) : 0 if. ttr.;Q zrt, u.i e rc't. The corresponding condition on the Dynkin diagram is connectedness: If tr, u € n, then there exists a sequence gair:04, %i2,"',clirc - zr such thatutraad tttr*, are connbcted in the diagram. If the Dynkin diagram is known, then all,(urar) will be known. We shall determine the Dynkin diagram$ for all the connected a.c. after a few simple observations as folloWs. 1. If S is a Dynkin diagram, the diagram obtained by suppressing a nurnber of Points and the lines incident with these is the Dynkin diagram of the a.c. obtained by dropping the uectors corresponding to the points. 2. If I is the nurnber of uertices (boints) of a Dynkin diagrarn, then the number of pairs of connected points (u, u, (u, a) + 0) is less than l. Then Proof: Let u : |!u.i. l 0 < (u,u) : I * ,Z(ttt, u)
l
If. (uru) + 0, then Z(ut,u) < -1. Hence the inequality shows that the number of pairs ilrr ui with (ur u) + 0 is less than /. 3. A Dynkin diagram of an a.c, contains no cycles.,(A cycle is a sequence of points ur, . . - , xrk such that ur, is connected to u;+r i = k - L and ur is connected to ur) Proofz The subset forming a cycle is a diagrami of an a.c. violating 2. 4. The number of lines (counting multiblicities) issiling from a uerter does not exceed three. Proof: Let u be a vertexl u1,'uzr, , ., ur the vertices connected to u' No two ur are connected since there are no cYcles' Hence
IV. SPLIT SEMI.SIMPLE LIE ALGEBRAS
131
(a;, u) -- 0, i + i. In the space spanned by u and the ?/t w€ c?r ' ' ' , orcare mutually choose a vector 2osuch that (ro, uo):1 and uo,ur, orthogonal. Since u and, the u;, i > \ are linearly independent, u : 2t@, r)t)ui, is not orthogonal to ?/oand so (u, u) * 0. Since u (u, u) : (tr, ai' * (u, ur)' + "' + (u, ur)' : L . Hence 2f @, rr)' 11 and 2! 4(u, a;)2I 4. Since 4(u, ui)z is the number of lines connecting u and I); wc have our result. 5. The only connecteda.c. containing a triple line i's A
ltz
i
o:o
This is clear from 4. 6. Let r be an a.c. and let ar l)zt ' '', ttr be uectorsOf n such that the corresponding points of the diagram form a simple chain in the sense that each one is connected to the next by a single line' Let t' be tke collection of aectors of t whi,ch are not i,n the simple chain Then n' i's an a'c' urt . - -, uy together with the uector u : 2!ur -1, : I, "', k - L' Hence (u, a) i Proof: We have 2(ai, ai+) Since there are no cycles (ut, a) : 0 if i < i h + Zin./a;, a). and u is a unit j i + L ' H e n c e( u , a ) - k - ( f t - t ) : t unless with at most is connected z Then vector. Now let u € n, tc * ar (u, u) : Then cycles' no are there ui, since say one of the uit -0,L,2 or 3 as required' (u,2!o) : (u,at) and 4(u, a)z: 4(u, a)' The diagram of z' is obtained from that of ft by shrinking the simple chain to a point: Thus we replace all the vertices ,i by the single vertex a and we join this to any ue n, tt + ?/dby the total number of lines connecting u to any one of the a1 in the original diagram. Application of this to the following graphs
(40)
O-O-O--O
...
oEo_o_o
. . oo _
O.--O::O
o\
"f iu;, u : Lljui. Since Z(ut,ui+r)- -1 and Z(ui,ui+t): -1 we tlave p^p-l
(43)
(u,u)- i i'- ti(i + L): P"- P(P- L)12
: oro + r)12 , (44)
(u,u): q(q+ l)12.
an)and Also (u, a) : fue(uo,on): @qlT)Z(ur, : (u, u)' P'q'12 (45) inequality By Schwarz's
(46)
ry'ryry'
:
ry. SPLIT SEMI-SIMPLELIE ALGEBRAS
133
Since fq > 0 this gives (P + L)(q + L) > 20g, which is equivalent to(P-rlcq_L) 1. Since r 2 2, this gives r :2. z€q | a n d -f.ot all p. holds which > 0 if. q Z, then the condition is that P-' in this Hence < 6. and > is U6 f lf. q:3, then the condition f-' p, q, are r : p for solutions the Thus 4, 5. 3, case r:Q:2' f arbitrary; (50) r:2, f:314,5. 4:3,
tu
LIE ALGEBRAS
This proves 10. The only connectedDynhin diagramsof the third,,typein (42) are o
I
(51)
0__o...
o-o_
I
D,
o
o
II I
o-o-o_o-_o
Es
o I
(52)
I
I
I
o+o-o-o-o-o
E, o II I
I
-o-o-o-o-_o
o-o
S'
We have now completed the proof of the following Tsponpu 5. The only connectedDynkin diagrarns are At, I >- L, Bt : Cr, l2 2, D7, I > 4 and the fiue "erceptional" diag/ams Gr, Fr, Eu,Er, Ea giuen in (42), (48), (51), and (52). We now re-introduce the weights on the diagrams. I This will give all the possibleCartan rnatrices: We recall that in the Dynkin diagram obtained from the simple system T : (dr, dzt - . ., &t), AriAi;, i+ j, is the number of lines connecting ar and ai. ,If. A.ii*O, Aii + 0, then Ai;lAu : (ai, a;)l(a1,a) and Au or Ai; + -I while the other of these is -1, -2 or -3. Since nothing is changed in multiplying all the d4 by a fixed non-zero real numbdr, we may take one of the at to be a unit vector. If the diagrarn has only single lines, then all the (a;, ar) :1 since the diagram is connected. Hence the weighted diagrams for At, Dt, Er, E7, Es ?te,
Att l-3...1-] d1
Dt:
d,2
..
l-o1 d1
&t-t
&2
''1
/>r, 4g
Lo,o' I
11
1
*ii *i-,'
t>4'
W. SPLIT SEMI.SIMPLE LIE ALGEBRAS
135
tlou Ea:
o-o-o-o-o d.1 dz
d4
1
11 I
1
I
dz
t
da,
dl
d's
4s
&a
d'r
45
tio'
Ea:l-t-l-l-Jt-L-l d.1
dz
ds
ds
d.a
d7
d.6
For Gr we have chosen the notation so that
.
Gz:3-l d4
Ol2
For Fr we may take the weights as follows:
.
F+i l_:_2"-? d..1
d.z
da
&q
For Br and Cr we take the following diagrams:
2
2
2
o-o-o...o-o, 41 d.0, dz
1
1
o-O.'.O-O:O, d1 4z
2
1
&t-t
&t
1
2
dt-r
d7
,_
122
,\
l>3
These diagrams give the possibleCartan matrices. 6.
Construction of the algebras
The time has now come to reveal the identity of the principal characters of our story-the split simple Lie algebras. With every connected Dynkin diagram of an a.c. which we determined in $ 5 we obtain a corresponding Cartan matrix (Arr) and there exists for this matrix a split simple Lie algebra with canonical generators ei ,fi h i , i: t , 2, . . . , 1, s u c h th a t l e rh l j : At& i . We shal l gi ve the simplest (linear) representation of the algebras corresponding to diagrams Ar, Br, Cr, Dr, Gr, F, and Eu. Later (Chapter VII) another approach will be used to prove the existence of split simple Lie
136
LIE ALGEBRAS
algebras corresponding
to the diagrams
E, and Er.
We recall that if I is an irreducible Lie algebra of linear transformations in a finite-dimensional vector space over a field of characteristic 0, then 8 : I, O O where I, is semi-simple and 0 is the center. Hence such an algebra is semi-simple if r and only if G :0. We note also that if ,8 is semi-simple and 8 contains an a b elian s ubalge b ra 6 s u c h th a t 8 :6 @ O e r@ OepO... w here &, 9,. . . are non-zero mappings of 6 into O such that Ieuhf : a(h)en, h e 8, then O is a splitting Cartan subalgebra for 8 and 8 is split. We shall use these facts in our constructions. Let O be the algebra bf linear transformations in an (, + t)dimensional vector space llt over O, I > L. It is well known (and easy to prove) that 0 acts irreducibly on llt. We have 0z : EL$AI where 8 = Gl is the derived algebra and is the set of linear transformations of trace 0. Evidently, aoy Gi-invariant subspace is @zinvariant. Hence 8 : Gi is irreducible. Also the decomposition CIz: Si O O1 shows that the center of Ol is 0. Hence @i is semisimple. We now identify G with the algebra O*r of. (l * l) x (/ + 1) matrices with entries in O,8 with O'r*r, the set of matrices of trace 0. We introduce the usual matrix basis (e;), i, j : l, ...,1* 1, in Or+r so that (53)
€;i€*n: 6i*Qi^
A basis for I is (54) hx:€rn-er+r,t+r, i+j -1,,",1+t. h3l; €;i, Set h - }l,rorh*. Then the set of h's is an abelian subalgebra 6 of / dimensions and lert, hl:
(55)
The l2 */
(or,- @r)on,
f er *r . , , h l- (r * a ,)o r+ r,,, f e, , r *r,h l- -Q * o 4 )e ,,s 1, linear functions h-1o)r-
T
-
x,, I
I
r + S:1,
"'rl
o)rt h--+T * @r, h-t+-(r
+ co,)
are distinct and are non-zeroweights of ads0. We haveE: O + >8' where a runs over these weights. It follows that 6 isl a splitting Cartan subalgebraand the a are the non-zeroroots. Set (56)
&r : dl-t
:
0)r 0)1,-t
0)z t -
(t)l
&z: ,
( D z-
dr,:T*at,
( D g s" '
,
ry. SPLIT SEMI.SIMPLELIE ALGEBRAS
Then a1:(tov-cDn+r)
arr*ctx+r* "'+
* (orsr",- ox+z) + "'
(57)
:y*o*t a; *
d;+t+
"'
*
h:lr2r"'rl-L,
(at -
al :
(58)
* Q * ttt)
o;+r)
*(ar;+,-to;+z)+"'*(at-@i+t) -
2, which are skew relatiae to a non' d.egenerate syrnmetric bilinear form of maxi'mal Witt index. Then g is a s\tit si.mhle Lie algebra of tyfe Bu C. Let lft be of dimension 21, I > L, (x, y) non-degenerate skew bilinear form in llt. Let I be the Lie algebra of linear transformations which are skew relative to (x, y). We can choose a basis
140
LIE ALGEBRAS
(trr, ur, .. ., uzt) so that the matrix q - ((u;, u)) is
(65)
o: (-1,
l')
As in B, one sees that I can be identified with the Lie algebra of matrices a e O27such that aq: -qat. This implies ttlat (oo)
a:
(a"
o")
\4zr
azz/
aii € or ,
',
where azz: -alr,
(67)
alz:
ap ,
alt:
azt.
Hence I has the basis h; : 0oyo6: (68)
€i; -
€t+;. t+i t
eii -
€g+z,i+t r
i + i i<j
€-ottr-to1:€i.itt.*ei,r*r, 0o4rtt1: €i+t,1* g*r.r:
€;,;+t,,
i < j
€Yt,t,
o z - Q : o i + r , i ,t
A s i n B , o n e p r o v e st h a t O = t > a r ; / z ; ) i s w h e r ei , i : L , 2 , . . . , 1 . a Cartan subalgebra and the subscripts in (68) define the roots relative to 6. Also 13has center 0 and So, by Lerrima 2, 8 is semi-simple. The roots (69)
oh :
ar -
o)zr "'t
dt-t:
form a simple system with proves
o)r-r -
Dynkin
o)t t
diagram
dt :2@t,
Ct. if
, > 3.
This
Tnnonpu 8. Let 8 be the Lie algebra of li.near transforntations in a Zl-dimensional space, I > 3, which a're skew relatiua to a non' degenerqte skew bilinear form (the syruplectic Lie algebm.) Then 8, is a sflit simfle Lie algebra of tyhe Cu D. Dt, 2/-dimensional, I > 2, (x, y) symmetric of maximal Witt index. Here we can choose the basis (zi) so that t - ((ur, zr)) has the form
(70)
t- -- (0, U,
1')/ 0
and 8 can be identified with the Lie algebra of matrices a satisfy' ing at : -ta'. These are the matrices
141
ry. SPLIT SEMI.SIMPLELIE ALGEBRAS
o : (1"
(?1)
a;i € or,,
, 1"\ azz/
\4zr
such that azz:
(72)
alrz: -aP
-a!r, ,
a!n:
t
-azr '
Then 8 has the basis Aa -
hi: (73)
Ar,+i,t+i
QoYol:eii-€++t'i+t'
i+i
eo4*ot1: €i+t,i -
7i+t,;,
i < i
^i,i+r,,
i < i
0-o4-o1 :01,r+t,-
where i, j :1,2, " ',1. 6 : {X ouh;} is a splitting Cartan sub' algebra and the subscripts in (73) define the roots. The center of I is 0; hence I is semi'simple. Set (74)
&t:
0)t -
Qtb "'
t
&t-r:
&)t-t -
0)r, r
dL:
1 a r ' l - r*
tot '
Then these ai form a simple system of roots which has the Dynkin diagram D, if. I Z 4. We therefore have Tnnonnu 9. Int g be the L|e algebra of linear transformations in a Zhdin ensional space, I > 2, which are shew symmetric relatiue to a non-degenerate symmetric bilinear form of marimal Witt index' Then if t > 4, I fs a sfli't simple Lie algebra of tyhe Dt. The four classes of Lie algebras A, B, C and D are called the ,,great, classes of simple Lie algebras. These correspond to the linear groups which Weyl has called the classical groups in his book with this title. It is easy to see directly, or from the bases, that we have the following table of dimensionalities
tybe At
(75)
Br, Ct Dr,
dimensi,onality
t(t + 2) t?t + t) t(zt+ L) t(zt- L).
The determination of the simple systems and the general isomorphism theorem (Theorem 3) and the criterion for simplicity (Theorem 4) yield a numhr of isomorphisms for the low dimensional orthogonal and symplectic Lie algebras. The verifications are left to the reader. These are 1. The orthogonal Lie algebra in 3'space (three-dimensional space
I42
LIE ALGEBRAS
llt) defined by a form of maximal Witt index and thb symplectic Lie algebra in 2-space are split three-dimensional sirirple and so are isomorphic to the algebra of matrices of trace 0 itr Z-space. 2. The orthogonal Lie algebra of a form of maximal Witt index in 4-space is a direct sum of two ideals isomorphic to split threedimensional simple Lie algebras. 3. The symplectic Lie algebra in 4-space is isomotphic to the orthogonal Lie algebra in S-space defined by a symmgtric form of maximal Witt index. 4. The orthogonal Lie algebra in 6-space of a formt of maximal Witt index is isomorphic to the Lie algebra of lindar transformations of trace 0 in 4-space. The remaining split simple Lie algebras: types Gr,4r, Ea, Et znd. Es zre called excepti,onal. We shall give irreducible representations for Gr, Fr, Eu but we shall be content to state the resirlts without proofs, even though some of these are not trivial. A complete discussion can be found in a forthcoming article by the 4uthor (t111). Our realizations of G" and F. will be as the derivatfion algebras of certain non-associative algebras, namely, an algeb4a of Cayley numbers 6 and an exceptional simple Jordan algebra i4f. Following Zorn, the definition of the split Cayley algehra or uectormatri.x algebra 6 is as follows. Let V be the threeidimensional vector algebra over O. Thus I/ has basis i, j,k over @ and has bilinear scalar multiplication and skew symmetric vector multipli' cation x satisfyingi i, j, h are orthogonal unit vectors and
(76)
ixi
: jx j:kxh-0,
ixk
:i,
kxi-j
Let 6 be the set of 2 x 2 matrices (a
(Tz)
-a\g )
\;
ix j:h,,
,
of the form
A
a,Peo,
a,beV.
Addition and multiplication by elements of O are as u$ual, so that O is eight-dimpnsional. We define an algebra product in 6 by
(?8) (;
'r)
(,
;)
: (ay-@,d)
\rD+ Pd+axc
m*Ba*Dxd\ Ps-(b,c)/
The split Cayley algebra is defined to be 6 together \iith the vector space operations defined before and the multiplicafion of (78). 6 is not associative but satisfies a weakening of thd associative
W. SPLIT SEMI.SIMPLE LIE ALGEBRAS law called the alternatiue
(7e)
143
law:
tc'! : x(xy) ,
!x2 : (Yr)x .
Let 8 be the Lie algebra of derivations in 6. The unit matrix fo r e very deri vati on 1 i s th e ident it y inG an d s i n c e 1 ' :1 , ID :0 (a * I : 0) coincides with D. The space 6o of elements of trace 0 the space spanned by the commutators [ry] : fri - !x, r, y e E. Hence GoDg 6o for a derivation. Thus 6o is a' seven-dimensional subspace of 6 which is invariant under 8. The representation in Go is faithful and irreducible. lf. T is a linear transformation of trace 0 in V, and T* is its adjoint relative to the scalar multiplication, then it can be verified that
(80)
(;
"u)-'(-our."[)
is a derivation in 6. The set of these derivations is a subalgebra 8o isomorphic to Olz. In any alternative algebra any mapping of the form Da,o: fafir] + [arbR] + lanbn), where a, b are in the alge' bra and az,en denote the leftand the right multiplications (x-+ar, x -, xa) determined by'a, is a derivation. In 6 any derivation has the form D"r,or,* Drz,tn a Do, where er:
(81) An:
(l3) (3 6)
€z:
(31)
bzr :
(: 3)
and Do € 80. If 6 is a Cartan subalgebra of 80, 6 is a Cartan subalgebra of 8. If we identify 8o with AL", we can take S to be the set of matrices of the form orht * ozhz, ltr: et - e$, hz: €zz- €ss, Then 6 is a splitting Cartan subalgebra of 8 and the roots of O in 8 are: t@r, *o)zt t( L. We define the inder of a monomial uir& uir$ . .' A uti^ 3s follows. For i, k, i < ft, set j,5 j* if f0
tr
if
j,>i*
and define the index (8)
ind(a1,&uir@"'@utn):
Note that ind - 0 if and onlY if
3 jol.
Monomials
I57
V. UNIVERSALENVELOPINGALGEBRAS having this property will be called standard. j* ) j*, and we wish to comPare
We now suppose
ind(w,E wr8"'8^un)
and
ind(utrA "' &uhnr&ut*A
"' 8u)
,
where the second monomial is obtained by interchanging ttrrc,tti*+r. Let Vlrc denote the ?'s for the second monomial. Then we have --' tJ ;,rc +4rt' r' * * r:4 i .* i f i < k; tL' t n 'tt: rl i t if i, ,j + k , k * I; T l * 4nrc+r-1'Hence I r a + t . iqr L + r , t : ' 4 r c .i if i > k + L a n d T L , * * r : 0 , ind(z;r4...8ryo):L - r i n d ( u n & . . ' A i l i n + r 8w r A " ' 8 u )
.
We apply these remarks to the study of the algebra tl : t/S which we prove first the following
for
Lnurrr.o,1. Eaery element of T i,s congruent mod I to a O'linear combi,nation of t and standard monomials. Proof: It suffices to prove the statement for monomials. We order these by degree and for a given degree by the index. To it suffices prove the assertion for a monomial u1r&uhA "'$un of the for those it for monomials of lower degree and lo "rru*e given monomial' the same degree z which are of lower index than ir*t. We Assume the monomial is not standard and suppose ir) have u t r & " ' A u l n : u 1 & " ' A % i n + r & u t * @" ' 8 U n + u 4 @ " ' I ( a l *& u t * + r - r t i t t + t 8 w ) A " ' & u t * ' Since uJkS uin+t- Itirc+t& ut* = lu1rul**rl (mod 9)' uh} "' A urnT u1& "' A ili*+r8 utrA "' & uo * u l r 8 " ' S l u t * u i * * , )I ' " I u o ( m o d S ) ' The first term on the right'hand side is of lower index than the given monomial while the second is a linear combination of monomi' als of lower degree. The resutt fotlows from the induction hypothesis. We wish to show that the cosets of 1 and the standard monomi' als are linearly independent and so form a basis for ll. For this ' ' ' tttnt purpose we introduce the vector space S" with the basis tt'trl,rtr : @ 1OS ' OS rC I " ' . i rsi r= . . . 3i1, , iie / , a n d th e v e c to r s p a c e$ The required independence will follow easily from the following
158
LIE ALGEBRAS
Lprurur 2. (g)
There exists a linear mappi.ng o of T. into * such that ( u n r & u r r A" ' @ m ) o : I t t ( r r z " 'u t n ,
lo :L,
if
(r0) tu"*"8r"|
ir3ir k + I, ll. l:k+1. I. Set z1* : r4, ltib+r: a, ujt, - tD, tclr*r: fr. Then the inductiOn hypothesis permits us to write for the right hand side of (11) ("'a 828
"'A
x8w 8 "')a
+ ( . . .8 o & u & . . . A W x l & . . . ) o + ( . . . L u u l A. . . 8 r 8 w & . . . ) o + ( . . . & l u a l A . . . & I w x lI . . . ) o. If we start with (jrjr*r) we obtain
( . . .A u & a A " . 8 r 8 b o I " ' ) o + ( . .. 8 u 8 u 8 . . . A W x l & " ' ) o : ( . . .8 a 8 a 8 . . . A x @ u 8 . . . ) o + ( . .. & l u u ) A . ' . I r 8 w & " ' ) o + ( . .. & u 8 2 8 . . . A W r l @ ' . ' ) o + ( . . ' & I u u ) A" ' & [ w x ] 8 " ' ) o .
V. UNIVERSALENVELOPINGALGEBRAS
159
This is the same as the value obtained before. II. Set z1r : ll, l,lin+r: u : uit, tlit+t: w. If we use the induction hypothesis we can change the right hand side of (11) to
(12)
( . . . w 8 u 8 u . . . ) o+ ( . . ' l u w l Quc"^' ) o + ( . .. a & l u w l " ' ) o * ( " ' t u u l & w " ' ) o
Similarty, if we start with ( . . . u 8 a 8 u - . . ) o+ ( . . . u @ l a w l" ' ) o , we can wind up with (13)
( . - . w 8 u 8 u . . . ) d+ ( . ' . w & l u a l " ' ) o + ( . - . l u w l S a . . . ) o* ( " ' u 8 [ a a r"]' ) q
Hence we have to show that a annihilates the following element o f. AL O I ' O " ' @, 8" -r:
(14)
(. . . lawl&u .. .) - ('" u I [utu]' '') + ( . . . a I l u w ). " ) - ( " ' l u , w l 8u " ' ) + ( . . . t u u l @w " ' ) - ( " ' w S l u u l" ' )
Now, it follows easily from the propertiesof o in Al @' ' ' @ 8"-r t h a t i f ( . . . a 8 D " ' ) e 8 " - r , w h e r ea , b e 8 ' , t h e n (15) (..'a80 "')o - ("'b&a "')o - ("'labl"')o -0. Hence a applied to (14) gives (16)
+ lluuTwl- lluwluJ+ llwulul + llualwl: 0, (16) Sincellawlul + luluw]|1
has the value 0. Hence in this case, too, the right hand side of (11) is uniquely determined. We now apply (11) to define a for the monomials of degree n and index e. The linear extension of this mapping to the space 8o,i gives a mapping on tLLO ' '' (E 8"-' @,8",i satisfying our conditions. This completes the proof of the lemma. We can now prove the following The cosets of L and the THponnr,r 3 (Poincar6'Birhhof'Witt). standard monomials form a basisfor 17: g/S. Proof: Lemma 1 shows that every coset is a linear combination of 1 * S and the cosets of the standard monomials. Lemha 2 gives a linear mapping o of. T. into !S satisfying (9) and (10). It is easy to see that every element of the ideal S is a linear combi'
160
LIE ALGEBRAS
nation of elements of the form u t r & " ' A i l r n - u t t D " ' A u i n + t @ u t * A" ' @ u r o - , t h A ' " Sl u t* u i * + ' l I ' ' ' & u t* . Since o maps these elementg into 0, So :0 and sor o induces a linear mapping of 11- !/S into S. Since (9) holds,i the induced mapping sends the cosets of 1 and the standarfl monomial u t* re s p ecti vel y. S i nce these u a 1 8. . . & u6 in to 1 a n d 4 c n rtttr... images are linearly independent in S, we have the linear independence in U of the cosets of 1 and the standardl monomials. This completes the proof. Conolunv 1. The maf\ing i of 8, into 17 es 1: L and) AL n 8i - 0. and Proof: If (u) is a b a s i s fo r I o v e r O , then 1= 1* 9 both : This independent. ui * S are linearly the cosets uii limplies statements.
We shall now simplify our notations in the followiflg way: We write the product in ll in the usual way for associativealgebras: xy. We write 1 for the identity in tl and we identify I with its image 8i in 11. This is a subalgebraof tl6 since the identity mapping is an isomorphism of I into tlr. Also I generatesI and lhe Poincar6' Birkhoff-Witt theorem states that if {uilj e I}, / ofdered, is a basis for 8, then the elements (17)
' ' ' cct,t L, ?t'rrct,t,
Ct
form a basis for ll. In particular, if 8 has the finite then the elements
(18)
u!,ut,... ufo*
ht
(ul - 1) form a basis for tt. The defining property Of U can be re-stated in the following way: If d is a homomorphiSm of I into llr, U an algebra, then 0 can be extended to a unique hofnomorphism d (formerly O') of 1l into !I. In particular, a representhtion R of 8 can be extended to a unique representation R of tl. fhis implies that any module trt for 8 can be considered in one {nd only one way as a right ll'module in which xl, x €fi, /e8, is as defined for I)t as !-module. Conversely, the restriction to I of a representa' tion of ll is a representation of I and any right ll'mddule defines a right 8-module on restricting the muttiplication to 8. In the sequel we shall pass freely frorn 8-modules to ll'modgles and con'
V. UNIVERSAL ENVELOPINGALGEBRAS
161
versely, without comment. The Poincar6-Birkhoff-Witt theorem (hereafter referred to as the P-B-W theorem) gives a characterization of the universal enveloping algebra in the following sense: Let I be a subalgebra of ?I1, ?I an algebra having the property that if {utlj e I} is a certain ordered basis for [J, then the elements 1 and the standard monomials uq24h... tttr, it S iz -< . . . S i,, form a basis for ?I, Then lt (and the identity mapping) is a universal enveloping algebra for 8. Thus we have a homomorphism of the universal enveloping algebra ll into ?I which is the identity on I!. The condition shows that this is 1: 1 and surjective. Hence ?I can be taken as a universal enveloping algebra. Now suppose that E is a subalgebra of 8 and let 1l be the universal enveloping algebra of 8. We may choose an o rd e re d bas is { uilj e I} fo r I s o th a t I: K u L, K fi L: @, h 1
Let ii - G(ll) be the associated graded algebra. It is ieasy to see from the definition of G(u) that, since 8 generates ll, E = g'r4l,or (oL + 8)lOt generates i[. It follows that if. {usli e 1y,l ordered, ls a basis for 8, then the cosets ili:ui+ AL i; d-gun.r"t. [. We have ilpy: uiu* * ll") and upl -- urt4 * ll(t) and ,ttrr, - ,ptr1 : luiurc\€ U(r). Hencr: frrui- uiilr. Thus the generatofs commute and consequently I is a commutative algebra. It follows that every element of I is a linear combination of the elements f(- 1;, ilrrilir "' il,r*, it 3 i, and the Poincar6-Birkhoff-wftt th.or.m that the different .,standard,, monomials indicated here form a basis for il. This means that the ui a,re algebraically independent and I is the ordinary algebra of polynomials in these elements. The general results we have derived and the properties of polynomial rings now give the following theorem on 11. Tnponnu 6 1. The uniuersal enuelopingalgebrallof any Lie atgebra g has no zero dioisors + 0. 2. If 8, is finite-dimensional, then lI satisfi,es the ascending chain condition for left or right ideals and \ has a left or right quotient
diuision ring. Proof : 1. Since I is a polynomial ring in algebraically independent elementsover a field, I has no zero divisors + 0. Consequently tl
V. UNIVERSAL ENVELOPING ALGEBRAS
167
has no zero-divisors * 0, by Theorem 4. 2. If 8 has the finite basis ur, uz, . . . t unt then I is the poly' nomial algebra in frt, iliz, . . . , ilr. This satisfies the ascending chain condition on ideals by the Hilbert basis theorem. Hence ll is left and right (Noetherian) by Theorem 4. Hence 11 has a left and right quotient division ring by the Goldie-Ore theorem. 4,
Free Lie algebras
The notion of a free algebra (free Lie algebra) genetated by a set x: {nl j e l} can be formulated in a manner similar to that of the definition of a universal enveloping algebra of a Lie algebra. We define this to consist of a pair (8, D ((88, i)) consisting of an algebra I (Lie algebra 88) and a mapping i. of. x into 8(88) such that if d is any mapping of. X into an algebra ?I (Lie algebra E), then there exists a unique homomorphism 0' of 8(&8) into lI(E) such that 0 : i|t. It is easy to construct a free algebra generated by any set X. For this purpose one forms a vector space Dt with basis ' X and one forms the tensor algebra 8(: !) : OLO rn O (m 8 tX) @ " into X based on Dt. The mapping f is taken to be the injection of 8. Now let 0 be a mapping of X into an algebra lI. since x is a basis for llt, 0 can be extended to a unique linear mapping of tlt into ?I and this can be extended to a unique homomorphism d of I into lI. Hence $ and the injection mapping of X into $ is a free algebra generated bY X. It is somewhat awkward to give a direct construction for a free Lie algebra generated by X. Instead, one obtains the desired Lie algebra by using the free algebra $ generated by X. Let 88 denote the subalgebra of the Lie algebra 8t, generated by the subset X. Let 0 be a mapping of x into a Lie algebra E and let 1l be the universal enveloping algebra of E, which (by the Poincar6'BirkhoffWitt theorem) we suppose contains E. Then 0 can be considered as a mapping of X into O, so this can be extended to a homo' morphism d of S into 11. Moreover, d is a homomorphism of &z into 1lz and since d maps Xinto a subset of E (g U), the restriction of.0 to the subalgebra 88 of 8r Senerated by X is a homomorphism of 88 into E. We have therefore shown that 0 can be extended to a homomorphism of 88 into E. Since X generates S8, d is unique. Hence &8 and the injection mapping of X into &8 is a free Lie algebra generated by X.
168
LIE ALGEBRAS
We note next that g (and the injection mapping) is the universal enveloping algebra of 88. Thus let 0 be a homomorphism of gg into a Lie algebra 216 2[ an algebra. Then there exi]sts a homomorphism 0 of. 8 into ?I which coincides with the restriction of d to X. Then d is a homomorphism of B, into llz and so the restriction 0t of. 0 to 88 is a homomorphism of 88 into ?[2. iSince x0' : x0 f.or tc e x and X generates 88, it is clear that 0t cdincides with the given homomorphism d of 88 into ?[2. Thus we have extended 0 to a homomorphism of I into ?I. since 88 genefates ff it is clear that the extension is unique. Hence & is the rfniversal enveloping algebra of 88. l The two results which we have established can be $tated in the following Tnponpu 7 (witt). Let x be an aTbitrary set and let 8 denote the free algebra (freely) generated by X. Let 88 denote tlw subalgebra of 8" generated by the elements of x. Then Ss rs a free Lie algebra generated by X and 8 is the uniuersal enueloping algebra of tg. For the sake of simplicity we shall now restrict our,attention to the case of a finite set X : {*r, xzr . .., )c,}. Then D t - o r , @ t L r z @. . . @ a r , and t! -OLeDtO(tX8tX)O..., and we write 8:O{x,,...,x,}. Th e algebr a I i s g ra d e d w i th T n ^ - tn 8 m8...A II (z-ti mes) as the space of homogeneous elements of degree /n. A basis for .. xt*, ii : this space is the set of monomials of the form rc;lc,;.2. L,2, . . . , ri hence dim 1)t- : ym. An element a e S is called a Lie element if a e 98. iWe proceed to obtain two important criteria that an element of 18 be a Lie element. we observe first that it is enough to treat the case in which the given element a is homogeneous. Thus 0onsider the collection of linear combinations of the Lie elements of the form (26)
l. . . llxEr4)xn,)... x.t^\ ,
ii:L,2,...,r, ffi:L,2,.... T h e J a c o b i i d e n t i t y s h o w st h a t t h i s subspace is a subalgebra of 8r. Since it contains the r it coincides with $8. we therefore see that every element of []8: is a sum of homogeneous Lie elements. Hence an element is a Lie element if and only if its homogeneous parts are Lie elements. we see also that if a is a Lie element which is homogeneous of degree m then
169
ALGEBRAS V. UNIVERSALENVELOPING
a is a linear combination of elements of the form (26). Let 8/ denote the ideal !n e (gfnI tlt) e . . . in 8. An element of I is in 8' if and only if it is a linear combination of monomials Since the different monomials of this type form rir . xi,n,m 2l. a linear mapping o of tl' into'?i8 such that we have ior a basis s' (27)
xid: x;,
(xor"'x;*\o: ["'fxirx;rl"'x;^l
,
ffi> 1'
We consider also the adjoint representation of 88. Since $ is the universal enveloping algebra of t!8, this representation can be extended to a homomorphism 0 of fi into the algebra 0(&8) of linear transformations in the space $8. We have
r (x t') "r " ?!::,' ^o' ;,'r :' ii,,i',,:,:!);i,;'
This implies that (28)
@u)o: (uo)(u9),
ue8',
0€ti
lf a, b € []8, then lablo - (ab)o - (ba)o : (ao)(b|) (bo)(a|) : (ao) ad b - (bo)ad a (9) lao, bl + la, bo) . Thus the restriction of 6 to 88 g 8' is a derivation in 88. can use this result to obtain the following criterion.
We
TsBonBu 8 (Dynki.n-Specht-Weuer)., If A is of characteristic0, then a homogeneouselement a of d'egreern ) o is a Lie element if and' onl'y if ao : vna where o is the linear ma?bi'ng of 8' into fi8, defined by (27). Proof: The condition is evidently sufficient (even for characteristic p, provided f tr m). Now let a be a Lie element which is homo' geneous of. degree m. We use induction on rn. We have seen that a is a linear combination of terms of the form (26). Hence it suffices to prove that ao : ma f.or a as in (26). We have f"'fxqx . ; r ] " ' t r " in tl l : [[" ' fx i rx ;r] " ' tc i m -rfdN, t^ l - [ [ ' ' ' lnr x u l " ' )c r^ -rf,x 6 o \ : ( m - l ) t . . . L x \ x h l. . . x t * \ + t . . . f r 4 r 4 l . . . x ; , * l - m l . . . l x 4 x ; 2 ) . .. r i * 7 ,
LTO
LIE ALGEBRAS
which is the required result. Our next criterion for Lie elements does not requite the reduction to homogeneous elements. This is Tsuonou 9 (Friedrichs). Let fi : 0{rr, ..., x,} be thetfree algebra generated by the rc; ou€r a fi.eld of characteristic 0. Let 6 be the diagonal mapping of 8, that is, the homomorphism of g into g gg such that r;6: ri81* 18r. T h e n a e g i s a L i e e l b m e n ti f a n d onlyif ad:a8I *18a. Proof: We have [a81+ 18 a, b8 t + lBD] : labl&l + l glabl which implies that the set of elementsa satisfying aB -cgl * lga is a subalgebra of 8r. This includes the x;; hence it contains gg. Let yv !2,' " be a basis for $8. Then since g is the universal enveloqing algebra of 88 the elements yf 'ykz . . . y!#, m arbifirary , h; ]_ 0 Oo,:1) form a basis for $. Hence the products
u!'yl' . .- r,:h& ul'yt, form a basis for I I8.
We have
u!'y!' ... y:ffid: (!rg I + 1 gy,)o'(y,g 1 * 1 gyr)*, "'(Y*8 1+ L&t)o^ (30) - y!'yf;'.. - y:,rg 1 + k,y!'-'y!, ... y!# g y, + kri't:z-' ... y:fgy, + ... + k^y!,...ylf*' @!* + * where I lepresents a linear combination of base elergents of the fo rm y r r ' y l' . . . - y j ' @ y l ry :r...i l , w i th X /,; > 1 . The se(ond through the (m * l)-st term do not occur in fh" expressions of this type for any other base element yltyrr'...!t'. It follows that in order that ad shall be a linear combination of the base elerhents of the form !l'...i,f i t i s n e c e s s a r yt h a t i n t h e el and lgyi'....yj' expression for a in terms of the chosen basis only bdse elements y:' ' . . y f f wit h o n e &; - 1 a n d a l l th e o th e r h :0 occur w i th nonzero coefficients. This means that a is a linear combination of the y;; henc e a e 98. H e n c e a 6 : a & l + 1 g a i f and onl l y i f a e gg. 5.
The Campbell-Hausdorff
formula
We shall now use our criteria for Lie elements to deritve a formula due to Campbell and to Hausdorff for the product of Exponentiais in an algebra. For this purpose we need to extend the free alge-
I7I
V. UNIVERSAL ENVELOPINGALGEBRAS
bra 8 : O{xt, ..., x,} to the algebra I of formal power series in the vcr. More generally, let !I be any graded algebra with ?Ii 3s the subspace of homogeneouselements of degree i, i :0,1,2, "'. Let fl be the complete direct sum of the spaces ?Ii. Thus the eleao * h I "' such that ments of fi ate the expressions }f,a': : A ddi ti on and 0 , !,2 , ' ' ' ' l, ai : l ,b i if and only i f 4 t: b t, i introduce a We scalar multiplication are defined component-wise. : where ()frar) 23c* multiplication in I by €i'ail cn: dobr* arbxt + "'
* anbo€ lIt .
It is easy to check that I is an algebra. Moreover, the subset of if i ) m, m:a,1,2" f l o f e l e m e n t sE a r s u c h t h a t a ; : 0 .is a of E(i) subset The lI. subalgebra which may be identified with 0' [(i): n and fr in ideal an is elements of the form at * d;+r + "' a n d 0 a + i f ' lal:2-i wedefineavaluationinIuysetting l0l:0' properties: following the have we Then a eE"', #8"*t'.
(i) (ii) (iii)
lal-0 lal=0, < l a l l b l . labl
i f a n d o n l Y i f a :0
.
l a + b l S m a x ( l a l ,l A t ) .
This valuation makes I a topological algebra. Convergence of s e ri e s xr* x z + . . . , 1, i€ [, i s d e fi n e d i n th e u s u a l w ay. The non' archimedean property (iii) of the valuation implies the very simple It i s " ' c o n v e rg e s i f a n d o n l y i f' l r;l -0' c ri te ri o n t hat x r lr z * ?I. in clear that the subalgebra lI is dense If the characteristic of the base field is 0 and e € [(1), then
(31) and (32) are well-defined elements of [ (that is, the series indicated con' verge). A direct calculation shows that (33)
e x p ( l o g ( 1 + z ) ): L * 2 ,
log(expz):2.
Moreover, i! 21,zz e 8\L'' and lztzz\: 0, then (34) and
exp zr €xP ^z2: €xP @r * z2)
172
LIE ALGEBRAS
(35)
log (1 * z'Xl * zz)- log (I * z) * log (L + zr)
We consider all of this, in particular, for ?I : I : O{xt, " ', x,}. The resulting algebra $ is called the algebra of.formal'power series in the rh. We shall also apply the construction to the algebra I A 8. Since I : XLo O 8r, where Sr is the subspace of homoWe have g e n e o u se l e m e n t s o f d e g r e e i , 8 4 8 : E 0 ( 8 r 8 8 ) i (8n I 8l) (8n' I 8r') G $i+r' I 8r*.,'. Hence, if we set i (8 I 8)r :
8 * 8 8 0 * 8 * - , 8 8 ' + " ' + 8 0 8 8 r , t h e n( 8 8 8 ) * ( 8 8 8 ) , ] - s ( 8 8 8 ) ' * , I t f o l l o w st h a t $ 8 8 i s g r a d e dw i t h and$88: >0(888)-. (8I8)r as subspaceof homogeneouselements of de$ree ft. We can therefore construct the algebra 8-6R-. I f a : I . f a n , a , i e8 i , t h e n X ( 4 r 8 1 ) a n d : ( 1 8 d d ) a r e e l e m e n t s of 8re=S which we denoteas a I 1 and 1 I a, respectively. The mappingsa -) a@ l, a-t I I a areisomorphismsand homeomorphisms We have [a 81, 1@ &] - 6 for afy a,b e$. of 8--i"to ffi. 0 case we have exp (a I 1) = exp a @ 1, the characteristic in Also, exp (1.I a) - 1 I exp a and similar formulas for the log function. Let S denotethe subsetof $ of elementsof the form b'* b, + . . ' where Dris a Lie elementin Si. It is clear that F it a subalgebra of 8r. The diagonal isomorphismd of I into $ I I h4s an extenwe note first that if sion to an isomorphism of s into ffi. ai e 8; then ai6e (8 I 8)t. This is immediate by induction on f, using the formulas$;: Ei=,4$_rt )cid: n$l + 1 & xt. Hsrce if a :2f at, a; € $;, then X a;d is a well'definedelementof I A 8' We denotethis as ad and it is clear that d: a -, a6 is an ibomorphism of $ into fi e 8. It is clear also from Friedand homeomorphism richs' theorem applied to the ai that in the charactedistic0 case a i € 8 ; , i s i n f f i i t a n d o n l y i f .a d : a & 1 + + 8 4 . a:}ai, We assumenow that r : 2 and we denotethe generdtorsaS f' y and write 8: a{x,y}. Assume also that the characteristicis 0. Considerthe elementexpt expJ of S. We can write this as | * z, wherez : zr * zr+ .,., zi€ 8;, and we can write exprexpY : 1 * z: exPLU (36) , where w : log (1 + e). We shall Prove 1. The elementw - log(exprexpy) is a Lie element, PnoroslrloN that i,s, ?rle 8T, Proof: Consider(exprexpy)d. We have
T73
ALGEBRAS V. UNIVERSALENVELOPING (exprexPY)d : (exPrdXexPYd) - e x p ( r 8 1 + 18 r ) e x P ( Y I 1 + 18 Y ) - exp (r I 1)exp (1I r) exp (v 81) exp (18v) * (expr I 1)(1I exp r)(expv I lxl I expv) : (expr I 1)(expyI 1Xl I exprXl I expv) - (exp.rexpYI 1)(1I exPr exPY). Hence(1 + e)d: ((1+ e) 81)(1 I (1 + e)) and so (log(1 + e))d- los(1 + e)D - tos (1 + e)I 1)(18 (1 + e)) -los((1 +z)81)*log(1 8(1 +z)) - tos(1 + e)I 1 + 18 log(1+ e) .
This shows that log (L + z) satisfiesFriedrichs' condition for a Lie element. Hence log (1 + e) e 8E'. Now that we know that ar - log (1 + z) is a Lie element, we can obtain an explicit formula for this element by using the Specht' Wever theorem. We have e :expf,expy-l:
sr
No
tcn
ilpIT'
f*q>o
and (37)
'^:An####
##'
f'l*q;>o'
Hence
(38) rog(1 *z)-4A,+##
##, fu*qt>0'
Since this is aLie element, if we apply the operator a to the terms of degree k we obtain ft times the homogeneous part of degree k in this element. It follows that we have the following expression
of log (1 + z) as a Lie element. (1 (39) loe * e)
:*,*,a#
It is easy to calculate the first few terms of this series and obtain
174
LIE ALGEBRAS
( 4 0 ) I o g (*1z ) : r + y + 6.
+ I r y l +$ . U * t l l - + I x y ) x ) + . . . .
Cohomology of Lie algebras. The standard complex
In this section we shall give the cartan-Eilenberg definition of the cohomology groups of a Lie argebra and we shall show that this is equivalent to the definition given in g 3.10. Tb obtain the cartan-Eilenberg definition one begins with tie field o which one regards as a trivial module for g, that is, one sets fx :0, E e o, / e 8. This module and all g-modules can be considened as right modules for the universal enveloping algebra ll of g, since every representation of 8 has a unique extension to a reprdsentation of u. The term ll'module will mean "right [-module,'ihrtoughout our discussion. we recall that a modure is called free if. it is a direct sum of submodules isomorphic to the module lr. one seeks a sequence of free ll'modules xo, xr, xr,. . . with ll-homomorphism e of Xo into b and ll-homomorphisms di_, of. & into &_, such that the sequence (41) Q+-oTXoTX,TXz+... is exact, that is, e is surjective, the kernel, Ker e -- irhage , lm do, and for every i > l, Ker d;-r: Imdi. A sequence(41) i; called a free resolution of the module o. Now let llt be an irbitrary g-, hence 11', module. Let Hom (x;,Tft), i >- 0, denote the set of llhomomorphisms of X; into llt. The usual definitions of addition and scalar multiplication can be used to make Hom (X;, Ut; a vector space over O. If n, e Hom (X;,Wt), then d,;nie Hom (Xo*,, Dt). We therefore obtain a mapping d{ n;-+d;rJi of Hom(X,,UD into Hom (xr*,, tJt), i = 0. This mapping is linear. Moreover, the exactness of (41) implies that d*dd :0, '> 0, and this implies that drdi*t - 0. Hence Im dr s Ker di+,. One defines the i-th: cohomotogy group of I relatiue to the module tn to be the fdctor space Ker dr/Im di-r, i >- L, and Ker do for l' : 0. The Cartan-Eilenberg theory shows that these groups are independent of the particular resolution (4D for O. We shall not go into the details of this theory here but shall be content to give the construction of what appears to be the most useful resolution (41). The space X = Xi"Xr we shalll obtain is
V. UNIVERSALENVELOPINGALGEBRAS
T75
called the standard, compler for 8. It will turn out that the standard complex has an algebra structure as well as a vector space structure. This has the consequencethat in suitable circumstances one can define a cohomology algebra in place of the cohomology groups. We now suppose we have a representation R of a Lie algebra 8 by derivations in an algebra lI. Let tl be the universal enveloping aigebra of B. Then we shall define a new algebra (!I, U, R), the atgebra of diferenti,al operators of the representation R of 8. The space of (lI, u, R) will be ?I8 u. since the mapping of elements -+ i e A into their left multiplications vr(u au) is an anti'homo' morphism, the mapping l'-+ lt is a representation of 8 acting in U. We now form the tensor product of this representation and the given representation R in 2I. The resulting representatibn maps / aSlu' into the linear transformation sending a@z into al* 8uU, which of a representation to We can extend this representation of 1l anti-homomorphism the zr is ztL where we shall now write as g, 11 into of anti-homomorphism L an is and l, I e such that tn the write We U. lI8 of transformations linear the algebra of : (a u)(l(trL)) have we Then & Lo. Z as 1l under e of a image aI"@u-&8lu;hence,
(42)
(a@u)Lr--a&lu-alo8u.
Next if DelI, we defineLatobethe linear mappingin ll81l such that (a8u)La:ba&u (43) The mapping D-, Ln is an anti'homomorphism of ?I into the algebra of linear transformations in ?I8 U. It follows from (42) and (a3) that (a & u)LoLr -- ba & tu * (ba)t* & u : ba I lu - blna & u - b(atB)& u (a & u)LtLo : ba I lu - b(aln) & u (a8u)Lora: (bla)a8u. Hence we have (M)
IL6LJ:
- LotB ,
which shows that the set of linear transformations lu is an ideal in the Lie algebra of linear transformations of the form L,, * Lt.
176
LIE ALGEBRAS
It follows that the enveloping algebra fi of this set of transformations is the set of mappings of the form \L"tLor (cf. 92.2). We have the canonical mapping of lI8 U into I sending 2arg4^ u, If I e 8, lln :0, since /8 is a derivdtion. Hence into \L*rLot. (l& u)Lt : L & lu and, consequently, (1 I o)L" : t&\ua, u,u € "0. This implies that (1 &t)L"L": a8 z so that, if > Lur\on:0, then Thus the mapping X ai& ui -+ ) L"rLdn is a vector E a;8 u;:0. space isomorphism. Since the set of mappings of the fdrm 2 LotLot is an algebra we can use this mapping to convert ?1811 into an algebra by specifying that our mapping is an a'lgebra antiisomorphism.. The resulting algebra whose space is lI S ll is the algebra (?I,U, R) which we wished to define. It is immediate from our definitions that the subset of (lI, U, R) of elements of the form a & L is a subalgebra isombrphic to ?I. We identify this with ?I and write a for c I 1. Similhrly, the set of elements of the form L&u, u€,LI, is a subalgebrh isomorphic to O. We identify this with tl and write u for I & U. The mapping of (lI,U,R):UBU i n t o I s e n d sa : a 8 l into L'L': L"and sends tr.: | & u into LuL, - L*. The formula (44) gives the following basic commutation formula in (lI,1l, R):
(45)
lbll:bl-lb:bln,
b€U,
/e8.
Since every element of f has the form > LutLu,, every element of (lI, U, R) has the form l, aiui, a; e \, h Q'/.7. Also I aru, : g if and only if > ae&u;:0 in U8U. We establish next a "universal" property of the algelbra (?I, U, R) in the following Pnorosrttorq 2. Izt E be an algebra and let 0r be a hoVnomorbhism of tl into 9, 0z a homomorfhisrn of 11.into 9 such that (4 6)
- (t0 ,)(b 0 ,)- (b f)0 ,, [ b0bl02] l= (b 0 )(t0 r1
b e ?I ,
/ e 8,
holds. Then there exists a unique hornomorphism 0 of (W,lJ, R) into D such that b0 : b0u u0 : u9z. Proof: We form (S, [, fi) e E which we consider a$ an algebra of pairs (x, d), .r e (?I, U, R), d e E, with component fddition and We have the homomorphisms zr, rz of this algebra multiplication. onto (?1,U, R) and 0 respectively defined by (x, d)rt : JC,(x, d)r, generated by the d. Let E be the subalgebra of the direct sum elements (a, a0r) and (2, u02), a € ?1, z c 11. The mapping zr1induces
V. UNIVERSAL ENVELOPINGALGEBRAS
L77
zz induces a homomorphism of u onto (?I, U, R) and the mapping generated by the D of subalgebra the onto a homomorphism of D the (46), have we (a5) and By uerl. ae}I, uTr, elements alrand in 9 following relation
(47)
(a, a0r)(1,10)-
- (aln, (alB)lr) Q,l0)(a, a0r)
indicated This permits us to carry out a collecting process such as ({n [, R) to algebra of the in the discussion of the construction aie \' form the in 2(a;,ai0r)(u;,ui0z), write the elements of E have we that (:(?l,U,R)) implies ?l8U property of u;e,ll. The sending |'aiui (?1, U, onto R) of 9 a vector space homomorphism implies into x (at,a;0)(ui,u;02). The existence of this mapping 0 : rl'r, is (U, Then U, R). onto of U isomorptrism an nr is that : a homomorphism oi (?I, U, R) into O such that a0 anl'rr: -* uniqueness : The (a, a\r)rz : a|t and u0 : trrclrftz (u, u02)rc2 u'02' (u, u, R). generate ll of d fotlows from the fact that tr and the conclusion tr then for generators Remark. If tn is a set of all b e Tft' (46) for holds provided that of Proposition 2 will hold for all this D satisfying of set the This follows from the fact that directly. verified be can / e B is a subalgebra of ?I-as We consider next an extension of the notion of a derivation: Dpr.rNrttou 2. Let tr and E be algebr?s1 s1 and s, homomorphisms an of ?l into E. Then a linear mapping d of. s into E is called (sy s)-deriuation if
(48)
(ab)d - (as,)(bd)+ (ad)(bs,)
just those which It is easy to check that the conditibns on d are insure that the maPPing
(4e)
o-,(3" :!,)
is a homomorphism of ?l into the two-rowed matrix algebra Ee' (A special case of this remark was used in the proof of 7 of Th' 1.) it U i. a subalgebra of $ and sr is the inclusion mapping, then we call d. an sz-d.eriuation and if sz is also the inclusion mapping, then we have a d,eri.uation of 2I into E. One can verify directly, or use (49) to see, that if d is an (s1,sr)'derivation, then the kernel of d is a subalgebra. It fotlows that d :0 if. Tftd: 0 for a set of generators TJt of ?I. Also if. d, and, dz are (sr, sz)'derivations then
L78
LIE ALGEBRAS
d, - d, is an (s,, ss)-derivation. Hence it is clear that dr : d, if xd1: xdz for every x in a set of generators. We consider again the algebra (?1,U, R) and we cah prove the following result on derivations for this algebra. , PnopostrtoN 3. Let e be an algebra, sr, sz homomorphisms of (2[,U, R) into 9. Let d, be an (sr,s)-deriuation of 2I into D and dz an (st, s)-deriuation of U inta I (s,, s, are the restrictipns /o ?1,1l resfuectiuelyof Sr,sz on (?1,U, R).) Supposethat for D e E, / e 8: (50)
(bs,)(ld) + (bd,)(ls) - (/s,XDd,)- (td,)(bs,)- (bt*)d, .
Then there exists a unique (sr, s2)-deriaatiand of (21,U, R) into D such th a t bd: bdr , ud: u d z . Proof: Consider the mappings dr and 0zof.2land11,respectively, such that
uf:,)'uor: (us' (3'' \0 "':
ud'\ . usz/'
Since dr is an (s,, sr)-derivation, d, and d, are homomorphisms into the matrix algebra Qr. A direct calculation shows that the condition (46) is a consequenceof (50). Hence Proposition 2 itnplies that there exists a homomorphism d of (?I,U, R) into Sz such that b0 : b01, u0 : r,t?z,c € ?I, u € \. It is clear that d has the form o: , -, (!t' \O
J\ xsz/'
Set y - xd; then since d is a homomorphism, d is an (s,, sr)-derivation of (?I,U, R) into D. We have bd,: bdr, ud: udz, aiqrequired. The uniqueness is clear. The proof of Proposition 3 and the remark following Proposition 2 show that it is enough to suppose that (50) holds for all D in a set of generators lJt of ?1. We shall need this sharper form of Proposition 3 later on. Let lIt be a vector space over @ and let E(IJI) be the exterior algebra (or Grassmann algebra) over fi. We recall thAt E is the difference algebra of the tensor algebra T. : O 1@tlt O (9nI tX) O. . . with respect to the ideal generated by the the elementsir @ r. It follows immediately that if s is ^ linear mapping of $lt into an algebra E and (rs)' : 6 for every t E 9Jt, then s can bb extended to a unique homomorphism of E into E. The canonic4l mapping
V. UNIVERSAL ENVELOPING ALGEBRAS
179
tIft of g onto E is an isomorphism on @1€) 'lt' One identifies aL + denote generates shall We ^E. with its image. It is clear that fi the multiplication in E simply as ab (rather than the more complicated e Ab which is customary in differential geometry.) lf x, y € tJt, thent1'y+yx:(x*y)'-tc'-Jt:0'ThealgebraEisgraded with \Jt- as the space E* of homogeneous elements of degree m. If. {utlj e I} is an ordered basis for IJ}, then the set of monomials of th e fo rm ili{ t iz ' . . ili^, i , < i r a " ' a i ^ i s a b a s i s for E . In i f. m > n a n d d i mE --(fr)i f parti cu l a r, if dim m - i, th .n E * :0 The exterior algebra has an automor' m S n. Hence dim E :2o. phism 7 such that xq: - N for all , e Uft. lf x^ e E^ then x*T : (-l) x* . We are interested in derivations and 7'derivations of E(9]t) into algebras E containing ,E as a subalgebra. The Z-derivations will be called anti-deriuations of E into E. We shall need the following criterion. PnorosrrroN 4. Let E be an algebra which contains the exterior al' gebra E: E(Tft) as a subalgebra. Let d be a linear maffing of Ift into E. Then d can be extended to a deriaation (anti-deriaation) of E into A if and only if x(xd) * (xd)r : 0. (r(rd) - (rd)x - 0) for all x e ,lt. Proof: The conditions are necessary since tr' :0 in -E implies x(xd) + (xd)x:0 or x(rd)- (rd)r - 0 according asdisa derivation or an anti-derivation. Conversely, suppose the conditions hold. In the first case we consider the mapping
o:x-,(x
'l)
and in the second, the mapping
o" '-' (;
4)
of TJt into the matrix algebra Ez. In both cases one checks that (r0)z : g. Hence 0 can be extended to a homomorphism of E. The rest of the argument is like that of Proposition 3. It is clear that when the condition is satisfied the extension is unique. We apply this first to the following situation. Let 8 be a Lie algebra and TJt a module for 8. Let E: E(IJ|) be the exterior
180
LIE ALGEBRAS
algebra defined by Dt. lf. r e lll, / e g, rl e llt so that x(xl) * (xt)x 0 in ^8. Hence there exists a unique derivation dr ini^E such that rd1 : rl, r € TIt. have rdtfttz - xdtl * xd4, rd,t - aldu rd7rrr27: _We x[d4d4l for x e Tn, a e O, l,lt, 12,e g, sincl tn is A" g_module. Since all the mappings considered here are derivations afid tjt generates ^8, we have d-rr:rr- f,\* dh, d61: M1, d.prt2J:fdtrdhf in E. Consequently, I - dt is- a represetttatiotr of 8 by derivatlo". i" EOn). we consider the special case of this in which rlt = g and the representation is the adjoint representation. we form the algebra 1: (E(8), u, A) where 4 denotes the extension of the adjoint r"pr"sentation to a representation by derivations in E(g). I There i, slight difficulty here in our notations since we now havF two copies " of 8, one contained in ll, the other in ^8. we shall {herefore denote the elements of the copy in u by l, I e g and we qfrite E: {D. we denote the corresponding elements in E by t anl we denote this copy as 8. In order to avoid ambiguity we shail denote the Lie composition in 8-which does not coincide with the Lie product in E"-by llr"lz7. Then we have the Lie isomorphisrn /--i of B onto E; hence we have WI - [lrtr] : I,l, - irlr. Since g and g generate ll and E, respectively, 8 u E generates X The following relations connecting these generators are decisive: U" l'l : [7=oI'zl, llr,irl : fllo 12!,
(5t1
12: 0
i
l /; € 8. The last of these is a consequenceof (a5). , The last relation in (S1) implies that if rr e E;, ther[ [r;, I]-g-Tir e Er. It follows easily from this that f,g, : E[17'i for OL+E'+-..+S7in ll. HencelJEr:Brg. If we sdt X;:E;II, then we have X;Xt : E;OEiIJ g E;*ilJ : X;+i. Also since X - E'8 u it follows from the properties of tensor pfoducts E11, that X:2 This shows that X is a grdded algebra OX, : X O Etl. with x;: E;11as the space of homogeneouselements df degree l. Pnoposnror* 5. There erists a unique automorphism u of x (E(8)' a, A) and a unique v-deriuation d of x into x such that
(52)
ln:-l ld:i,
Moreoaer, Td + dT :0
,
in:i id:0,
and d' :0.
/e8.
V. UNIVERSALENVELOPINGALGEBRAS
I81
Proof: Evidently we have a homomorphism 4r of. .E into X such that 14 - -1, / e B and a homomorphism \z of 11into X such that in - i.' The condition (46) for 6 : Ir, I : ir, 0r-:4r, 0z:42 reads : -ll ro l 2)' Thi s lir nr,l rn rl f lr olglz r whic h i s e q u i v a l e n tto [- l r,i r) to prove the applied be 2 can holds by (51). Hence Proposition the mapping that evident It is of T. existence and uniqueness :0 t i i t : U l ) : l loll S i n c e X . O i n t o dz:0 is an 7'derivationof the condition of Proposition 4 is satisfied and so there exists an anti-derivation d., of. E(8) into X such that til: i, / e 8' We now c heck (5 0 ) f or b : lr , I : l r, s r : 1 , s z : L d t: d r, dz: dz' The left-hand side is /,0 + l,i, - iri, - o(-/,) - [i,irl . The right-hand side is Ll,ir)d, - U, o lr7d.,: WTI : lirir). Hence (50) holds for these elements, so by the remark following Proposi' tion 3, there exists a unique ?-derivation d in X satisfying the condit ions (5 2 ). W e hav e X od:0 a n d Xd tr X;-r w h e re X t : E ;LI, i > l ' Also This can be established by induction since X;: X;-rX'' (-L)'xid xfl\d: Hence (-l)ixiif. X;. )e e that r;n: is immediate it : If x,y € X, lx-v)dz Hence nd + dn:0. and x;dn : (-l)i-tx;d. : (xd')(vu\ (x(yd) + (xd) (yil)d : x(td') + (xd) (ydd + @d)(vrtd) + x(yd') + (xd'z)(yq'z\.Since T' : I this shows that d' is a derivation. Hence d':0' On the other hand, (52) implies that ld':A -id'This concludes the proof. denote the restrictionof d to X;.- Then L e t d . ; - r ,i : 1 , 2 , . . ' , d;-r mdpS X; into Xr-r and, d,id*r : 0. Since M :0, d and the di commute with the right multiplications by elements of U. Hence if we consider X; in this way as a ll-module, theh di is a ll-homo' by the a n d X: l @X ;, m orp h i sm . W e hav e X ; - E ;1 7 -Er8 i l properties of tensor products. Hence any @-basis for E; is a set of free generators for Xi relative to U. Thus every X; is a free ll-module. We have Xo : U. If U' denotesthe ideal in U generated l f { u 1 l i e I} i s a n ordered basi s by th e I, / e 8, t hen lll l J ' = A. ' ' iiit,,i, S ir 3' ' ' 3 i,, for 8, then we know that the elements1, r7;rll;r' . [ti,, then we de' : a) a;r...;rr1;rt1;r... lf. 11. a * | form a basis for fine a mapping e of U into 0 by ae: a. This mapping is a ll' homomorphism of Xo : 1i into O considered as ll-module where rt17':0, al-_ d, a e A. Evidently e is surjective. Let b e Xr. Then D: X up1, {u} the basis for 8, Di € U. Then S f l o e : 0 . C o n v e r s e l yl,e t A e X o a n d bdo:bd:lilp1e)J'and
182
LIE ALGEBRAS
T h e n a : 2 a , ; r . . , ; r i i ; r . . . f i ; ,€ U ' a n d , d = b d of o f a s s u m eZ e : 0 . D : X rr1,() a;, iriliz. .. u;r). We have therefore shown that S +- O *; & *tu X, is exact. It remains to show that Ker d;-r : Im di, i > 1. For ihis purpose we consider first the case in which I is abelian; hence 1l is the polynomial'algebra in the 17;which commute and are algebraically independent. Also in ithe abelian case [''li] --lmorl : 0, ttr,/ e 8. Hencethe elements of '11commute with those of E(S). Consequently, X is the tensor pfoduct of. E and tl in the sense of algebras. Now let 8, - llt O Tl where 9ll, yl are subspaces,hence:subalgebras of the abelian Lie algebra 8. Let F: E(9lt), S : tl(ili); Y : X(n)' G : E(Tt), [8 : U(Yt), Z: X(\l). It is easy to see that F can be identified with the subalgebra of E generated by Dl, E with the subalgebra of u generated bV fft : {m e glt} and Y with the subSimitar statemen]ts hold for algebra of. X generated by llt + ft. These results follow easily by looking at bfses. Simi' G,L\,Z. larly, we leave it to the reader to check that X : YZ where Y and Z are the subalgebras we have indicated and that we have a vector space isomorphism of Y&Z onto X sending y&z intoyz, I e Y, z € Z. This result implies that if p is a linear mapping in Y and y is a linear mapping in Z then we have a unique linear mapping i, in X such that (yzil - Q1t)(zv)We now regard the augmentation e of. Xo into O as, a mapping of Xo into the subalgebra OL of. X and we extend this: to a linear mapping in x s u c h th a t x ;e :0 , i > 1 . T h i s i s an al gebra homo' morphism of. X onto the subalgebra @. We now provel PnopostttoN6. There existsa linear mapping D in X: X(8), 8 abelian, such that Dd + dD = 1 - e. Proof: Suppose first that dim 8 : 1 and (u) is a basis for 8. We have tt,L6i,ilil',"'. Then X has the basis l,il,iiz,It',"'i l , t l i ' t e : 0 , u u i e : 0 - I i l e n c ei f w e {rnd:0, uilid: fii'|, iZ0;le: d e f ine D t o be t h e l i n e a r ma p p i n g s u c h th a t LD :0, i l i t' D --tti l i , u uiD : 0, i 2 0, th e n o n e c h e c k sth a t d D + D d:1 - e as requi red' We note also that TD + Dn -- 0 if A is the automorphisnripreviously defined in X (ilin : {ii, utliq : - uu'). Moreover, we have tfre relations a n d e T : e : n e . N o w s u p p o s e, 8- l l l @ I t i w h e r eT t i s ed:0:de one-dimensional. Then X : YZ, Y = X(9ll), Z : X(tt) wihere Y and Z are che algebras defined before. It is clear that Y and Z are invariant under d, e and T, and that their restrictiorls are just
ALGEBRAS V. UNIVERSALENVELOPING
I8:}
the corresponding mappings in X(St) and X(gt). Let Dz be the mapping just defined in Z and let Dr be any mapping in Y such that dD, + DLd: 1 - e in Y. There exists a unique linear mapping D in X such that (53)
(yz)D:y(zD)*(tD')(ze), veY,
zeZ.
Then we have (yz)Dd,- y(zD2d)+ (vd)(zD*i * uD)(zed) + (tDd)@e7t) - y(zDzd)+ (td)QDni + UD'd)(ze) (yz)dD-(y(zil+Ud)kd)D : y(zd'D,)* (vD)(zde)+ (td)(zvD) * (vdD')(zr1e) : y(zdD) + UA@rtD) * (ydD,)(ze). (yz)(Dd + dD): y(z(\- e)) + (y(1 - e)Xze) - yz - y(ee) + y(ee) - (ye)(ee) ' - y r(!- e ) . The inductive step we have just established implies the result in the finite-dimensional case by ordinary induction and in the infinite' dimensional case by transfinite induction or by Zorn's lemma. X, Xr This proposition implies the exactness of Xo T 7 (-.... Then r-- r(l -e): Thus let r € X;=rXisatisfyxd:0. -- Im dr i > L. r(d.D + DA - @D)d e lm d. This implies that Ker d;-, We consider now the general case of an arbitrary Lie algebra 8 and we introduce a filtration in X using the space of generators Thus we define yrit-oL+!t+sts+"'*Ttj: It-8+E. ytit is a subcomplex. Also 1o*tstEngl. Since (Ebg\ds Ei-,3&+r, it is clear that X(i) is a sum of homogeneous subspaces relative to the grading in X. We can form the difference complex Xtit lyti-rt which has a grading induced by that of X. lf. {uili e I} is an ordered basis for 8 then the cosets relative to Xtr-t) of the base ele me n ts u\ . . . uioilt 1. .. i l ;* , i , < i r. h + k: j form a'basis for Xrit lyti-tt. We identify these cosets titl;gtJ-r) is with the base elements. Then we can say that d in X determined by the rule
(54)
(utr'''
t t ' 5 o i l t ',' ' i l t * ) d
184
LIE ALGEBRAS
where the ^ indicates that u,o is omitted, and the ordbr of the subscripts in the product of the z's is non-decreasing. This differenti' ation does not make use of the structure of 8. Thus it is the same as that which one obtains in the abelian Lie algebra. In the latter case the space spanned by the monomials on the left.hand side of j fi x e d fo rm a s u b compl exof X and X i s (54) s at is f y ing h + k It follows a direct sum of these subcomplexesfor j :0,1, '''. Xr+t i is in and that if r is in one of these subcomplexes , > 1, and y given con' is the Xi and where x:!d, e then 0omplex tcd:0, ! taining r. The result we wished to prove on the exadtness of (a1) (i) by dhe following will now follow by induction on the index j in x Lpuu.r, 3. Let Y be a subcomplexof a graded com\lex X :27ae
Xd
such that ts: xlo( y n xr). set z - xlY - l.::o'.s^Zo, Zt: Kerdn-r: Im dr i2l, holdsin"Y and i,n Z. Xil(Y n X,). Suppose Then this holds also in X. H ence T h e n (x* Y )d:0. P r oof : Let x € E t> ' X i s a ti s fy rd :0 . T h u SN ' d : x * ! , t h e r e e x i s t sx ' + Y s u c h t h a t ( r t + Y ) d : t * Y . y e Y. Then yd: x'd'- rd:0 and so there exists a, !' e ts such that y : !'d, Hence r : r'd - y - r'd - !'d : (r' * y'rd. We have now completed the proof of the following THponpru 10. Let X: (E(8),11, A) be the algebra of diferential operators determined by the exterior algebra of E(8) Iand the er' tension of the adjoint re|resentation of 8 to E. Iat d be the anti.' deriuation in X defined in Proposition 5 and let X; j E;17. Then A* , Xo* ^ Xr+ "' l's a free resolution of the nlodule tDIt remaini"to show that the cohomology groups dedned by this resolution coincide with those of S3.10. Let there he a ll-homomorphism epof Xt into a (right) ll-module llt. Let (lr,,lr, ' ' 'r /r) h an ordered set of i elements of 8 and define a nlapping f of where (i times)into llt by f(|r,"',l):(lJr"'1,)p 8x...x8 lrl, . . . l; € Ei tr X;. It is clear that this / is multilinef,r and alter' nate and it is easy to see that if / is any multilindar alternate mapping of 8 x ..' x 8 into fi then there exists a linear mapping cpof E;(8) into l}t such that (lJz "' l;)tp: f (l', '",lr)- t This lp has a unique extension to a ll-homomorphism of Xi into IIt. We therefore ti"ve :l 1 : 1 linear isomorphism of the space of ill'homomorphisms of. Xi into !]t and the space of multilinear alternate mappings of the e-fold product set I x ... x 8 into 1Il. Ifr/ is a map-
V. UNIVERSAL ENVELOPING ALGEBRAS ping of the latter
type we define fB by
(55)
( f l) ( lr , " ' , l ;* r) : ((/' " ' l * r)d )g .
185
We have 6+r
(lr. -- lr*r)d: t
(-l;t*'-or, . -. io..' l;*r,
and, by (45), iolo*r"'
lr*r :
lq+r"'
lr*rlo
:
l q + t . . .l r * r l n +X 1 " . r . 1 ) p ; , 1 . . .l ; * ,
:
lq+ r .. . lr *r l n + > (-1 )t* t* ' ro + r. .. tr. -. l i +rfl ql ri.
r>.t
_
Hence -. io... lr*rlo H(-l;i*'-or r.
(lr - -. l*r)d:
q=l ,+1
*
. " ' in' '' t,''' l;+rtlql,l "F=rt-1)T*n/,
It follows that (56)
(f|)(tt,... l;*,):
ti
(-!)i*'-of (lr, , tr,
which are the same definitions as given in $ 3.10. 7.
Restricted Lie algebras of eharoateristie p
In many connections in which Lie algebras arise naturally one encounters in the characteristic 2 * 0 case structures which are somewhat richer than ordinary Lie algebras. For example, let ?l be an arbitrary non-associative algebra and let O(?l) be the set of derivations of ?I. Then we know that E(U) is a dubalgebra of the Lie algebra of linear transformations in ?I. We note also that one has the Leibniz formula
(57)
(ab)Dk: * Oao'>(bDo-)
186
LIE ALGEBRAS
for any derivation D. This can be establishedby induction on fr. Now assumethe base field of [, hence of S(?l) is of characteristic p and take k - p in (57). Then the binomial coefficients (f) : 0 if 1 < i = p - l. Hence (57) reducesto (58)
(ab)DP :
(aDe)b * a(bDP) ,
which implies that De is a derivation. Thus S(U) is closed under the mapping D -, De as well as the Lie algebra compositions. Similarly, let ?I be an associative algebra and a + a an antiautomorphism in ?1. Let 8 be the subset of skew elemOnts relative to a -+ a. Then we know that 8 is a subalgebra €Er,r Moreover, i f the c har ac t er i s ti c i s P th e n a - - a i m p l i e s that ae:d.P :(-a)n: -ao. Hence ao e 8, so again we have a Lie algebra closed under the p-mapping. The systems S(?l) and 8, just considpred are examples of restrlcted Lie algebras which we shall defind abstractly. For this purpose we need some relations connecting ap with the compositions in a Lie algebra ?[2, U associative of characteristic p. We recall first the following two identities in 0U, pl, f,, tt algebraically independent indeterminates, @ of characteristic p: (l-p)':lP-tte
(5e)
.
(t - p)o-' -f
xt-l
I oo-r-i
t'=O
The first of these is well-known and the second is a fonsequence of the first and the identity ),e- pe:(i - p)(>tr:t^tpot'-'). These relations imply corresponding relations for commutirtg elements a, b in any associative algebra. In particular we may take a : bn, b : bz the right and left multiplications determined by an element D e ?I. These give (b" - br)o: f" - b!,: (bn)" - (bn)" p-l
p-l
(b* - br)o-':'>_- b'*b?-'-t: t (bn)*(bo-'-')" , t,=0 C=u
Oft p
(60)
L ' ' ' l a b l b l ' ' 'b l : f a b o l ,
(6t1
t . . . t a b l b l . . . b:5 l
be-L-iabi
I87
V. UNIVERSALENVELOPINGALGEBRAS Also it is clear that (aa)o :
(62)
aeaP ,
and we shall use (61) to Prove that @ * b) e: a P* b ' + T s i (a ,b )
(63) where
,
is;(a,o)is the coefficient of ,'-: ," a(ad,Q,a+ r))n-' ,
(64)
I an indeterminate. To prove this we introduce the polynomial ring ll[.i] and we write * D" +5 Q,a-l b)' : ),PaP
(65)
s;(a,b)]i ,
total degree 2' where s;(a,b) is a polynomial in o,i='* differentiate (65) with respect to i we obtain p-l
.
o-1
Tflo * b)ta(la* b1o-i-r !
If we
is;(a,b)^'-''
By (61),tfri, girr.. (66)
a(adQ,a* b17n'r:5 is;(a,b)^'-' . t.:r
Thus we see that is{a,D) is the coefficient of ii-r in a(ad (}.a * D))o-'. on the other hand, substitution of I : 1 in (65) gives the relation (63). It is clear that s,i(a,b) is obtained by applying addition and commutation to a,b and so is in the Lie subalgebraof.\Ir. generated by a, b. For examPle, if s'(a,D) - labi s,(a, D) llablbl ,
f :2 ; Zsr(a,b) : llablal
if
f : 3,
s,(4,b) : fillablblDlrl, : llllablalblbl+ tlltablblalbl+ ll[lablb]blal, Zsz(a,b) + llllablblalal, D): [t[tcb]alalbl+ Llllablalblal 3ss(a, :5 . b): tt[tab]alalal if 0 4s,(a, These considerations lead to the following p + 0 Dsr,ll,rtrtoN 4. A restricted Lie algebra f] of characteristic a defined is there p which in is a Lie algebra of characteristic
188
LIE ALGEBRAS
mapping a -t srnr such that
(i) (ii)
(oa)l'l - ooolnl
(a + b)rot: 6rn!a 6tr.'+ 5 s;(a,b), i:r
where is;(a,D) is the coefficient of ,ld-' in a(ad(ta * 6))1i-'and (iii)
[obc'11: a(adb]' .
If ?I is an assrciative algebra of characteristic p, then the foregoing discussion shows that ?I defines a restricted Lie algebra in which the vector space compositions are as in A, Wbl: ab - ba and at't : a?. We use the notation ?Iz for this restricted Lie algebra (as well as for the ordinary Lie algebra). A homomorphism S of a restricted Lie algebra into a second restricted Lie algebra is, by definition, a mapping satisfying (a * b)s : 4s * bt, \(aa)t : aas, fab]s - laubul,(arn:ru- (as1tol. Ideals and subalgebraslare defined in the obvious way. A representation of 8 is a homofnorphism of 8 into the restricted Lie algebraEz, @,the algebra of linear trans' formations of a vector space llt over O. If R is a representation acting in the space tjt, thln !ft is an 8-module relative to xa = x&n, reTlt, a€,8,. The module product ra satisfies the usual Conditions in a Lie module and the additional condition that ,otil - ( " (xa)a) "'a, f a' s . We consider now the following two basic questionsf (1) Does every restricted Lie algebra have a 1 : 1 representationl (2) What are the conditions that an ordinary Lie algebra be redtricted relative to a suitable definition o1 oIorT In connection'\,tiith (2) it is clear that a necessary condition is that for every a e 8, the derivation (ada)P is inner; for, in a restricted Lie algebra (ada)p : ad'ar'l. We shall show that this condition is sufficient. In fact, we shall see that it will be enough to have (ad u;)e inner for every u; in a basis {u;} of. 8,. We note also that if I is restrictedl relative to two p-mappings a-+4rn\ and a-'+sferz, thenf(a) - rtn\l- otnJzis in the center of 8 since lb,-f (a)l : lbafttrl - lbat'tz1- 6(ad a)'i- b(ad a)p : 0. It is clear from (ii) and (i) that r (67) .f(aa): aof(a). f(a+b) -f(a)+f(b), A mapping of one vector space of characteristic P + 0 into a second one having these properties is called a p-semi-lincw mappingConversely, if 8 is restricted relative to a-trtrh apfl 4-+f(a) is a
V. UNIVERSAL ENVELOPING ALGEBRAS
189
2-semi-linear mapping of 8 into the center G of 8, then 8 is also restricted relative to Q -+ qtnJ,- otrh + f @). The kernel of. a psemi-linear mapping is a subspace. Hence if f(u;) -0 for u; in a basis, then / : 6. It follows that if two p-mappin gs a -, arr}, and a -+ srn72making 8 restricted coincide on a basis, then they are identical. Suppose now that 8 is a Lie algebra with the basis {uli e I} where ^I is an ordered set and let ll be the universal enveloping algebra of 8. The Poincar6-Birkhoff-witt theorem states that the "standard monomiats" u!]ul] -.. u::, i, I i, basis for U. We have the filtration of U defined by 11tlcr OL+ 8+8'+...+8&. I t i s e a s y t o s e e ( i n d u c t i o no n f r a n d t h e ... usual "straightening" argument) that the monomia.ls "f: such that h * hz *... + h, < k form a basis for ll'e). "fi"i; We assume now that for each base element ui there exists a positive integer fli ?\d an element ze in the center 0 of tl such that (68)
a t:
u \t - z ;
is in U(ni-r). Then we have the following Lpuul
4.
The elements of the form
(6e)
,i;"ii...ziiui:ui:... "i:
such that L 1 iz l a basis for 1J. Proof: We show first that every element ,ry, fu > 0 is a linear combination of the elements (6g). rf. "f:... k - k, * kz + ... a k,, then we employ an induction on h. If every li 1. /tt1, then the result is clear- Hence we may assume ki 2 nt, for some /. Then we may replace uTlt 6t h1 * zt, ?Dd obtain
"f:...
u!;-
. . u!1-""..- + uii... u!1-"o,rn, ",,ufl-."f: "f:
. .. u?,1-"" . . . The elements and the secondterm on the right "f: "f: are in qt&-t). Hence these elements are linear combinations of the elements (69). Since the set of elements in (69) is closed under multiplication by any zi, the result asserted is clear. We show next that the elements (69) are linearly.independent. we replace zr, b!
Ztr: uTlt- ut, in ,!i... ,ii,.ti...ui:. Thisgives
190
(70)
LIE ALGEBRAS
-.-,?:,i: ,ii--.,i;"i;.--"i: : ,!;"i;"!;"i: ...
- 1uT!, u,,)o'"i: fuT), a,|n'uli = u!]"tr*\ulzntz+\z . . . y!,n!t)',
-r,; (modtU,ft
i f h : 2i- , ( hit 4t 1* ti ). H e n c e th e e l e m e n t t?: ... ti !" i i -.. ul ; xt .. u!fltf u!;"',*^" + * , where * is a linear combination of standard monomials belonging to U('c-r). It follows that if we have a non-trivial linear relation connecting the elements in (69), then we have such a relation for terms for which the "degree" fusame coefIii(h;tu1* ii) is fixed. This gives a relation with the tr' ' ' ' u!:"t'*^'. ficients in tlie corresponding standard monomials el|'"it we must independent linearly Since the standard monomials are have hyhl * li - ht, i - 1, . . ., rt for the elements of (69) which appear in the relation with non-zero coefficients. Sitrrce ),i 1 rh1 this implies that hi, f,i are determined by the equation hifttl * i1 : This is h1. Hence there is just one term in the relation. impossible in view of (70). Hence the elements of (69) are linearly independent and so form a basis for 11. We can now prove the following Tnponpu 11. Iat 8, be a Lie algebra of characteristic f + 0 with ordered basis {u;} such that for euer! u;, (ad u;)p is an inner deriaa' tion. For each u; let utot bn an element of 8,such that (adu)p - adu?r. Then there erists a unique mapping a -->rrnt of g into'g such that ulo' it as giuen and I is a restricted Lie algebra relatiue to the ma\' bing a -+ sttr. Proof: Let 11 be the universal enveloping algebra ff 8. Since (adu)e: ad ulor, zr: u! - rrln] com*utes with every t e g. Hence zi is in the center of tl and u'{ : z; * ut't where ul't e g. We can therefore conclude from the lemma that the elements
(7r)
,i;ri;...,i;"ii .. .
"::
such that ir < iz < U. Let E denote the ideal in tl generated by the z;. Then it is clear that the subset of our basis consisting of the elements (71) with some hi > 0 is a basis for A. Hence the cosetSof the elem ent s uli. . . uil ,0 ( i i < p - 1 , fo rm a b a si s for the pl gebra [,: U/4. Since thecanonicalhomomorphism x-+E:x* $ is a homomorphism of 11, into [r, the restriction to 8 is a homo{norphism of
V. UNIVERSALENVELOPINGALGEBRAS
I91
Since the ili:u; * E are linearly inde' 8 onto E : (8 + 8)lg. is an isomorphism of 8 onto E. We note pendent, t-rl:/+E next that E is a subalgebra of [z considered as a restricted Lie ul'r + I e E' It algebra. Thus we have alt : (u;+ bf : al * E: proves the asser' (f,aiili)e E which (62) (63) e that and follows from I as a permits consider us to E 8 and of The isomorphism tion. - 1c. Then we have restricted Lie algebra by defining |tt\ fiy 1tt\ ulotr- ul +E:uf,or *E so that ulpt'-uI't as required. We now write l/tnr-;trlr 41d the result is proved. We recall the result of $ 3.6 that if 8 is a finite'dimensional Lie algebra with a non-degenerate Killing form, then every derivation of 8 is inner. It is clear also that the center of I is 0. Hence if the characteristic is 2, then for every @e I there exists a unique --ad.at't. It foltows element atol such that the derivation (ada)p p-operator in 8 in one and only one way that we can introduce a so that B is a restricted Lie algebra. We therefore have the following Conor,r,.lny. If I is a finite'dimensional Lie algebra of characteris' tic p + 0 with nan-degenerate Killing form, then tlwre is a uniquc p-mapping in 8, which makes I restricted. We suppose next that 8 is an arbitrary restricted Lie algebra and we prove the following Tnponpu 12. Let 8 be a restricted Lie algebra of characteristic P + O, 17 the uniaersal enueloping algebra, E the ideal in 1I generated by the elements ap - ar'l, a € 8, [ : 1l/l8. Then the mapping a --+a : a * E fs an isomorphism of 8, into the restricted Lie alge' bra fr.". If S is a hornomorphism of 8, into the restricted Lie alge' bra \1, W an atgebrg (associati,uewith L), then there exists a uniqrc homomorPhism of tr into ?I such that a -- at. It 8, is of finite dimensionalitY n, then dim-Ll.: f". Proof: Since otnt - ap : ap * E, a --+a is a homomorptrism of the lf. {u;} restricted Lie algebra I into the restricted Lie algebra [r. is a basis for 8, over O, then it is clear from the rules for p-powers and the operation a --+nEorthat ap - ottt is a linear combination of the elements af - ulot. Hence E is also the ideal generated by the elements ut - uE'\. The proof of the preceding theorem shows that the cosets il;: ui * E are linearly independent; hence a -+ d is an isomorphism of I into [2. Now let S be a homomorphism of I
LIE ALGEBRAS
192
into a restricted Lie algebra ?Ir, U an algebra. Thqn we have a homomorphism of U into ?I sending a e 8 into a8. r Under this mapping a.p-- (au)o: (ot't)t. Thus oo - orot is in the liernel and so A is in the kernel. Consequently, we have an inducbd homomorphism of [ : 1l/E into ?I such that a -- a,u. If 8 has the finite basis ut, xtz,. . . , u* then we have seen that the cosets of the elements ul'ul' -.. uh", 0 3,ir < p -1 form a basis for [. ] This proves the last statement of the theorem. The algebra [: WE of the theorem will be called the u-algebra of the restricted Lie algebra 8. It plays the same fole for 8 as restricted Lie algebra as is played by 11 for 8 congidered as an ordinary Lie algebra. In particular, a representation of 8 as restricted Lie algebra defines a representation of tl and conversely. Since I has a faithful representation it follows that evdry restricted Lie algebra has a faithful representation. Moreover, iif 8 is finitedimensional, then I is finite-dimensional and so has a faithful representation acting in a finite-dimensional space. Consequently this holds also for 8. 8.
Abelian restricted Lie algebraa
] An abelian restricted Lie algebra is a vector spabe 8 with a mapping a -+ ap (we use this notation for the p-operatpr from now on) in 8 such that (72) (a * b)e -- ao + b' , (aa)o : dPaP. The theory of these algebras is a special case of the theory of This is equivalent to ,a theory of semi-linear transformations. modules over certain types of non-commutative polynorhial domains (cf. Jacobson, Theory of Rings, Chapter 3). In the prepent instance the polynomial ring is the set of polynomials do * tar + . -. * t*a^, di€O, / an indeterminate such that at:tap. If @lis perfect it can be shown that the ring has no zero divisors an[ every left ideal and right ideal in the ring is a principal ideal, The study of this ring and its modules is a natural tool for studying abelian restricted Lie algebras. However, we shall not undertdke this here. Instead we shall derive one or two basic results on lthe algebras without using the polynomial rings. TnsonpM 13. Let 8, be a finite-dirnensional abelian testricted Lie algebra oaer an algebraically closedfield of characteristif F. Assume
V. UNIVERSALENVELOPINGALGEBRAS
193
that the p'mapping in g i's L:1. Then t has a basi's(ar,ar, "',a*) such that alt : a;. Proof: Let a + 0 be in 8 and let m be the smallest positive integ";'J; $;; d* : &ta * azap+ '" * d^so^-'. Then a,an, "',ao^-' are linearly independent and every aek is a linear combination of -- 0, set Ft : ailp, i : 2, " ' , ffi' Tfen these elements. If dr + : 0 which implies that aF*-' : pn-rap*-t)' (oo*-' Fza . . to the choice of. m. Hence ar * 0. contrary + B^-ran*-' Bza *. linear combination of ao,ao', "'. .We is a a that implies This : 9fi * Bzae*... + g*6e*-r +0 b a exists there that show now shalt such that bP: b. This will be the case if the Fr satisfy the following system of equations:
(73)
9r: Be*a, B;:
Bo*a;* Bl-t ,
i:2,,..tffi,
and not all the B; are 0. Successivesubstitution gives (74)
g^: go#ol^-'+ go#-'al^-'+ ... + \o*a^
Since dr * 0 and O is algebraically closed this has a non'zero solution for F-. Then the remaining Fi can be determined from (73) will hold by Q$. Now 1 and the equation for i:rn f.or iSm. a' which are linearly az, ", au determined already we have suppose independent and satisfy alt : ai. Then 8, : E Oai is a subalgebra of 8. Suppose a is an element of 8 such that ae € 8r. Since a is Thus w e a lin e a r.com binat ionof a o ,a o t," ' i tfo l l o w s th a t a € 8 ,. have shown that ,8/8r is a restricted Lie algebra satisfying the hypotheses of the theorem. Hence if 8 + 8,, then we can find a b 4 g, such that be = b (mod 8'). Thus b' - b Z\T;a;. We can -L,2, " ', r. Then ar*r : di * T;: 0, i determine di so that df b + > dia; satisfies 4"+r Q 8r, a?+t: ar+t. Hence the result follows by induction. Tnnonpu 14. Izt 8 be a commutatiae restricted Lie algehra of f.nite-dimensionali,ty oaer Gn algebraically closed rt'eld. Suppose the p-mapping a *+ ap is 1 : L and tet TIt be a fi,nite-dimensional module for B. Then !)t rc completely reducible into one'dimensional sub' modules. If (ar, ctzt.. -, ao) is a basis such that al : a; and a : 2 f,;a;, then euery weight in Wt has the form A(a) \ m;tr;, m; in the prime fi.eld. Proof: Let the basis (ar, ar, . . -, ao) be as indicated and let
194
LIE ALGEBRAS
a; + A; in the given finite-dimensional representation. Then AI : A; so the minimum polynomial of At is a factor of )p - f,: flfi:'r(f - m). Thus the minimum polynomial of Ai lhas distinct roots and these are in the prime field. It follows that ithere exists a basis (xr, xr, . . . , xn)for D? such that xiAi - t'nrixi, rnri inr the prime field. Since the A; commute we can find a basis which has this property simultaneouslyfor the Ai, i = 1,2,' ' ' ,n. Then l*i(X f,;A) : (1, m;itl.).f,; so that 9lt is a direct sum of the irreducible invariant subspaces Ox5 and the weights are Li: l1rft;il;. Remarh. It is easy to extend the first part of rheorem 14 to arbitrary base fields of characteristic 2, that is, complete reducibility holds if the p-mapping is 1 : 1. On the other hand, it has been shown by Hochschild (t4l) that if all the modules for a te' stricted Lie algebra (everything finite-dimensional) are completely reducible, then 8 is abelian with non'singular p-mapPtt* Exercises 1. Let 8 be a Lie algebra over a field of characteristic zero, U the uniShow that every element of tl is a linear combiversal enveloping al{ebra. powers of elements of 8. nation of
2 (Witt). Let 38 be the free Lie algebra with r (free) generatoisot, t2,' ' ' , xr over a field of characteristic 0, 8 the universal enveloping algejbraof 88. Let (Bg)o= S8 fl 8n, 3n the space of homogeneouselements of dfgree n in 3Show that dim (8ll)n
=!
r p(d,)r,,ra
n dln
where p is the Mtibius function. 3. Let ! be finite-dimensionalof characteristic 0, Il the nrl radical of 8. Show that the collection of linear transformationsexp (adz), z e Tl, is a group under multiplication. 4. Show that if Z is a nilpotent linear transformation in a finite dimensional vector space over a field of characteristic 0, then exp Z [s unimodul ar (det exp Z : 1). Show that if Z is skew relative to a non'de$enerate sym' metric or skew bilinear form, then expZ is orthogonal relativelto this form. 5. If ?I is an algebra there exists a unique automorphism r pf period two in !I @ lX such that (o I b)t : b I o. Show that if 1l is the universal en' veloping algebra of a Lie algebra, then lld' is contained in the bubalgebra of r-fixed elements of tl @ 11. 6. Let ll be the universal enveloping algebra of a Lie algebra and let 11* be the conjugate space of tl. lf e,* € l1*, then there exists a nrnique linear Define function s@* on 11@1l such that (sI*tf>ui$tut):Z,t@d*hti).
V. UNIVERSAL ENVELOPING ALGEBRAS
195
U' eV € U* bV (,p*Xa) : (e @g)(ud), d the diagonal mapping of 1l into U E This makes ll* an algebra. Show that this algebra is commutative and associative. 7. prove the following analogue of Friedrichs' theorem in the characteristic if and only if c p + O c a s e :a n e l e m e n t o o f S s a t i s f i e sa d - - a 8 1 + 1 @ c generated by the rt. is in the restricted Lie algebra B. Let 8 be a finite-dimensional Lie algebra, 8r and 8s ideals in I which are contragredient modules for I relative to the adjoint representation. Let (ut,...,%n), (ut,...,?ro) h dual basesfor 8r and 8z as in the definition of a is in the Casimir element of a representation ($ 3.7). Show that 7 = lulut center of the universal enveloping algebra tl of 8. g. Let the notations be as in 8. Let R be a finite-dimensional representa' a € tl' Show that the element tion for I (hence for tl) and let f(a) : tr 7t'R t N1,..,lt:I
-fQur*tr'"
utr)ulrutz" ' u'tr
More generafly, sbow that if e
is in the center of lI. L,2, -.., f, thgn n
2
tt...trr-L
f(uqeary "'
utre)utrutt "'
'is
a permutation of
ttr6'
is in the center of Il. 10. Determine ttre cerrter of ttle u,niversal enveloping algebra of the threedirnensicrat split sirnple algebra over a field of characteri:stirc aero' lret o be a field, altil= 1frfh,tz,.. -,,t L be the_algebra af polynomials ll. in indeterminates te, Q 1ta)} : il 1 tt, tz, "', tr') : c:[fl ttre ctoeure of' @ltil regarded as a g.raded algebra in the ustlal way (an e*cr;remfiis' hsmogeneous of degree k if. it is a homogeneous polynomial of dcgree & in ttrc usual sense). its quo' O < tt > is called the algebra of forrnat puwer sefies in the tc md has a o'1 tt ) the tr.. putir in series tient field P is the f,etd of fwmol -", atr homogene' valuation as defined in $5. Thus if a: $x * ar+t * ar+z * ous of degree i, a* + 0'then lal : 2-n. This valuafirm has'a t'miqUeextension to a valuation in Psatisfyinglobl: ioll'D'|. LetPnbethe a'lgebraof nXn -maxlaUl. Shsqrthat this defines matrices with entriesinP. Define l(ou)l -.., matrices with entries are Dr a valuation in P" and show that if Dr, Dz, of the algebra homomorphism continuous unique a exists there in o, then g:@{rr, "',frr!
of $5 into Pomappingilt-+ttDui:I,Z,"',T' lZ. Let I be a finite dimensional Lie algebra over a field @ of characterDr be derivations in istic 0 and let P be defined as in 11' Let Dr' Dz' " " grover by D.1:,D2,";,D, again. to P [! over 1l and denote their extensions ' automorphism defined $rGIl is i1' exp trD, exp exp trDr tzDz" that G Show in 8r and that G : exp D where D is in the subalgebra gencrated by the Dl in the Lie algebra of derivations of 8r.
196
LIE ALGEBRAS
l3- Let ti be a finite-dimensional restricted Lie algebra in iwhich every element is nilpotent; apk: 0 for some b > 0. show that the #algebra of g has the f.orm 1[1* S where S is the radical. f4. A polynomial of the form f,pm+a1f,nm-r+... +amlid called a ppolynomia,l and it is called regular if. an * 0. Let g be a restficted Lie al_ gebra (possibty infinite-dimensional) with the property that foi every o € g there exists a regular p-polynomial po(i) such that pq(a): e. Show that if c is an elernent of ! such that all the roots of po(t) are in dhe base field @, then c is in the center G of 8. Hence show that g is abellan if o is algebraically clobed. Show that any finite dimensional nonabelian hestricted Lie algebra over an algebraicalry crosed field contains an ereme$t a # 0 such thatap:0. l 15- Let 8 be restricted with the property that ap : ae. a fixed and +0. Prove that I is abelian. 16. use 14 and 15 to prove that if o,oz: a in a restricted Lie algebra, then lJ is abelian. conjecture: lf apn@)- a, n(a) > 0, then g is abelian17. Prove that if I is restricted of characteristic three andior:0 for all a, then any finitely generated subalgebra of g is finite-dimensiional. Conjecture (probably false but probably true under addition]l hypotheses): If u is finitely generated and every element of g (restricted of characteristic p) is algebraic in the sense that there exists a non-zero p-pdlynomial p6(l) such that p,^(a)- 0, then g is finite-dimensional. 18. Call a derivation D of. a restricted Lie algebra rcstrli,cted,if. aeD = @D)(ada)n't- Note that every inner derivation is restricted,r Show that a derivation is restricted if and only if it can be extended to I derivation of the. u-algebra. Show that if I has center 0, then every defivation is restricted. 19- Let I be an abelian finite-dimensional restricted' Lie hlgebra over a perfect field. Show that I : !o O,[Jr where 8o is the space o[ nilpotent elements (an* - 0) and [!1: flI]ar. 20. ! as in 19, base field infinite and perfect. Assume go 4 0. show that ll is cyclic in the sense that thereexists an element b such ttiat g : ZOwi. 2l' Let !l be the group algebra over a field of charact.j,.istic p of the cyclic group of order p. Then !I has the basis (1, &, az, .. ., nn-ry with cp : I' Show that the derivationalgebraD of r{ has a basisD1, i :0, l, . .., p -1, where 0D1:
st+r.
Verify that
(i - j)Dt+t lDDi: Dt:D,,, D!:0,
i>0
Prove that S is simple. S is called the Witt algebra. 22. Generalize 2l by consideringthederivation algebra of ithe group algebra ?I of the direct product of r cyelic groups of order p,i0 of characteristic p. These derivation algebras are simple too.
V. UNIVERSAL ENVELOPING ALGEBRAS
L97
23. Prove the following identity f.or Lie algebrasof characteristic p * 0: p-2
b-I (ad a)o-z-t(ad c)(adoX t=0 p-2
: c'rZ_r(ado,)n-z-;(adDXad o)d . ZL. Let I be a nilpotent Lie algebra of linear transformations in a finite' : 0. Show dimensional vector space over a fietd of characteristic p I 0 such that 8p that if /, B €8 then (/ + Do : le + Be. Use this to prove that if the base field is algebraically closed, then the weight functions are linear' Xt. Let to be the Lie algebra of n X n triangular matrices of fface 0 over a field of characteristic p * 0, p X n. Prove Eq complete. (Hintz Show that every derivation is restricted if fo is considered as a restricted Lie algebra and study the effect of a derivation on a diagonal matrix with distinct diagonal entries. ( O may be assumed infinite.))
CHAPTER VI
The Theorem of Ado-Iwasawa In this chapter we shall prove that every finite-dimensional Lie algebra has a faithful finite-dimensional representation. We shall treat the two cases: characteristic 0 and characteristic p separately. The result in the first case is known as Ado's theorem. For this we shall give a proof which is essentially a simplification of one due to Harish-Chandra. For the characteristic p case the result is due to lwasawa. The proof we shall give is simpler than his and leads to several other results on representations in the characteristic p case. 7.
Preliminary
reeults
If R is a homomorphism of a Lie algebra I into ?I1, where ?I is a finite-dimensional algebra (associative with 1), then wq know that R has a unique extension to a homomorphism R of ll into 1I. If X, is the kernel, lllt,: Un g U so UIX is finite-dimensional. In general, if It is a subspace of a vector space 9Jt,then the dimension of ltt/t will be called the co-dimensi,onof Tt in Tft. Thus R determines an ideal f in 1l of finite co-dimension. The homomorphism R is an isomorphism of I if and only if I n I * 0. Conversely, let I be an ideal in U such that f n I : 0 and I has finite codimension in ll. Then the restriction to I of the canonical homoIiI is an isomorphism of lJ into the finitemorphism of lt into [: Since any finite-dimensional algebra has dimensional algebra [". a faithful finite-dimensional representation it is clear that a Lie algebra I will have a faithful finite-dimensional representation if and only if the universal enveloping algebra U of IJ contains an ideal t of finite co-dimension satisfying * n I :0. We recall that an element a of. an algebra lI is called algebraic if there exists a non zero polynomial g()') such that s(a) - Q. This is equivalent to the assumption that the subalgebra lI generated by a is finite-dimensional. Consequently, every element of a finite' [1es]
LIE ALGEBRAS
2OO
dimensional algebra is algebraic. If X, is an ideal in l?I we shall say that a e W is algebraic modulo X. if there exists ]a non-zero polynomial tp(),)such that rp(a)e X". This is equivalentl to saying that the coset d. : a * I is algebraic in fl - U/I. It follo*vs that if I is of finite co-dimension, then every element of lI is algebfaic modulo f. We now state the following criterion for the univer$al envelop' ing algebra 1l of a Lie algebra. Lpuur 1. Izt 8, be a fini,te'dimensional Lie algebra, (Iru 0az,''', ilo) a basi.sfor 8,17 the uniuersal enaelopi,ng algebra of 8, X, nn ideal in U. Then X, is of finite co'dimension in ll if and only if \eaer! ui is algebraic modulo X,. Proof: The necessity of the condition has been establi{hed above. Now let ei(D be a non-zero polynomial such that cpt(ur)€ f and let Then every ul is congruent modulo f [o a linear ni:dega{D. combination of the elements 1, ua rlt,, " ', %Ti-t. The set df standard monomials uftut'...uh*, ki2 0, is a basis for 1l and the rlemark just made implies that these monomials are congruent modrllo I to a with 0 < tn 1 n;. linear combination of the monomials u!'ul'"'ul* Since this is a finite set, lUf; is finite-dimensional. Lnnur 2. Same assumptions as Lemma !, X, and E ideals in 17. If X, and P are of fi.nite co-dimension, then Ig is of finitv co'dimen' sion in lI. Proof: Let p;U), 0. If bb D, e E and brD"r : 0 brD"z then (br + br)D" : 0 for n: max(nr, n). (ab)D"r :0, a e @tand
: o (b,b)DN:t(T) @,D\(b,D'-') if N : lh * nz - l.
Hence E is a subalgebra and
l
sinpeE = . B ,
N. THE THEOREMOF ADO.IWASAWA E : l[. This proves the first statement. mediate consequence. 2.
The characteristic
2OI
The second is an im-
zero case
The key lemma for our proof of Ado's theorem is the following Lsurul 4. Let e be a finite-dimensional soluabte Lie algebra ouer a field of characteristic zero, n the nil radicat of 6,lr the uniuersal enueloping algebra of g. sufbose x, is an ideal of lr of fi,nite codimension such that eaery element of n is nilpotent modulo *,. Then there exists an ideal 3 in lI such that: (l) B g X, (Z) g is of finite co'dimension, (3) 3D g 8 -for eaer! deriaation D of E (extend,ed to U), (4) eaery element of n is nilpotent modulo g. Proof: Let D be the ideal in ll generated by r and !t. Then U = f; and Etx is the ideal in u/ff generated bv (gt + x)lx. since (tt + f)/r is an ideal in the Lie algebra (6 + x)lx and the elements of this ideal are nilpotent in the finite-dimensional enveloping associative algebra lUX. of (g + X)IX. it follows from Theorem 2.2 t}rrt (tt + I)/f is contained in the radical of ElX,. Hence E/f; is in the radical. This implies that there exists an integer r such that lf. le 6 and D is a derivationin€ then we know 3=U'gf. that lD e n (Theorem 3.7). It follows that llD g E. Hence tDD g D, which implies that 3D:v'D s g':8. Hence (3) holds forBand we have already noted that (1) holds. since p contains r, U is of finite co-dimension and 8 -- V' is of finite co-dimension in ll. This proves (2). rf z e fr, z" e f; for some positive integer z. Hence z" e V and z"' € 9' : 3. This provei (4). gr wh.eree is a soluableideat and gris Trponnu 1. Let g:g@ a subalgebra of the finite dimensionat Lie atgehra g of characteristic 0. suppose we haue a finite dimensional representation s of 6 such that zs is nilpotent for eaery z in the nil radicat rt of @. Then there exists a finite-dimensi.onal representation R of g such that: (l) if NR:0 for r in 6 thcn .rf :0, (Z) y" is nilpotent for eaery t of the form ! : z * u wlure e e gt and u € gr is such that ad,gu is nilpotent. Proof: s defines a homomorphism of the universal enveloping algebra u of 6 whose kernel r is of finite co-dimension. Also if e e rt then (e")' : 0 so z" e fr and f satisfies the hypothesis of
202
LIE ALGEBRAS
Lemma 4. Let 3 be the ideal in the conclusion of ithis lemma. We shall define the required representation R in 1l/8. We first define a representation R' of 8 - 6 @ I' acting in the space 11. If s e 6 we set sa' : snr the right multiplication in llr determined by s. If / e ,8, w€ define lR' to be the derivation in 11which extends the derivation 5 -+ [s/] of. g. The R"s defined on 6 and on 8r define a unique linear transformation R' on 8 which is a representation of 6 and 8, separately. To prove that R' is a representa' tion for [! it suffices to show that [s/]8' : [so', /t'], t e g, / e 8,. Now [s/] e 6 so [sI]"' : [s/]n. On the other hand, if D is a derivation in 11and a e'n, then the derivation condition gives [anDl: (aD)nHence we have [s/]t' : [s/]n : (s/8')" : [sn, l*'l: ls*' , l*'f , as required. Since 3 is an ideal in 11such that 3D tr 3 for any derivation D of 6, 3 is a subspaceof ll which is invariant relative to the representation R' of I acting in tl. Hence we have an induced represeiltation .R in the finite-dimensional factor space ll/3. Let r e 6 satisfy xe :0. This means that r8 maps 11into 3. Hence x e g, r € f ar[d so .trs= 0. Let e € It. Then, by Lemma 4, z is nilpotent modulo 3. Hence e" is nilpotent. Since Tt is an ideal in I (Theorem 3.7) it follows that z* is in the radical ft of the algebra of linear transformations a w h e re e € ft, u€' E t' andadga g ener at edby S " . N o w te t y :z * is nilpotent. Since e* € Dt, in order to prove that yB ls nilpotent, it suffices to prove that uR is nilpotent. By definitiod u,*' is the derivation in 11 which coincides with adgu on 6 and ladga is nilpotent. Since 6 generates tl it follows from Lemmh 3 that for every a e lJ there is an integer z(a) such that a(un')"'o' = 0. Hence Since 1113 for every a eall we have z(z) such that a(u*)"'i':0' is finite-dimensional this implies that u* is nilpotent. Thus R satisfies the conditions (1) and (2). We can now prove Ado's theorem. Euery finite-dimensional Lie algebra 8' of charac' teristic zero has a faithful f.nite'dimensional representation. Proof: We recall that the kernel of the adjoint representation A is the center G of 8. It will therefore suffice to prove the existence of a finite-dimensional representation R of 8 wtnich is faith' ful on the center O. For then we can form the directl sum representation of R and ,4. The kernel of this is the interse[tion of the kernels of R and of A. Hence this representation is faithful as g well as finite-dimensional. We proceed to construct R. Let
VI. THE THEOREM OF ADO.IWASAWA
203
b e t h e r a d i c a l ,I t t h e n i l r a d i c a l . L e t T t r : 6 c l t z c... cTtu:Tt where each lt; is an ideal in the next and dim lti+r : dim lti * 1. Such a sequence exists since rt is solvable and contains G. If dim 6 : c then in a c * l-dimensional vector space there exists a nilpotent linear transformation e such that e' + 0. Then G is isomorphic to the Lie algebra with basis (t,z',...,2'), so G has a faithful representation by nilpotent linear transformations in a finite. dimensional space. Since each Il; is nilpotent and }ti+r : IL @ Ou;+t where ou;*, is a subalgebra, the preceding theorem can be applied successively to obtain a finite-dimensional representation ? of !t by nilpotent linear transformations such that ? is faithful on 6. Next we obtain a sequence of subspaces, Er : It c 6g c ... c 63 : @ such that 6i+r is an ideal in 6; and dim 6i+r : dim 6; * 1. Then €;+r : 6; @ Ou;+r. Also It is the nil radical of every 6i (Theorem 3.7). Hence the theorem can be applied again beginning with r to obtain a representation S of 6 which is finite-dimensional, faithful on G' and represents the elements of 9t by nilpotent linear transformations. Finally we write 8 : 6 @ 8r, 8r a subalgebra (Levi's theorem). Then we can apply Theorem 1 again to obtain the required representation R of 8. Remark: The R constructed has the property that el is nilpotent for every e e tt. The same holds for the adjoint representation. Hence the direct sum has the property too. we therefore have a faithful finite-dimensional representation such that the transformations corresponding to the elements of !t are nilpotent-and hence are in the radical of the enveloping associative algebra. 3.
The charaeterietie p eqse
we recall that if o is of characteristic p then a polynomial of the form dolo* * arlon-r + ... * anl, a; e O is called a p-nrllynomial. lt p(]) is a polynomial of degree m, then we can write
(1)
Joo= p(t)q{t) * r;(}) ,
, fll,
where the rt(i) are of degree < rn. since the space of polynomials of d e g re e < m is z e- dime n s i o n ath l e re e x i s t a ;, i :0 , ...s/rrt not all 0 such that xair;(l): O. Then (1) implies that >;a;]rr' : p?')(2a,tr,{^)). we have therefore proved that every polynomial is a factor of a suitable non-zero F-Wlynomial. Now let I be a finite-dimensional Lie algebra over o, u the
204
LIE ALGEBRAS
universal enveloping algebra. Let a e g and let p(t) be a nonzero polynomial such that p(ad a) : O. Such a polynomial exists since the algebra_o! transformations in g is finite-dimerrsional. Let m(1) - ^o* + at^pn-t + ... * a*r be a Apolynomial divisfible by p(r). Then we have (2)
(ada)'* * ar(ade)r*-'+ ... * a^(ad,d :a
.J
In other words, for every 6 e Il we have Pn-r --ry- | . r_
(2')
[ - . . I b a ) . . . a*) a r [ . . . t b ; ] . . . " 1 * . . : r d ^ [ b a ] + 0 .
on the other hand, we know that [ ..wF:;1 of this gives
- [bao). Iteration
pk
(3) Hence(2') implies (4)
f "WTlal:1ba'*)
I b , a o ^ * a r 6 o * - r+ . . .
*a*aJ:0,
D e 8, which irnplies that the element (5)
z=a'^*a,rsn*-r +...
*o*a
is in the center G of 11. we have therefore proved thp following Lpuur 5. Let 8, be a finite-dimensional Lie atgebra buer a fietd of characteristic p + 0 and let o be the uniuersal enuelop{ngatgibra. Then for euery a G 8, there erists a porynomiat m,e) tutyrlhol*"61 is in the center E of \. The result just proved and Lemma 5.4 are the mdin steps in our proof of Iwasawa's theorem. Euery finite-dimensional Lie algefua of characteristic p + 0 has a faithful finite-dimensional represenfation. Protif: Let (ur, t4z,. . ., u) be a basis for I and let m;(l) be a p-polynomial such that m;(ur) : zi e 0, the center of thE universal enveloping algebra. If degmre) : p^i then zi : ulni 1l ut where u ;6llo^i- t Hence ,b y L e m m a 5 .4 , th e e l e m e n t sz?tzl z.-.zf" q1' ...uf;o, h;20,0 s in < p*o form a basis for tl. Let E be the lideal in U generated by the a;. As in the proof of rheorem b.ll,i the cosets o f t he elem ent s u l ' ...u tr" ,0 < f,i < p ^ i f.o rm a basi sforl l /8. H ence this algebra is finite-dimensional and the canonical mapping a -+ a : c * E, a e 8,, is an isomorphism of lJ into [r, [ : ll/U. It follows
vI.
THE THEOREMOF ADO-IWASAWA
205
that there exists a faithful finite-dimensional representation of 8. We shall show next that in the characteristic p + 0 case there is no connection between structure of Lie algebras and complete reducibility of modules. In the following theorem we shall need a result proved in Chapter II (Theorem 2.10) that an algebra ?t of linear transformations in a finite-dimensional vector space which has a non-zero radical cannot be completely reducible. We shall need also a result which is somewhat more difficult to prove, namely, that if z is an element of a finite-dimensional algebra and a does not belong to the radical of the algebra, then there exists an irreducible representation R of the algebra such that zR + 0 (See, for example, Jacobson [3], Theorem 3.1 and Definition 1.1.) THnonnu 2. Euery finite-dimensional Lie algebra ouer a fi.eld of characteristic b * 0 has aL:L finite-dimensional representation which is not completely reducible and a r: r fi.nite-dimensional completety reducible representation. Proof: Let the u; and zi: tni(ut be as in the proof of Iwasawa's theorem. Let Er be the ideal in ll generated by (z?,22,...2n). Then the argument shows that zr * E, ;a 0 in Ul8, but (2, + E,)' :0. Hence zr * Er is a non-zero center nilpotent element in the finitedimensional algebra 11/E'. The ideal generated by such an element is nilpotent. Hence 11/8, is not semi-simple. Hence any 1: l representation of this algebra is not completely reducible. Since (8 + E'yEr g€n€rates 11/E' this representation provides a representation for I which is not completely reducible. The argument used before shows that the canonical mapping of I into u/8, is an isomorphism. Hence the representation we have indicated is 1 : 1 for 8 and this proves our first assertion. Next let a be any nonzero element of 8 and take h : a in the basis (ur, trr, . . ., uo) for 8. Let a, +0 be in 0. Then mr(l) -a is not divisible by I and this is the minimum polynomial of. a * Ez in ll/Ez where Ez is the ideal generated by m(u) - d, ?/tz(az),. . .,r/tn(un). Thus a * Eg is not nilpotent and so it does not belong to the radical. It follows that there exists a finite-dimensional irreducible representation of tl/Ez such that a * E' is not represented by 0. This gives a finitedimensional irreducible representation R" of 8 such that aR" + 0. Let S, denote the kernel of R, (in 8). Then flreg,Fo:0. Since $ is finite-dimensional we can find a finite number (rr, az, ,Q^ Of
206
LIE ALGEBRAS
the a's in 8 such that Or'S"r: 0. We now form the module !!lt which is a direct sum of the rn irredacible modulesfi1 corresponding to the representations R r. Then evidently llt is completely reducible and the kernel of the associatedrepresentatiorfis 0S,r-0. Hence this gives a faithful finite-dimensional completely reducible representation for 8. Exercises
I
l. Show that any finite-dimensional Lie algebra of charactbristic p has indecomposablemodules of arbitrarily high finite-dimensionalitiies. Exercises 2-4 are designed to prove the following theorem: I Let lI be an algebra over an algebraically closed field of characteristic 0, 8 a finitedimensional simple subalgebra ot Vt which contains a non-zero algebraic element. Then the subalgebra of ' { generated by 8 is finite dimensional. We may as well assume that this subalgebra is ![ itself and:it suffices to show that 8 has a basis consisting of algebraic elements. ] 2. Show that I contains a non-zero nilpotent etement e. (Hintz\Use Exercise 3.11.) 3. Show that 8 contains a non-zero algebraic element I which is contained in some Cartan subalgebra 0 of 8. (Hintz Use Theorem 3.17, Fnd Exercise 3.13.) 4. lf e, have the usual significance relative to 6 show that there exists a root a * 0 such that ho, ao, a-a are algebraic. Then show that this holds for every root a and hence that I has a basis of algebraic elements. Use this to prove the theorem stated. 5. Extend the theorem stated above to 8 semi-simple under ithe stronger hypothesis that I contains a set of algebraic elements such that the ideal in 8 generated by this set is all of 8. 6. Extend the result in 5 to the case in which the base fieldt is any field of characteristic 0. 7. (Harish-Chandra). Let 8 be a finite-dimensional Lie algebrh over a field of characteristic 0 and let B be a faithful finite-dimensional rQpresentation of g by linear transformations of trace 0 in lllt. Let 80, i: t,?,, "' denote the representation in llll I xlt €) . .. I ![4, i times and let Ie denote the kernel in ll of .Br. Prove that flifre = 0. 8. Show that every finite-dimensional Lie algebra has a fafthful finitedimensional representation by linear transformations of trace 0.
CHAPTER VII
Classification of lrreducible Modules The principal objective of this chapter is the classification of the finite'dimensional irreducible modules for a finite-dimensional split semi-simple Lie algebra 8 over a field of characteristic 0. The main result-due to Cartan-gives a L: 1 correspondence between the modules of the type specified and the "dominant. integral" linear functions on a splitting cartan subalgebra e of 8. The existence of a finite-dimensional irreducible module corresponding to any dominant integral function was established by cartan by separate case investigations of the simple Lie algebras and so it depended on the classification of these algebras. A more elegant method for handling this question was devised by Chevalley and by Harish-Chandra (independently). This does not require case considerations. Moreover, it yields a uniform proof of the existence of a split semi-simple Lie algebra corresponding to every Cartan matrix or Dynkin diagram and another proof of the uniqueness (in the sense of isomorphism) of this algebra. Harish'Chandra's proof of these results is quite complicated.* The version we shall give is a comparatively simple one which is based on^an explicit definition of a certain infinite-dimensional Lie algebra 0 defined by an integral matrix (A;t) satisfying certain conditions which are satisfied by the Cartan matrices, and the study of certain cyclic modules, "e-extreme modules,, for 0. The principal tools which are needed in our discussion are the PoincareBirkhoff-Witt theorem and the representation theory for split threedimensional simple Lie algebras. 7.
Definition
of eertain Lie algebras
Le t (A;), ' i,j : 1, 2, . . . , 1 , b e a ma tri x o f i n te g e rs A ;i havi ng the following properties (rn'hich are known to hold for the cartan matrix of any finite-dimensional split semi-simple Lie algebra over * Chevalley'sproof has not beenpublished. 12071
208
LIE ALGEBRAS
a field of characteristic 0) (a) Ar; : 2, A;i :0 implies Ah : 0. A;i * 0 if. i + i, (p) det (A;.r) + 0. (r) If (ar, dzt ..., a,t) is a basis for an /'dimensional {ector space .bi over the rationals, then the group W generated byl the / linear I transformations S,, defined bY
(1)
aiSal- d'i-
Aiiat,
i
-
1, "',1,
i is a finite group. Let O be an arbitrary field of characteristic 0. We rshall define a Lie algebra 5 oner o which is determined by the rhatrix (A;)' we begin with the free Lie algebra 88 ($ 5.4) generfted by the ' ,/, a n d l e tS be thb (Li e) i deal free gener at or so i ,frl t;, i ,:I,2 ," generated the elements by in 88 lh;hil
(2)
le$il - driht leihil - Ai$i [f+hil * Ai,f, l
tet S - &8/n. Let 0 be the subspaceof 88 spanned by the ht and let a; be the linear function on 0 such that (3)
a{h)--Aio,
i-1,"',1.
The condition (9) implies that the / at form a basis fop the conjugate space b* of 0. any Since S8 is freely generated by the ei,frh'i, i -1,4,"',1, hi--+Hi of the generators ]into linear mapping €t+Et, ft-Ft, transformations of a vector space defines a (unique) representation of 88,. In other words, if f is any vector space with basis {ui}, then f can be mdde into an fi8-module by defining the module products u&t, utfr, utht in a completely arbitraty mafiner as elements of f. We now let I be the free (associative) al$ebra generated by I f.ree generators xr,l'z, . .., fit. Then X, has the basis Let 1: l(h) 'be a linear L , x r r " ' x t , y iti : I r 2 , " ' r l , f : L r Z , ' ' " function on 0. Then the foregoing remark implies that we can turn f into an S8,-module bY defining
VN.
CLASSIFICATION OF IRREDUCIBLE MODULES
2W
Lh:/,1, - a+)x\. (x\ " . rt,)h - (1 - 4\ Lf;: x;, fr\. . . xqf; : X\ . .. Xt,X6i
(4)
.. tci, i
le;- 0 , fr\. .. Xtr€i-- (nr. .. xir_r€;)X4 - 6;ro(A- 4rL - air-)(hr)ror. - . rir-t . In these equations and in those which will appear subsequently we abbreviate /.(h) by A, etc., but write in full A(h;), etc. Let S' be the kernel of the representation of 88 in f. we proceed to show that s' = s, the ideal defining 5. This will imply that f can be considered as defining a representation, hence a module, for 5. The linear transformation in fi corresponding to & has a diagonal matrix relative to the chosen basis. Hence any two of these transformations commute and so [h;hil e S'. We note next that the linear transformation corresponding tofi is the right multiplication ri* in f determined by re. The last equation and fourth equation in (a) imply that
(x "
{j,,il,:Z, ;)xn,nir_, ="_'' ; ::,rY,u,
This implies that [e;-fi - diihi is in the kernel S'.
We have
- *x; rrr;ht n*':(/t- a;)h =Y'!,;!-t!,ti, , xrr "' rr,tf;hl:
ttr... r4(fih - hfr)
- It\ . .. Xtrh;h - a,1r)I4 . .. ItrX; U - &ar - &+ - ai)rq .. frr,r; : (A - dq - (A - str - a+)rq ". rr,ri - - d.irq .. XtrX; - tra1... Xrr(difi) . Hence lfthl + aif; e, fti. This implies that if r is an element of f; such ttrert xh - M(h)r : Mtr, then (rf,)h : (M - a;)(tf), or (rx;)h: (M - a)(rr;). We now assert that (rr,,... n,€t)h : (l - 4tr dr, * a;)xq. - - frt,€;. This is clear for r: 0 if we adopt the conventio'n that the corresponding base element is 1. We now assume the result for r - L. Then
210
LIE ALGEBRAS
(x\. .. xqe)h : (Qc\ "' rrr-r€;)x4)h -
-
:(1 :
This
6;,t(tl
-
s\
Bnro(tl -
(A -
-
sit)(hr)xrt
dir_t*
at-
&rL
-
&rt -
s\
*
a; -
d;rrt)x\'"
x;r-r
that
., n,(e,th - he;) - 4tr * a;)xq.. . xtr4;
- (l - s\ d;lCt1 "
utr
a6)(xq
and it implies
!c\ . .. tci,leihT- r\. : (A - 4\
:
-
dir-)(hn)U
dir_t
proves our assertion
a,6r)((nr... x;r-ror)nr)
-
aq)xtr "'
xtrQ;
' Xtr0;, .
Hence le;.hl - a+€,t€ fii . We have therefore proved I that all the generators of S are contained in S/. Consequently $ g S/ and so fr can be regarded as a module for E - 88/4. We ca4 now prove Tuponpu 1. Let 88. denote the free Lie algebra gdnerated by 3l elements e;,fr, h;, i :1,2, "',,!, let R' be the ideal in 88' generated by the elements (2), and let E - 88/n. -Thenz (il The cananical iso' homornorphism of 8S onto ! maps 2laet + Zloft I 2toh morfrhically into i,, so we can identify the corresponding subspaces. (ii) The subspaceg; = ae; t afn * ah; of 8 is a subalgebra which is a s\tit three-d,imensional simple algebra. (iii) The bubalgebra iof i, generated, by the .f; is the free Lie algebra generh(d bt these elements and a similar statement hotds for the subalgebrd 8+ generated is an abelian subalgebraand we h4ue the uector by the er Q = \lfih space decomposition
(5)
0:0OE-g>0*.
(iv) If iI is the uniuersal enueloping algebra of i, then tl - Sfl*[where E is the subalgebragenerated by b,lI* the subalgebragenerated by i* and iI- the subalgebra generated by 8-Proof: For the moment let i: r * ft, t e 88. Consider the : 0 representation R of ! determined in f by -the linear fqnction I w h i c h i i m p l i e st h a t then fr.*:h*:0, o n 0 . I f .h e Q a n d h : 0 , (at , * . . . + a4 )(h ): 0 fo r a l l c h o i c e so f th e aor. S i nceidt, dz,' ' ' ,dr form a basis for the conjugate space this implies that h = 0.
lf,frl- -2f0, ldi,f;]: rt"i"-n-.Iis r:r. we have[eth;-l:26;, fi', henceodt* of t,+ oltt is a subalgebraof s whi4r is a homo-
VtI.
CLASSIFICATION OF IRREDUCIBLE MODULES
211
morphic image of the split three-dimensional simple Lie algebra. Because of the simplicity of the latter the image is either 0 or is isomorphic to the three-dimensional split simple algebra. Since h.i,+ 0 by our first result, we have an isomorphism. This proves (ii). We have Wn'\ - O for any h, h' e 2 Aht, and ldthl : a;(h)6;, If ofi: - ai@)f;. Since the linear functions 0, * a; are all different the usual weight argument implies that a relation of the form h'eZoht i m p l i e sf i : 0 , 2lir6o+Zlrhf,t+h'-0, Ti:o for alt Then h' :0 by our first result. Thus we see that f and fr,':0. x --+fr is an isomorphism on 2 Ae; + 2 O.ft * 2 Oht We make the identification of this space with its image and from now on we write ot, fr, hi etc. for d6,fr fir etc. We write Q : \lhh; and w€ have seen that this is an /-dimensional abelian subalgebra of 5. We have noted before that in the representation R of E acting in fr, ff : rcintthe right multiplication in f determined by rr. Let [denote the universal enveloping algebra of 5-, the subalgebra of E generated by the f;. Then we have a homomorphism of [- into the algebra of linear transformations in I mapping ,fr into rrn. If we combine this with the inverse of the isomorphism a - an of I (the regular representation) we obtain a homomorphism of [into I sending f, into xr. On the other hand, since fr is freely generated by the fri we have a homomorphism of t into Il mapping n into f;. It follows that both our homomorphisms are surjective isomorphisms. Since the free Lie algebra is obtained by taking the Lie algebra generated by the generators of a free associative algebra it is now clear that 0- is the free Lie algebra generated by the fr and fi- is the free associative algebra generated by the fo. Also the Poincar6-Birkhoff-Witt theorem permits the identification of fi- with the subalgebra of fi generated by 0-, hence by the f;. The basic property of free generators implies that we have an automorphism of $8 sendiag e; -.f;, .f; + €;, hi n - h;. This maps the generators (2) of S into ,ft and so it induces an automorphism in 8 - 88/n which maps f; into ei. Since the subalgebra generated by the fi is free it follows that the subalgebra E* generated by the e; is free and its universal enveloping algebra l1* is the free associative algebra generated by the e1. This algebra can be identified with the subalgebra of fr generated by the €.i. It remains to prove (5); for once this is done then the relation fr. : Sfr.*fr- follows from the Poincar6-Birkhoff-Witt theorem by choosing an ordered basis for E to consist of an ordered basis for
212
LIE ALGEBRAS
0 followed by one for 0* followed by one for 0-. To prove (5) we show first that E, = O + E* + CI- is a subalgebrb of 5. The argument is similar to one we have used before ($ 4.3).: We observe first that every element of E* is a linear combinatidn of the elem e n t s f e r r e q . " e r , l : [ . . . l e t r e 6 J . . . 0 n , ) a n d e v e r y e l e r n e n t o f0 - i s a linear combination of the elements [/c, . .. .fn,]. The Jacobi identity and induction on r implies that (\ 6- /)
Ilenr"' eqJh): (atr * a6z+ "' I a6)fetr "' un,l aq+ "' * at,)tftr"'fr,l llfnr"'.fn,lh): -(atrl
.
Hence tE,0l S !,. We have Ie;fi e 0 and by inductibn on r 2 2 we can show that [[er, .. . en,)fil e E*. It follows that, Erad.ft =ir. Iteration of this and the Jacobi identity implies that [0,CI-Js 0,. Similarly, [0,0*] tr 0,. These and [0,0] s 0, imply thatr5, is a subalgebra. Since e;,fr,hie i, it follows that E, -S, {hat is, 00* + O + E-. Equations (6) imply that 5 is a direct sum of root spaces relative to A and the non-zero roots are tlle functions + (at, + . . . I an,). It is clear that E* is the sum of ther root spaces corresponding to the roots et, * - . . * aq and E- is the sum of those corresponding to the roots - (ar, + ... I otr). It follows that 0 : 0* O 0 O E-. This completes the proof . 2.
On certain eyclic modules for i
A Lie module llt is cyclic with generator x if. Wl is the smallest submodule of !)t containing r. If U is the universal enveloping algebra of the Lie algebra, then vll - {rulu e U} is {he smallest submodule containing x. Hence llt is cyclic with r as Senerator if and only if !n - A1. The module f; for E - 88/A which we constructed in $ 1 is cyclic with 1 as generator since lfq .- . .fr,: r\ . .. xt, ?r7d these elements and 1 form a basis for ff. We shalt call a module !)t for i, e-extrerne if. it is cyclic and the generator r can be chosen so that xh - A(h)x and fi€r:O, i I t is .a p p a re n t fro m (4 ) th a t f is e-extremew i th 1 as L ,2, . . . , 1. generator of the required type. Thus we see that for every linear function A(h) on b there exists an e-extreme module fof which the generator r satisfies xh - 1(h)r, x€i: Q. We shall now consider the theory of e-extreme modules for 5. A similar theory can be developed for /-extreme modules which are defined tO be cyclic with generator y such that yh - A(h)t, lft:0, i, : I,2, .- .,1. We
vu.CLASSIFICATIoNoFIRREDUCIBLEMODULES213 shall stick to the e-extreme modules but shatl make use of the corresponding results for /-extreme modules when needed. Let llt be e-extreme with generator r satisfying rh = ,l(hy, 1c2i: 0. We know that the universal enveloping algebra ll of 8 can be iactored as ff. : St*fi- where S, [* and f- are the subalgebras - ril: rSfi.*[-. generated by O, E*'and E- respectively. Then Dt -: Sin." rh * 1(h)x, rE : Or and since r01:0, r11* Or. Hence !y1 Moreover, [- is generated by the elements fi. r8fr.*fr.- - ril-. of llt is a linear combination of the elements element Hence every
(7)
xf\... f,, ,
L if. r - 0' where we now adopt the convention that fi, "'fr,: (induction on (ao,+ "' + a.-)ftr'" fn, We have lfor "'fn,, hl (rfrr that "' f,,)h : r) and xh : 1r and these relations imply Hence a4)rf4"'.fr,' (ActdL xlftr-..f0,, h|+'@h)fq"'.fr, (8)
Thus Dt is a direct sum of weight spacesrelative to 0 and the weights are of the form
(e)
t,
l-(a;r*aq+"'*aq):A
-Zk;at, I
where the fti are non-negative integers. Also it is clear that the restriction to a weight space of the linear transformation corre' sponding to any h is a scalar multiplication by a field element. It is clear also that the weight space llt,r corr€sponding to I has r as basis and so is one-dimensional. The weight space l}tr corf€sponding to the weight fu[ : A - > k;ai is spanned by the vectors (7) such that d;r *' ' ' + 4tr l, kia;. Clearly there are only a ' ' ' * da,: finite number of sequences(lr, ir, ''', a) such that a;r + 2 k;ai where the ki, are fixed non-negative integers. Hence lltr is finite-dimensional. The weight. l can be characterized as the only weight of 0 in fi such that every weight has the form A - > h;ai, hi non-negative integers. We shall call A the higltcst weight of 0 in fi or of fi. If !t is isomorphic to fi, then !t is also e'extreme and has the It follows that two e'extreme 0-modules having highest weight l.
distinct highest weights cannot be isomorphic. Let gt be a submoduleof Dt : > e 'ftr where Str is the weight * llltrp where module correspondingto M. ll y e It, J e lltr, *
2t4
LIE ALGEBRAS
Hence {Mr, . . ., Mo} is a finite subset of the set of weifhts. y e tt fl (TJt', + ... * Iltr*) and this is a submodule iof the finitedimensional F-module lf=,I}tyr. Such an Q-module ib split and is a direct sum of weight modules whose weights are ini the set {M} (cf. $ 2.a). This means that It n (> Tft*1): X (It n Tlta ). Since y is any element of Tt we have also that Tt : X O Ttrr where ltr : Il n lll,u. If ltll * 0, M is a weight for 0 in It. At any rate, it is clear that It is a direct sum of weight modules and the weights of b in Tt are among the weights of 0 in llt. Now let [Jt' be the subspace of m spanned by the Dt'( with M + I and, assume that the submodule !t +Yft. In fhis case, we must have Ttt :0 since, otherwise, Iln -- IJlz which iis one-dimensional. Then r € Tt and Tt - )tI: IJt contrary to assurfiption. Thus we see that any proper submodule Tt : Z*+oftn g IJl/ c l]l. It folfows from this that the sum S of all the proper subfirodules of TJI is contained in TJt' c TJt and so this is a proper subrnodule. This proves the existence of a maximal (proper) submodule lB of !]t. Moreover, T is unique. We consider again the module I constructed in $ 1 *hich we shall now show is a "universal" e-extreme module with highest weight I in the sense that every module fi of this type is a homomorphic For this purpose we define 0 tobe the lihear mapping image of-I. (xnr...tccr:1i1 of t onto 9ft such that (x;,...nr)0:rfh--.fn, r: A ) . T hen ( r t ' ' " x t' f)0 : (x " ' tctrtc;)0- rfnr "' frrf,. (1 0 ) , -- ((xr, "' xu,)0)ft
(r;,
,xq)o
(11) : (xf1r...fi,)h : ((xo,
n,)0)h
( x nr ' ' ' x t r o6)0 - ((r,, "' xi,-t€i)n)0 - 6o,dA_ : ((r;, . .' x;r-tet)for)0- Brrr(l : ((x4. .. x;,-t€i)0)fn,- dr,u(A_
&ir
rir-r)0
gir
xl,r-1)0
t/,ir-r)0
If we use induction on r we can use this to establislr the formula (12)
(xir "' th,ei)0: (x\ " ' h,?Pi .
Since the elements e;,f;,,ft; gen€rate E, equations (10), (11) and (12)
VII. CLASSIFICATIONOF IRREDUCIBLEMODULES imply that d is a module homomorphism of f onto fi. The result just established shows that any e'extreme 0-module with highest weight I is isomorphic to a module of the form I/Tl, If lDt is irreducible we have It a submodule of the module I. of f . We have seen submodule I/$ where S is a maximal [!l it is clear that any Hence submodule. one such that there is only highest weight same the with modules e-extreme two irreducible module e-extreme irreducible of an The existence are isomorphic. gJl : I/!S module for, the also, A is clear weight with highest satisfies these requirements. We summarize our main results in the following Tuponpu 2. Let the notations be as i,n Theorem I and let A(h) be a linear function on b. Then there exists an irreducible e-extreme F,-moduti with highest weight A. The weights for such a module are oJ the form i - >, krai, h; a non-negatiue integer. The weight space corresyonding to A is one-dimensional' and all the weight - h e b acts as a scalar multi' iporn, are finite-d,imensional. Euery plication in eaery weight space. Two irred'ucible e-extreme E' modules are isomorphic if and onty if they haue the same highest weight. 3.
Finite'dimensional
irred'ucible modulea
We shall call a linear function A on .b integral if l(h;) is an int e g e r fo r ev er y i : 1, 2, ..' , /, a n d w e s h a l l c a l l a n i ntegral l i near function dominant if A(hi)>:Q for all i. In this section we es' and tablish a I : I correspondence between these linear functions modules irreducible the isomorphism classes of finite-dimensional for 0. In view of the correspondence between the isomorphism weights classesof irreducible e-extreme E-modules and the highest things: prove two to it suffices 2 em Theor in which we established with e-extreme is module irreducible (1) Every finite-dimensional e-extreme (2) irreducible Any integral, dominant weight highest is module with highest weight a dominant integral linear function finite-dimensional. We.prove first THponnu 3. Let i be as before an1 let Dl be a finite'ditnensional is irreducible i-module. Then Dt is e-extrenteand its highest weight S)' a dominant integral linear function on' 81 : Proof: 9Jt is u finit"-aimensional module for the subalgebra
2t6
LIE ALGEBRAS
Oet * Af; * Oh; which is a split three-dimensional sir{rple Lie algebra. Hence lll is completely reducible as 8;-module. The form of the irreducible modules for 8i ($ 3.8) shows that there exists a basis (x!" , rl" , . . ., ,1'l') for IJt such that rtt h; : llti*frt) where the m;* arte integers. Since [hrhi] - 0 the linear transformations associated with the different hi commute; hence, we can find a basisi(frt, fr2,- . . , JCr) such that x*h : fltikgk,i : l, ..., l, k : 1,..., N. lf. A* denotes the linear function on 0 such that l*(ht) : ,n;*, then 1*iis integral and (13)
x*h:
lrr* ,
h : L , 2 ,. . . , N .
Then 1* alre the weights of S in !]t. Let 8i be the fational vector space spanned by the linear functions c;. It is easy to see that a linear function d e 0f if and only if the values a(h) are rational for i : t,2, . . .,1 (cf..the proof of XIII in g 4.2). He4ce the weights lk e Qt and so we may pick out among these weights the highest weight I in the ordering of ,pi which is specifiediby saying that Zllra > 0 if the first li * 0 is positive. In this lase it is clear that I * a; is not a weight for any ai. Let x be d non-zero vector such that xh -- tlx. Then (xe;)h - (1 a ar1@e;) ,2ci xe;:O by the maximalftY of. A. Since llt is irreducible and rll iis a submodule of l}t it is clear that Dt : ril. Hence fi is e-extreme with I as its highest weight also in the sense of the last section. The results of $ 2 show that'every weight is of the form I - > k;a,;, kt a non-negative integer. On the other hand, the proof of the representation theorem, Theorem 4.1, (applied to S *lAe; + Af) shows that if M is a weight for 0 in l)1, then M - M @)ai is also a weight. Hence for each i, ,l - A(h;)a; is a weigh[ and so has the form A - 2 kia;. It follows that A(ht): ft; 2 0. Thus we see that I is a dominant integral function. This completds the proof. Next let l be any dominant integral linear function on 0 and let Dt be the irreducible module furnished by rTheorem 2 with maximum weight 1. The weights of A in !n have the form /1 - 2, k;a;, k; integral and non-negative. Hence these are integral and so they can be ordered by the ordering in Qfr. We shall prove The proof will bd based on several that ![t is finite-dimensional. lemmas, as follows.
Lsurtra,1. L e t 0 ; i : f { a d f ) - a i i + t , i + j - L , 2 , " : , 1 . Then lfl;ierl: 0 , k : I , 2 , . . .,1, and Tft?;i - 0 for any e'extrvme irreducible Emodule TJt.
VII. CLASSIFICATIONOF IRREDUCIBLEMODULES Proof:
lf. k + i, [fie*]:
0; hence ladf, ad e*l : 0.
2I7
Then
l0iie*) : f /61df )-aii+t ad,et,: f 1ad e*(adf ;)-ait+r : lf ptlkrd f i)- ^ti*' - - 6i*h t(adf )- a ii+r If. k +j this is 0. lf. k :, we obtain l0;grl: - Ai"fr(ad/t;-rti' 11 A;t :0, At, - 0 by the hypothesis (a) of. $ 1, so the result is 0 in this case. Otherwise, -A;r ) 0 and i(ad 7'1-tti - 0' Next let Then l|;ie*l:f1@d,fi)-'{ii+r ad,ei. We recall the commutah:i,. tion formula: akr: xak -(!)r'ao-t + (!)r"ok-z - "', wher" rt -lxal, x" : lx'a\, . . . . (eq. 2.6). If we make use of this and the table:
leii - hi, lIe1;Lfi:2f;, llleJif;lf;l : 0, we obtain
-f t@dfr)-^i;+r ad et fi adet(adf)-a,i+l - 1- Au + L)fi ad,h;(adf)-^it , if A;i : 0 - f 1@det)@df;)-a;i+t 1- Au + L)fi ad hi(adft1-^;t * 21-A;r + lX-
Adfi@d2f;)(adf)-tu-t
,
if -Au
> 0'
The first term in both of these formulas can be dropped since If we have fiadlt-fffii]:0. If A;t:0 frads;:ffp;l:0. --A;t Auft and > 0 we use thesecond formula fiadhi-fffii: t o o b ta i n - q- A ; i* l x -A r )fi @ d fi )-a ;i * q-A;t + 1X-4r r)f {adft)-/'i : 0 . Now let llt be an irre' This completes the proof of l0;p*l:0. ducible e-extreme E-module and, as before, let III' be the subspace spanned by the weight spaces corresponding to the weights other than the highest weight A. Consider the submodule \ll?;;lJ": !ltdriS[*[where E,'[* and fi- are as in Theorem 1. If we use ad h is a derivation and that lfthl - - aif; we see fact that the that 0;1h:h?u*[0uhl - hoit + (-at t, (Au - L)a)oit , which implies that IJld;.rS: Wt2u. Alry 0;i0tc:.erdrr and this im' plies thaf Tftflriil* - T!10u. Hence Dhzufr: Tftl;tll-. If. x is a canoni' ,/, then c a l g e n e rat or of f i s uc h th a t rh - tl x , x € ;:0 , i ,:t,2," every element of IJt is a linear combination of the elements of the and every element of. wl' is a linear combination form xf\...fr, of these elements for which r Z l. It follows from the definition of 0;1 that Thliis IJt/. Also it is clear that lft'fi- g tJt'. Hence
LIE ALGEBRAS
218
TftLrt g llt' and so Tftarii. is a proper submodute of ItJt. Since l)t is irreducible we must have yll?;t:0 and the proof ,is complete. Lpuur 2. Let Tft be an irreducible e-extreme module for i whose Then for highest weight Li.s a dominant integral linear functiun. any y e tlt there exist positiue integers /;, s; such that yeii :0 yfii, i:1,2,...,1. Proof: Let x be a generator of IJt such that rh)- Ax, xei : 0, i: L,2, '. -,1. Then every element of llt is a linedr combination of elements of the form xf;rfir. .. fr,. It suffices to prove the res u l t f o r e v e r yt : x f ; J t r . . . f n , . W e h a v e l h : M y , M - A - Z h i a r h an integer 2 0. Then (ye!i+')h - (M * (h; + I)aDlteli+'. Since M + (h * l)ar is not a weight, yeli*' - 0 which provds the assertion for e;. (This argument is valid for arbitrary e-extreqne 0-modules.) L et 1( h) : / n, i20. W e have W e s h o w n e x t th a t x.f{ i * ' = 0. xhi: lltflc, x€i : Q. Hence if we apply the theor! of e-extreme modules to the algebra g+: Oe; * O.f, * Oh; (in placb of 5) we see that the 8;-submodule fii generated by r is the spa0e spanned by x,xf i, t c f ! , . - ' . Su p p o s eth i s h a s a p ro p e r s u bmodul el Tt.r + 0. S i nce - (m, - 2ilxf!, the spaces oxf! are the weight spaces relakf!)h; tive to Ohr Hence lii is spanned by certain of the subspaces Axf! and k>- L since Sti c TJti. Moreover, if. xf! € Tte then xfl e Ttr for all q > h. It follows that It; : Zqzk>r0xf: where & iS the least posi' tive integer such that xf! € }ti. Evidently Tti S !Jtl. We now ob' serve that TIih I Tti since (xf{)h: (A - qa;)(xff) andl Tl;e*9 Tti since this is clear for h: i and it holds for h * i since eo : rer,f,iq: A. nlf - ft,8fi*fi- It follows now that Tt,E - Iti and It,fi* - ftr. Then "/f nrf- g !:tt'. Thus Tti[ is a proper E-submodule oi TJt+ 0. This contradicts the irreducibility of fi and so provesi that IJtr is 8r' irreducible. Now in g 3.8 we constructed a finite-dimensional irreducible 8.i-modulewith a generator N' such that xthi,: mixt, x'e.r:0 and x'f{i*t : 0. It follows from the isomorphisrnl result on irreducible e-extreme modules (Theorem 2) that this module is isomorphic to IJti. Hence we have rfli*' - 0. Now sqlpposewe have l f i ,: I thi s i ma n int eger m 20 s u c h th a t (x fn r" ' .f+ -)fi :0 . - 0. lf i,: j + i we use plies that (rfi, ..-f;,)f{ lhe relation -.An'\ff-^,,-' (m
[email protected]) fi.f{-"ot -=r t f!"-a;i Jf,r ' \+ L
+
+ (* \
-
- / r d^o"\frf i /
)tt
o@of,)-o" (mdddri)
vII.CLASSIFICATIONoFIRREDUCIBLEMODULES21-9 in the where d;r. is as in Lemma 1 and the congruence is used Lemma from It follows |ri. generated by ff. in ideal the sense of 1 that xfrr "'f,,-rf
Ji-aii
: xfir "'frr-rf{-^'if
+ ''' * (:t,oo)rrn
t
- 0' "' f,,-,f{fi@df,)-a;i
This proves the assertion on f; by induction on r' Lnuur 3. Let \It be as i,n Lernma 2. Then if M is a weight of ' " ,1 ' Q i n In , M - M ( h) ar , is a w e i g h t fo r i : L ,2 , Proof: Let y be a non-zero vector such that yh - M(h)y' If M ( h r ) 2 0 , t h e n w e c h o o s eq s o t h a t z : y e q i * $ , y e l * ' : 0 a n d ? n - 0. This can be done by Lemma 2. Then so that zf{ + 0, zf{*' the determination of the finite-dimensional irreducible modules for g; : Oe; * Af n * Ohi (S3.8) shows that Ef= obzf! is such a module and zh,t: mz. On the other hand, yh: U(h)y implies zh: (yeon)h qa)(h;): (M + Hence (M na)&)zf!. (M+qd)(h)z and.Qfhn + Qdi M(h) * 2q -- rn and
M+qa;.,
M+(q-l)at"',
M+(q-m)a; 'we
have M -M(h;)ar-are weights (corresponding to z, zf ;,. . . , zfr). m S 4 since M(h) - tn - 2q ZO rn 3 U M + (2q rn)ar and q M(h)at, is in the displayed sequence. If and q Z 0. Hence M M(ht) < 0 we reverse the roles of. ei, and fi and argue in a similar fashion. We can now prove Tnponuu 4. Let Tft be an irred,ucible e'ertrerue module for E such that the highest weight /l is a dominant integral linear function' Then Dt elsfi'nite-dimensional. Proof: Let s"o denote the linear mappinE f -' E E(h+)a.;in the functiorls 4t linear space 0i of rational linear combinations of the -that S", is so A;td; d,i Ain. We have a'iSal such that a;(h) (r) of $ 1. axiom our in specified one of the linear transformations is finite. genetated S,n the by group W that the This axiom states of Q weights of set the that implies 3 Lemma hand, On the other images J of set the consider We now under 17. invariant fi is in under W of. the maximal weight /1. This is a finite set, so it has a least element M in the texicographic ordering we have defined
220
LIE ALGEBRAS
in 0f (at the beginning of this section).* Let y be a non-zero vector such that yh - My. Since M - M(hi)ar e E, M,3 M - M(h;)a; so that we have M(ht)0-t,.fr-fr, h-+8,i. Since E ls semi-simple it can be identified with a completely reducible Lie algebra of linear transformations in a finite-dimensional vector space and the homomorphism of E onto E can be considered as a rppresentation. Since So is mapped into 0 in any finite-dimensionjal irreducible representation, the homomorphism of E maps ,Qoindo 0 and so we have an induced homomorphism of I onto E sucti that 0r -> dr, h;- E.t. Since 6 is /-aimensional the homombrphism maps ft-ft e isomorphically. If ,Rr is the kernel of the homurilrorphism of ,B onto E, since S, is an ideal, it is invariant under ade 0. It follows that if St ;E 0 then it contains a non-zero element of 0 or it contains one of the (one-dimensional) root spaces 8,,, a * 0. The first is ruled out since Sr fl 0 : 0. If S, 2 8r, S, 3 [8"8 -rf * 0. Since [8,8-"] s Q this is ruled out too. Hence Sr :0 ahd the homomorphism of 8 is an isomorphism. This proves (4). Theorems 3 and 4 establish a 1: 1 correspondence between the isomorphism classes of finite-dimensional irreducibls modules for the (infinite-dimensional) Lie algebra E and the colledtion of dominant integral linear functions on O. Also it is clbar from the definition of 8 that any finite-dimensional irreducible E-module is
VII.CLASSIFICATIONoFIRREDUCIBLEMODULES22S an 8-module. The converse is also clear since 8 is a homomorphic image of !. Hence we see that Theorems 3 and 4 establish a I : 1 correspondence between the isomorphism classes of finite-dimensional irreducible modules for the finite-dimensionalsplit semi-simple Lie algebra and the collection of dominant integral linear functions on a splitting Cartan subalgebra Q of 8. We recall that a set -X of linear transformations in a vector space llt over A is called absolutely irreducible if the corresponding set (the set of extensions) of linear transformations is irreducible in lJlr for any extension P oI the base field. This condition implies irreducibility since one may take P: 0. The term "absolutely irreducible" will be applied to modules and representationsin the obvious way. It is clear from the definition that a module llt for a Lie algebra 8 is absolutely irreducible if and only if Dlp is irreducible for 8r for any extension P of, 0. Now let ll be a split semi-simple Lie algebra over 0 as in Theorem 5 and let llt be a finite-dimensionalirreducible 8-module. We assert that llt is abso' lutely irreducible. Thus we know that 9jt is an e'extreme module w i t h g e n e r a t o rr s u c h t h a t x e t : 0 , i : 1 , 2 , " ' , 1 , a n d x h : l ( h ) x , h e b where A is a dominant integral linear function on sJ. we know also that the weight space 9J?r corresponding to I coincides with Ax. Consider lltr as module for 8p. Since this is finite' dimensional and ,8r, is semi-simple, lltp is completely reducible. Every irreducible submodule tt of gltp decomposes into weight modules relative to 0r. Hence every weight module for Q. in 9ltp can be decomposed into submodules contained in the irreducible components of a decomposition of 9lt" into irreducible 8p-modules' )r : Px. Since this is one dimen' In particular, this holds for (9Jt.r sional it follows that r is contained in an irreducible submodule It of !ltp. Since r generates lJtp we have Tl : lJlp and lltr is irreducible. This proves our assertion on the absolute irreducibility of !lt. 5.
Exietenee of Et and Eg
In $ 4.6 we established the existence of the split simple Lie algebras of the types corresponding to every Dynkin diagram except E, and Es. (Our method-an explicit construction for each typeadmittedly was somewhat sketchy for the exceptional types Gr, F, and Er.) An alternative procedure based on Theorem 5 is now
LIE ALGEBRAS
available to us. This requires the verification that the Cartan matrices (Au) obtained from the Dynkin diagrams sati$fy conditions (a), (F), (r) of $ 1. We shall now carry this out for the diagrams E and Er and thereby obtain the existence of these Lie algebras. We remark first that in any such verification (a) iis immediate once the matrix is written down and (P) is generdlly easy. To prove (7) one displays a finite set of vectors in Si *trictr span 0i and is invariant under the Weyl reflections S",. This will prove the finiteness of the group W generated by the S"n. Ea. The Cartan matrix is
2-L 0 -1 2-t (15)
(Ai) -
00 00 0-1 2- 10 0 0-1 2-L 0 0 0- L2 000 0-1 000 00 0-1 000
000 000 000 000 -1 0-1
2-r -1
2 002
0 0
One can see that det (A;1) : l, so (9) is clear. parent. We introduce the vectors
A[so
(a) is ap-
f,, : 3(a, * a, + ds * dt * rrr) * 2au + dt *',.de f,z : 3(az * a, + dt *
at) * Ztut + dz * de
f,s :3(ds
* a, + a) * 2*u+ dt * de j(ar.1 ar) * 2au * at * ae h : lr:3drlZao*ar*aa ]a:Zae*ar*aa f,t:-de*dz*de .le:-do-Zaz*ae
where (dr, dr, . . ., a,a)is a basis for an 8'dimensional vector space over the rationals. It is immediate that the (ft form a basis. We write Si for the Weyl reflection S', defined by (1). We can verify that Si, L < i 3 7, permutes li and li+' and leaves the other li fixed while , l ; S e :i r t * ( i u
*,lr *,le), .icse:t(,t.-2h-zAE)
1< i< 5
VII. CLASSIFICATIONOF IRREDUCIBLEMODULES
225
irS, : +(-2),6 + ^7 -U,B) irsr: *(-2^6-2f,2 *,la) Let 2 be the following set of vectors: t(ti*
li-h,
\*)*), l * * t i :t(ir * * h* l'' * l"), *.(2),i* ir' * lr * h+ i- + l" + lp + lq)
where the subscripts are all different and are in the set (1,2,''',8). It is easy to see that these vectors generate the space. Since the Si, i = ?, are permutation transformations of the I's it is clear that these Sr map J into itself. One checks also directly that Ss leaves .5 invariant. This implies that the group W genetated by the S; is finite. Et. Let a,1,dzt '' ', ao ba a simple system of roots of type Ea in the split Lie algebra Es. The matrix (Aii - (Z(ai, al)l(ai, a)), j, k : 2, . . ., 8 is the Cartan matrix Ez. This satisfies (a) and (0). Moreover, (r) holds since the group generated by S"r, '' ', S", is finite; hence the group generated by the restrictions of these mappings to the subspaceof 0f spanned by ar, ''', dt is finite. This proves the existence of ^8. A similar method applies to Ee. We remark also that it is easy to see that if e;,f;,h;, i:1, "',8, is a set of canonical generators for Es, then the subalgebra generated by €i,fi,ht, i :2, "',8 is Er. (Exercise 1, below.) 6.
Basic irredueible
modules
Let 8 be a split finite-dimensional semi-simple Lie algebra over a field of characteristic 0, and, as in the proof of Theorem 5, let llli be the finite-dimensional irreducible module correspondjng to dii, the dominant integral linear function li such that l;(h): I f r ; i s d e fi n e da s i n th e p ro o f o f Theorem 5 and i, i :1 ,2 ,.. . , 1. absolute r is as in (14), then the argument used in $4 to prove ,tLl
irreducibilitym I shows that the submodule of ffit,
O
...4ffir8...4l]tr g e n e r a t e db y r i s t h e i r r e d u c i b l em o d u l ec o r r e sponding to A 2 m;Ar. The problem of explicitly determining the irreducible modules of finite-dimensionality for 8 is therefore reduced to that of identifying the modules fi6, which we shall
226
LIE ALGEBRAS
call the basic irreducible modules for 8. We shall now carry this out for some of the simpte Lie algebras. Others will be indicated in exercises. At. As we have seen in $4.6, Ar is the Lie algebra of (/+1) x (/+1) matrices of trace 0 over O. lf (e;) is the usual m4trix basis for the matrix algebra A*, and the Cartan subalgebrh and simple system of roots are chosen as in S4.6, then a se{ of canonical generators corresponding to these is (16)
0z:
€ g 2 ," ' r 0 t :
€I+r,,
€t:
ezt ,
ft: hr:
-on , *ozst ''',f t : -ot,t+t fr: l l r : € z z- € g s t' ' ' , h t : € L L- V I + r , L +, r €t € z z,
The Cartan matrix (A;r) is given by (38) of Chapter IV and we O t h e r W i s e ;T h i s C a n b e Ai,*r, A;i:0 have A;t:2, A;*r,;: -!: checked by calculating lethil: A1$r, using (16). The simple root The Weyl reflfction $at is d.i is specified by: at(hi): An. i E-E-€(h)at We consider first a representation of At by the IJie algebra of linear transformations of trace 0 in an (/ + l)'dimdfrsional vector space !lt. We can choose the basis (ur, ur, '' ', th+r) so that %;hi: Lt.;,tJt+rht= -ui+rt tlih;:0, j + i, i + L. Then we havg uth - lt(h)ut where the weights l; are given by the table: t , ( hr ) - 1
(17)
It follows A'i.Sq:
i + 1,
'lr(h):0,
tlr(h+r): - 1
lr(h) : I
A*r(h) -
At*r(h) :0
from
-1 ,
tr,(h) 1+' -l,i ,
this and the table for the
Ai+r
A*rSr,t -- tli ,
ltSq
-+I
that - A5
if
j+
+1
Thus s"o interchanges A;, and, lr+r 'od leaves fixed lthe remaining weights li. The group of transformations of the weights A; generated by these mappings is the symmetric grodp on the I + I weights. Now let r be an integer, L S r s /, and form thp z-fold tensor product U[ I tn I ... I iln. Let 9Jt, denote the subspace of skew symmetric tensors or r-aectors in tn I ' ' ' I tX' By definition, this is the space spanned by all vectors of the form
(18)
ly r , y r,
=
}
t !ir8-tr, A ... I !;,,,
VII.
CLASSIFICATION OF IRREDUCIBLE MODULES
227
where the summation is taken over all permutations P - (iiz "' i,) of (1, 2, .. ., r) and the sign is * or - according as P is even or odd. It is easy to see that if (ur, ttr,' ' ', zr+r) is a basis for fi : fir then the (tl') vectors (19)
fuir,utr"',utrl ,
itlir
2 then If l 4. 6. Show that the minimum dimensionality for an irreduclble module $? for .Eosuch that NtEa * 0 is 27. Hence prove that .&'oand ,Bo, and .oo and G are not isomorphic. 7. Prove that the spin rep;esentationof Dt in the secondClifford algebra (of even elements) decomposesas a direct sum of two irredpcible modules. Show that these two together with the spaces of r-vectors I S r 3I - 2 are the basic irreducible modules for Dt, I > 4. 8. Show that a basis (gr, gz, ..., gt) can be chosenfor S sdch that [e$i= 6 i i e i ,I f c l t l : - d u f i . L e t h : 2 Z l g t Zp&r wherethe p'LeO. Let e= Prove that (e,,f,h) is a }Tpr where every 7i;* 0 and let /: Zft'ptft canonical basis for a split three-dimensionalsimple Lie algpbra 0. Prove that adg$ is a direct sum of I odd-dimensional irreducible representations for S. (The subalgebra S is called a "principal three-dimensionalsubalgebra" of 8. Such subalgebras play an important role in the cohomology theory of !. See Kostant I3l.) 9. Determine a S as in 8, for .4r. Find the characteristic roots of adgh and use this to ob.tain the dimensionalities of the irreducibld components of adg0. f0. Let Sl be a finite-dimensional module for 5J,let !+ be the (nilpotent) subalgebra of 8 generated by the et and let I be the subspdceof Dl of elements a such that el =0, I € !+. Show that dim8 is the riumber of irreducible submodules in a direct decomposition of lJl into irreducible submodules. (This number is independent of the particular ddcomposition by the Krull-Schmidt theorem. See also S 8.5.) ll. Let lll and Il be two finite-dimensional irreducible mddules for 8 and let !Ji* be the contragredient module of $Jt. Let .E and S, fespectively, be the representations in [Jl and !?, 8* the representation in FJI*. Show that
VtI.
CLASSIFICATION OF IRREDUCIBLE MODULES
237
the number of submodulesin a decomposition of Sl* @ It as direct sum of irreducible submodules is dim 8 where I is the subspace of the space G(m' n) (= fi* I m by $ l) of linear mappings of 9[t into [t defined bv 8: {ZlZ e G(Vt,n),FZ : Zls} . tZ. (Dynkin). If Ilfu and $lz are finite-dimensional irreducible 8'modules with highest weights h and Az and canonical generators sl and rr, respec' implies ozll=A tively, then Sls is said to be subord,innteto Wt if nru:0 for every a in the universal enveloping algebra 1l- of the subalgebra 8- of 8 generated by the /a. Prove that $?z is subordinate to $h if and onlj' if tlz * M where M is a dominant integral linear function on s*. (Hi'nt: Ir: Note that if tt is the finite-dimensional irreducible module with highest weight M and,g is a canonical generator then II|1 can be taken to be the submodule of Sts@It generatedby rz8A.) lg. Note that the definition of subordinate in 12 is equivalent to the fol' lowing: there exists a ll-homomorphism of IIh onto ![lz mapping cr onto tr2. Use this to prove that if a finite-dimensional irreducible module St with Stg + 0 has minimal dimensionality (for such modules), then III is basic'
CHAPTER VIII
Characters of the Irreducible
Modules
The main result of this chapter is the formula, due to Weyl, for the character of any finite-dimensional irreducible module for a split semi-simple Lie algebra 8 over a field of characteristic 0. If the base field is the field of complex numbers and R is a finitedimensional irreducible representation, then the character of R is the function on a Cartan subalgebra O defined by (1)
x(h)-trexPh&,
If 6 is a connected a + a z l z !+ " ' . where,as usual, expa:1* semi-simple compact Lie group, then (1) gives the character in the ordinary senseof an irreducible representation of O. Thus in this case b corresponds to a maximal torus and it is known that any element of 6 is conjugate to an element in this torus. Then (1) gives the values of the characters for elements of the torus. We know that hn acts diagonally, that is, a basis can be chosen so that the matrix of lzR is (2)
diae {A(h), M(h), ' ' '} ,
where /l(h), M(h),. . . are the weights of b in the representation. T he n th e m at r ix ex phR i s d i a g { e x p A (h ),e x p M(h )," '} ; hence (3)
x(h) : Znn exP M(h)
where r4p is the multiplicity of the weight M(h), that is, the dimensionality of the weight space !ItN. One obtains a purely algebraic form of the definition of the character x(h) by replacing the exponentials exp M(h) of (3) by formal exponentials which are elements of a certain group algebra. Weyl's formula gives an expression for the character Xt of the finite-dimensional irreducible module with highest weight A as a quotient of two quite simple elementary alternating expressionsin the exponentials. Weyl derived his formula originally by using integration on i23el
LIE ALGEBRAS
240
compact groups. An elementary purely algebraic rqethod for obtaining the result is due to Freudenthal and we shhll follow this in our discussion. A preliminary result of Freudenthal's gives a recursion formula for the multiplicities zr. Weyl's formula can be used to derive a formula for the dimensionality of the irreducible module with highest weight l. It can also be used to obtain the irreducible constituents of the tensor product of trl,'o irreducible modules. 7.
Some propertiee of the WeUl group
In this section we derive some properties of thd Weyl group which are needed for the proof of Weyl's formuli and for the determination of the automorphisms of semi-simple Lie algebras over an algebraically closed field of characteristic 0 (Chapter IX). As usual, 8 denotes a finite-dimensional split semi-sijmple Lie algeb r a ov er a f ield @ o f c h a ra c te ri s ti c z e to , e ;,f;,h;,,i --L,2,...,1, are canonical generators for 8 as in Chapters IV and VII, and O is the splitting Cartan subalgebra spanned by the &r. Let O* be the conjugate space of 6, OI ttre rational vector spdce spanned by the roots of S in I]. If. a is a non-zero root therl the Weyl reflection is the mapping
(4)
S,:t -+{ -z-(E'al o a) ld,
in 6i. Here (E,d is the positive definite scalar product in Oi defined as in $ 4.1. The mapping S' is characterizedby the properties that it is a linear transformation in OJ whigh mapsa into - a and, leaves fixed every vector orthogonal to a. The reflection S, is an orthogonal transformation relative to (t, z) aild S' permutes the weights of 6 in any finite-dimensionalmodule for 8. The S" generate the Weyl group 17 which is a finite group. lf. T is an orthogonal transformation such that aT is a root for some root a * 0, then a direct calculation shows that (5)
Sro
In particular, this holds for every T e W. We noti also that if a is a root then - a is a root and it is clear from (4)ithat S': S-o. the such that ai(h): Air, (Ai The roots d,it i:L,Z,...,1, Cartan matrix, form a simple system. The notion of a simple
MODULES OF THE IRREDUCIBLE VIII. CHARACTERS
247
system of roots is that given in $ 4.3 and dependson a lexicographic ordering of the rational vector space OJ. We recall that / roots -t- ()&ta;) where form a simple system if and only if every root a : the k; are non-negative integers. This criterion implies that if ft : {ar, dzt . . ., a,t} is a simple system and ? is a linear transform' ation in SJ which leaves the set of roots invariant, then rT: {arT, "',dt ?} is a s im p l e s y s te m o f ro o ts . T h e s e t P of posi ti ve roots in the lexicographic ordering which gives rise to the simple system z is the set of non-zero roots of the forrn Dk.a;, h >- 0. On the other hand, if the ordering is given, so that P is known, then z is the set of elements of. P which cannot be written in the form 9 + r, 3,r e P. Thus P and rc determine each other. Any simple system of roots defines a set of canonicalgeneratorset,fthi (cf. Sa .3 ) . b e a s i m p l es y s t e m o fr o o t sa n d L e r r l M .1c., L e t n : { d r , d 2 , . " , * , } let d be a positiue (negatiue)root. Then aS", > 0 ( 0. We have a - lhpi,
(6)
ogai
-*#)*' E u , o , *( u ,
.
lf. a + a;, then since no multiple except 0, t a of. a root is a root, a * k;ai and so k1 + 0 for some j + i. Then the expression for aS,. has a positive coefficient for some a1 and hence aS'o > 0. If a: &it then aS, . : - a; 10. This completesthe proof. Our results on I;7 will be given in two theorems. The first of these is Tnponpu 1. If rc : {dr, dz, . . ., at} is a simple system of roots, then the Weyl group W (relatiue to O) is generated by the reflections S';, dt€tr. If a is any non-zero root, then there erists ai€r and S e I7 su c h t hat a5S : a . Proof: Let W' be the subgroup of 17 generated by the $cr. We show first that any positive root a has the form aiS, ai € n, S e W . We recall that if a - }k;a; then X&r is the level of a, and we shall prove the result by induction on the level. The result is clear if the level is one since this condition is equivalent
242
LIE ALGEBRAS
to a e tt. If Xft; ) 1, a 6 r, and there exists ani,ar such that (a, a) > 0. Otherwise (d, o) { 0 for all i and sinlce a - Zhrai, h i> 0 t his giv e s (a ,a ) < 0 c o n tra ry to (a , a) > 0.r C hoose4; S o that (a, a;) ) 0. Then I : a,Sq ) 0 by Lemma 1 land we have B : Zi+ikp1 * @t - l}(a, a)l(a;, a;)f)a;. Since (a, a,r|> 0 it is clear that the level of F is lower than that of. a. Hencer the induction sh ows t hat F : d i S' , d i e r, S' e W' . T h en a: dtS l sdi as required. Since a: atS where S e W', S' : S-tS'rSte W' . Since S, : S-, this shows that every S, e W' and so Wl - W. It re' mains to show that if a is any non-zero root then at: a;5, dl € tc, S e 17. This has been shown for a > 0. Since &So: - a the result is clear also for a 10. Tsnonpu 2. Let T and, TE' be any two simple systernsof roots. Then there exists one and only one element S e W such that zrS: r'. Proof: Let P and P', respectively, denote the sets of positive roots determined by r and, r'. It is clear that P and P' contain the same number 4 of elements, since this is half the number of non-zero roots. It is clear also that P : P' if and lonly if 7r : rc' and if P + P', then z Z P' and n' * P. Let r be the number of elements in the intersection P n P'. If / : q, P 4 P' and S : 1 satisfies zrS: z', so the result holds in this case. We now employ Then m a y a s s u m er < q , o r P + P ' . i n d u c t i o no n q - r a n d w e there exists a; € n such that at d P' and hence - di -- atS't e P' . I f 9 e P n P ' , F S , n € P , b y L e m m a 1 . H e n c e0 S [ , e P n P ' l " r . Also er: (- ar)Sr; e P n ,rSoi. Hence P n P'Sr. contains at least r * | elements. The simple system corresponding td ,r Sor is z'Sr. so the induction hypothesis permits us to conclude lthat there exrrSar Then S: ?S"t satisfies zS: ists a T e W such that rTrc'. This proves the existence of S. To prove uniquFness it suffices to show that if S e W satisfies nS : r, or equivalently, PS: P, then S: 1. We give an elementary but somewha,t long proof of this here and in an exercise (Exercise 2, below) we shall indicate a short proof which is based on the existence theofem for irreducible modules. We now write S; : S". and by Tfreorem 1, S : a d r s t r s i n c e m : h a v e c a n n o t S n r S o r " ' S 0 . i,i : 1 , 2 , " ' , 1 . W e ! - air (, tt and if rn :2, then drrSlr : - a\ ?r'td sir[ce (- a;r)St, : a;,S;,S;, ) 0, iz : i, by Lemma 1. Thus S : S3,! 1' We now suppose rn > 2 and we may assumethat if T: SirSir"'Si", / 1ffi,
VIII.
CHARACTERS OF THE IRREDUCIBLE MODULES
J
' 'Su,'-r, so that S : S'St-' and,rT: zr, then T: !. Let S' : S;,Srr' PS'+P S i n c eS + S r * , S ' + 1 a n d s i n c e S ' i s a p i o d u c { o f m - l S i , n such di € an exists there Then hypothesis. induillon by the *iS/ : that implies 1 Lemma ) 0, aiS/SrSincl < 0. arS, that - ai,n Similarly, if g is any positive root such that FS' < 0, then : - airl: alS'i hence I : at. Thus FS,S- > 0 implies that FS' 'd;m alfld PS' > 0 for S' has the following two properties: o3st : every positive I + ai. Set S;o 1. Then aiSiq > 0 and aiSioSr.' "So--r
:
aiS' -
-
ct'int< 0'
)0 s u c h th a t a r' ScoS ,r" ' S ;* -, H en ceth e r e ex is t s a k , L < k < rrt-L , (a u 1 S6' ' ' S i " -)S r* :0' ( 0. a n d a ;S;s ' ' ' S r* -r> 0 S in c e a l Sro S nr ' ' ' S r * and then SroStr"'Sn*--r' ?: set we if Hence in*. arS;0"'S,*-, - ?Srr. Then, if T' :,Sr**t"'9t-:l for T-tSr? -*So* or S;T : H e nce h < m- r ana T' :1 for k = m- 1, S'- ?S;rT' SiTT'' I f a i s a p o s i t i v e r o o t* d i , t h e n 9 - a S i i s a p o s i t i v e TT':SlS'. : cr'SiS' : root * ott. Hence a,TT': aSrS' : PS/ > 0. Also aiTT a product is : TT' since z and (- ar')S' : d;*) 0. Hence nTT' : Then I' gives TTt hypothesis | Sr;t the induction of m case the by 1 S: : Hence : S StrSi. and Sr.S' 1 and so S/ Sr before. which we considered m:2 2.
Freudenthal's
formula
Let llt be a finite-dimensional irreducible module for 8 with highest weight A: 1(h) a dominant integral linear function on 6. We know that llt is a direct sum of weight spaces relative to 6 and that the weights are integral linear functions on 6 of the form 1 - Zkiat, k; ? non-negative integer. lf. M - M(h) is any integral linear function on 6 we define ttre muttiflicity np of M in !]t to be : O if. M is not a weight and otherwise, define rau dim ![1,r, where str is the weight space in IIt of b corresponding to the weight MWeshall We recall that for the highest weight A we have fl,t:\. expresses now derive a recursion formula due to Freudenthal which np ifr terms of the ftp, for M' > M in the lexicographic ordering ' oiOil determined by the simple system of roots 't : {d,, dz, " ' a''}' 8-, so -zero e 8a,, 0-a .b eo e choose and root of Let a be a non on form Killing the (x,y): xady, trad 1 where that (€,,0-n): 8. Then we know ($ 4'1) that le'e-'\ : h' where' in general for p e b*, ftp is the element of 6 such that (h, ho)- p(h). As in S4.2,
244
LIE ALGEBRAS
let 8(') be the subalgebra 6 + Ae" * Oe-,. Then we know that llt is completely redtrcible as 8'''-module (Theorem 4.1). we consider a particular decomposition of fi as a direct sum of irreducible 8'''-submodules and we shall refer to these ast,the irreduci,ble 8t't -constituents of Dt. Let rt be one of these. Thed we know that 9t has a basis (!0, yr, . . ., !^) such that ( M - i a )y , ,
lih: (7 )
!;o-o
j
i:0,L,..-,tfl
!;+r ,
We know also that (8)
m : 2(M, a)l@, a) where (p, o) p(h,) : o(hp), as in $ 4.1. Equations (?) imply that
(e) we note also that rt is a direct sum of weight spaces, that the weights are M, M - d., M - 2a, ..., M - md and the weight spaces \t*-r, : \Jlx-p, n Il are one-dimensional. Let M be a weight of s in Dt such that M + d i$ not a weight. T h e n t h e a - s t r i n g o f w e i g h t s i n l J t c o n t a i n i n gM i s M , M - d , . . . , M-ma, where 7n:2(M,a)l@,a). Let 0 < p<m. tThen M- pa is a weight and Wly-p, is a direct sum of the I[ - pa. weight spaces of those irreducible 8(''-constituents which have this as rveight. If 0 < p = (M, / (d, d), then these are the irreducible ") 8'"'-constituents having maximal weight M, M - &, . . ., M - pa. L e t f f ii, 0< i < (M ,a )l (a ,a ), d e n o te th e number rof i rreduci bl e 8''''constituents of highest weight M - ja. Then it is now clear that
(10)
lln-jo:
lllo *
m, +
*mi,
so that, (11)
l'fli :
lla-ia
-
nN-
0-ru
If 0 S j S p s (M, a)l(a, a), then the weight space corresponding to M - pa in an irreducible 8'')-constituent having highest weight M - ja is spanned by the vector !p_i in the notatioh of (Z). The dimensionality of this module is 2(M-ja,a)l(a,a)+l -(m-Zj)*L.
VIII.
CHARACTERS OF THE IRREDUCIBLEMODULES
Hence we may rePlace I bY
(12)
p-i
245
to obtain
andmbY
:
lp-Plre|
We can use this to computetrgqr-, ,e!neY,the trace of the to T!tr-r, of.e!,e!. We remark that it is clear that Wt"-rrLre, E Wtr-,r*r,ne| g lltr-po, so that yftr_p, is invariant under e!,e|. By (12), the contribution to trg4r- ,,e!re| of the m1 itreducible 8'(')-constituents with highest weight M-iais
*
(f -i+t)Lf -m+i)
tret
(a,a);
2
hence,
trss,-n,e!,g*=
(a, a)
o *^* W
(a' a)
: f,tn"-io - rtr-.i-,,') W i--o : i
rtr-io
Qi = m) (a, a) '
i=O
t'
Since mlT : (M, a)l(a,. a) we have the formula (13)
try1"-rne!,o2 - - i
nx-n(M-ia,a),
for o =P=(M,a)l(a,a)Next let (M, a)l(a, d) < p 3 m. If we apply the weyl reflection -we obtain M - pd - 2((M - 0d, a)l(a, a))a S, to M - fa We have 03m-P< since m:2(M,a)l(a'a). M-(m-0)a, (M, a)l(a, c) and we see that M - pa, is a weight for the irreduciUt" 8'"'-constituents having the following highest weights: M, M - dt " ', M - (m - ila and only these. The reasoning used to establish (13) now gives (14)
trg1"-o'elneT : -^f'n"-in(M
- ia, a)
i=o
f.or (M, a)l(a, d) < p < fit.
we recall that if. M is a weight, then
LIE ALGEBRAS
MS, is a weight and /tN: nilso (Theorem 4.1). Hence nx-ia: rtx-w-ita. On the other hand, (M - ja,a) + (M - (m - j)a,a): (2M - ma,, a,) : Q. Hence (15)
nr-n(M - ja,d)+
/tx-rm-jto(M-(m-
j)*,d) - 0.
This and (14) imply that (13) is valid also for p > (M, a)l(a, a). we recall that M was any weight of 0 in TJt such lhat M * a is not a weight. Hence M - pa can represent any w€ight of O in Dt. We now change our notation and write M f.or IWt- po. If we recall that np, : Q if M' is not a weight, then we can re-write (13) as (16)
trgl*e2,e3 :
-in**t,(M
+ ja, a)
j:0
for any weight M and any root a * 0. we consider next the casimir element defined by the Killing form. This is /' - ZiuludR where (u;), (ui) are duhl bases for 8 relative to (x, !). We know that laell : 0 for all o b g. Since R is absolutely irreducible it follows from Schur's lemma that r : TL, T e A (cf. Jacobson, I*ctures in Abstract Algebra II, p. 2Z6). We choosedual bases in the following way: (hr, . . ., h), (h,, . . . , h') are dual bases in 0 relative to(x,y). Then since (er,e_d) - -l ande, is orthogonal to 6 and to every €p, g * -d, the follor,ying are dual: (ht, ..., hr, €-o, a-g, . . .) (ht, ..., h', -
€o, -
€Fr...) .
Then
(17)
r : i h fh i "
d*0
If we take the trace of the induced mappings in llt1, M a weight, we obtain (18)
Tna -- Irtry1*hf;h'*
-
\trry*e!,u*
. l
Since hn is the scalar multiplication by M(h) ir, iUt, we have tryi*hf hiR : naM(h)M(ht), so the first term on the iight hand side of (18) is ny2;M(hi)M(h'). We proceed to show that ErM( h)M(hi) (M, M). T'hus we write lru : Zp,ihr. Then (M, M) = Z(hr h)p;pt and M(h):(hn, h):Zipi(hi, h;) and M(ht):(hr, ht):Epi(h1, hi): pr.
VIIT. CHARACTERSOF THE IRREDUCIBLEMODULES
247
It follows that Hence ZM(hr)M(h;1 : l(h;, h)prpi : (M, M). If we use this result and (16) in (18), 2;try17*hf h'^:no(M,M). we obtain (1S;
r?tx:
(M, M)nw + >
in**t,(M
+ iaL a) .
0 r=0
The terms nn(M,a) and np(M, - a) cancel in this formula so we The resulting formula is valid also if may replace XLo by )i='. M is not a weight. In this c?s€ 2r : 0. If , for a particulat d, no M * ja, j > L, is a weight then nx+io: Q and the particular If. M + ia, is a weight for some * ia,a):0. sum !pfiN+i,(M j >- L, then no M ka is a weight for k > | since the a-string containin g M+ ja does not contain M. Hence the set {M+ka I e>1} Then containing M + id. contains the complete a-string : 0, since + ia, a) \i:nx+ir(M f lp+ ir ( M + ia, a ) : Thus in all cases )i:fia+i,(M argument shows also that (20)
i
-/tw + i a ts r((M + i a )S " ,a) ' and (19) holds. The
+ ia, a):0
n**n,(M + ka, a) :0
holds for any ,","*r", U"ear funct ion Mon O. This implies that if D?:flx-r,(M - ka, - a) : nru(M, d) + }l:,'nx+x"(M + ka, a), so (19) obtain we with in Xt', we substitute Tltx : (M, M)np +:
nn(M, d) + ,7no*0,(M
+ ka, a) '
Setting 5 _ (1l2X>ln>oa)this gives the following formula: . TttN : (M, M * 26)nN+ 2i nn+r,(M * ka,a) ' L1r'o then nN:1 and /tx+*o:o inlll, of weight 6 the highest lf M:A gives (21) Hence & 1. a, > 0, for = -r : (/1, A + 26) (/l + d, A + d) (d' d) (2f1
and substitution in (21) gives Freudenthal's recursion forruula: (22) ((A + d, tl + d) - (M + 6, M + B))nx - ztna+*'(M
L=',,
* ha, a) .
LIE ALGEBRAS
We shall now show that this gives an effective [rocedure for calculating flr beginning with ltr : L. For this we rdquire a couple of lemmas. LpMu,c,2. Let B -- (U2)2*>od. Then D(/zi): 1, i * 1,2, . . .,1, if e;,ft h; is a set of canonical generators for 8. Also lf S + 1 is in W, then d - dS is a non-zero surn of distinct positiue ,roots. Proof: If a is a positive root we know (Lemma 1) that aSi = a$ri:a-a(h;)d,;)O unlessd,:di, in which case d;Si:-d;. Hence
: (+,t_,)- + d,: 6- d;. dsi: G \ z iio E")r, / \, ,*,n Also, (dSr,dr) : (d, aiSi) : (d, - a,r). Hence (d - ar, wr): (6, - a,r) and 2(d,ar) : (ai, a). Thus 6(hi) : 2(0,dil(db dt) : L,' i : t,2, . . ., l. Since any S e W maps roots into roots we evidently have dS: A - >p, where the summation is taken over the P - - aS > 0. If there are no such p, then aS > 0 for all a > 0. This implies S: l, by Theorem 2, contrary to hypothesis. Lpuu.r, 3. Let I be the highest weight of 6 in Tft. )Then
(23)
( M + d , M + d ) < ( ' l + d ,l + d )
for any weight M + A of 6 in Tft. Proof: We shall prove that there exists a weight'M' > M such that (Mt + d,Mt + d) > (M + 6,M + d). This will prove the result by an evident ascending chain argument. Assume firqt there exists an i such that M(h) < 0, hi in the set of canonical generators. Then we take M' :'MS;: M - M(h;)a; > M and wg have (M' + d, M' + d) - (M + d, M + 6) - - zM(h')(M + 0, a;) * M(h)'(d;, d;) - - ZM(h)l(d, d) , since M(h;) - 2(M, a;)l(ai, a;). Since (d, ai) > 0 this shows that (M'+B,M' +d)>(M+6,M+d). N e x t s u p p o s eI U I ( h ) > 0 , i 1,2,...,1. If no M*a; i s a r o o t a n d r + 0 s a t i S f i e sr h : M x , then rei :0 for all i and r is canonical generator o! an e-extreme 8-module. This must coincide with Dl, so M is the hJighestweight, contrary to hypothesis. Now let a; be one of thQ simple roots such that Mt - M * c; is a weight. Then M' > M'and
VIII.
CHARACTERS OF THE IRREDUCIBLE MODULES
( M ' + B , M ' + d ) - ( M + B , M + d ) - 2 ( M + d , d i )* ( a ; , d + )Since M(h;) > 0, (M, a;) 2 0. Also (6, a) > 0 and (ai, a;) ) 0, so again we have (M' + d, M' + d) > (M + D,M + d). It i. now clear that (22) can be used to determine the weights and their multiplicities. Thus we begin with n,r: L. Suppose that for a given fu[ - .4 - |,k;&, &i floil-n€gative integers with. at - y,k!;a;' least one hi > 0, we already know rtx' for every M' : A right the on term every Then < M. < M' + kr, hl integers, 0 k| (A d) : I 0, * if + Moreover, known. is hand side of Q2) (M + d,M + d), then, by Lemma3, M is not a weight and,n* -9. Otherwise, the coefficient of ny in (22) is not zero so we can solve for nx using the formula. 3.
Weyl's character formulq'
In order to obtain a general formulation of Weyl's formula valid for any field it is necessary to replace the exponentials which appear in this formula by "formal" exponentials. This notion can be- made precise by introducing the group algebra over the base field of the group of integral linear functions on s. we recall that an element M e b" is called integral if M(h;) is an integer wher e e ;,ft,h ; AtE c a n o n i c a l g e n erators for 8' f . or i :L ,Zr. . . r 1, linear functions is a group under addition' integral The set S'of the direct sum of the cyclic groups generis that 3 and it is cleal such that t';(h) : dir. We now introduce li of elements S ated by an algebra ?l over @ with basis {e(M)l M e 3} in 1 : 1 correspondence with the elements of S in which the multiplication table is
(24)
e(M)e(M')-e(M+M')
: Then ?I is the group algebra over O of ,S and e(0) 1 is the identity exponentials to formal the fl. are element of ?1. The elements of X of a the character define We now which we alluded before. exponential formal the to be Ift finite-dimensional module
(25)
x:\nre(M) u
,
where ny is the multiplicity of M e 3 as defined before: ltN:o the dimensionality of the if M is not a weight and rtr:dimfir weight space lnr if M is a weight. The summation in (25) is taken over all M e S. This is a finite sum since flx * 0 for only a finite
250
LIE ALGEBRAS
number ot M. Let rr : e(f,;), ]o(h) - drj. Since any integral lineat function can be written in one and only one way as M - Zmth lwhere the mi, are integers, the base element e(M): e(Zmif,;): e(mrlr)...e(mt)): e () , ' 1r " ' e( ) t l*t : i { ' -..x ft. W e h a v e c a l l e d M domi nafrt i t M(hr)20 f.ori:L,2,...,1. T h i s i s e q u i v a l e n tt o r h 2 0 . Thd set of linear combinations of the e(M), M dominant, is a subalgetjra of ?l which is the same as the set of linear combinations of the monomials tcYt...xTt, mi20. The set of these monomials obtalned from the sequencesof integers (mr,...,mr) with lntt- 0 are linearly independent. Hence the subalgebra we have indicated caq be identified with the commutative polynomial algebra (0fx1,x2,.,. ., xtf in the algebraically independent elements xr, . . ., xr. Every element of ?I h a s t he f or m ( x [ , ...x i t\-rf w h e re th e rr> 0 a nd f e A fi xr,...,.rr). It is well known that a[w . . - r rr] is an integral domair[. It follows that lI is a commutative integral domain. with each element s in the weyl group w we associate the linear mapping in ?I such that e(M)S - e(MS). Sincd (e(M)e(M'))S:
(e(M + M'))S - e((M + M)S) - e(MS + M'S) - e(MS)e(M'S)- e(IW)S^(M,)S,
it is clear that s is an automorphism in ?I. The set of s in ?I is a group of automorphisms isomorphic to w. An eleilrent a € lI is called symmetric if as: a, s e W, and alternating if i as - (det S)a, S e W where det s is the determinant of the orthogohal transformationSin6i. Thus detS: rl and if S:S, the Weyl reflection determined by the root a then det S, - * 1. In partidular, det S; : -1 for Sr : Srr. Since the S; generate W, a is symmetfic if and only if. aS.t:a, i:L,2,...,1, a n d a i s a l t e r n a t i n gi f a n d o n i l yi f a S r : - a f.or all i. The set of symmetric elements is a subalgebra; the set of alternating elements is a subspace. The product of two alternating elements is symmetric and the product of an alternlating element and a symmetric element is alternating. Since Wus : ltx for S e W, it follows that the character x:}nxe(M) ist a symmetric element. We note next that the element
(26)
Q - e(- d)Jl(e(a)- 1) : e(d)Il(l - e(- a)) ,
where a - (1/2)(2,rpt) the integral linear function defined in g 2, is an alternating element of ?I. Thus we have seen (prodf of Lemma 2)
VM.
CHARACTERS OF THE IRREDUCIBLE MODULES
that dSi: I - ar so (- d)S; - - d + d,t and e(- d)Sr e(- d)e(a)' di ?nd sends * positive roots Also we know that S; permutes the atr Hence ai into
/\ _ l)si : (Il,trtol -t))(e(- a) - r) fl(r@) d>0 o*u5
and
QSo: ( n Crt'l- r)t MSi : M - M(h;)u'i,, that is, we may suppose that M(ht) Z 0 for i : 1,2,' ", l. Suppose M(h;,) : 0 for some i, or equivalently MSr = !l[. Then e(M)o - - e(M)Sto : - e(MS;)o : - e(M)o and e(M)o - 0. We therefore conclude that every alternating element is a linear combination of the elements e(M)o w her e M ( h) > 0, i : I,2 , " ' ,1 . we now apply this argument to the element o defined in (26). If we multiply out the product in (26) we see that I is a linear
252
LIE ALGEBRAS
combination of elements e(M) where fu[ - d - p and p is a sum of a subset of positive roots. Thus M: (ll2) o>oEcdWhere eo: t 1 and any conjugate MS of. M under the Weyl group again has the form d - p' : (Il?)2r>oeLa where p' is a sum of positive roots and et,: + t. If we apply the projection operator .(Llw)o:,tothe expression indicated f.or Q we obtain an expression f.or Q {s linear combination of elements e(M)o where M is of the form Bl- p, p a sum of positive roots. We have seen also that we may assume that M These conditionb imply that satisfies M(h;) > 0, i : L,2, ...,1. M : 6. Thus, if fu[ - d - p, a - Zkrai where the k; are nonnegative integers, then 0 < M(h) : 6(h;) - p(h;): | - p(ht). Since p e 3, p(ft) is an integer. Hence we must have p(h) < 0. On the other hand, 0 ( (p, o1 : I,hr(di, p) - Qlz)>k;(a;, ai)p(h) < 0. Hence (p, p): 0 and p : 0. Thus we have shown that O :'qe(6)o, T e O. Now e(6)o : )"(det S)e(dS). By Lemma 2, dS + d if S + 1. Hence e(6)o- e(6)+ >:t e(M) where M:6 - 0, Q* 0 and a sum of positive roots. Also it is clear from (26) that Q - e(d) + > * e(M). Since Q - ae(d)o it follows that tt : l. We have therefofe proved the following Lonrnr.r4. Let Q - e(- B)[I,>o@(a)- 1). Then (28) S)e(dS). Q - e(d)o:"$(det
,
We shall next introduce the vector space O* SrlI and we shall define certain linear mappings of this space and of the algebra ?I. p;le On, a; € V. The elements of O* 8?I have the form 2a@at The algebra composition in ?l provides a linear mapping of ?IQpU into ?I such that a I b -, ab. This gives a lineaf mapping of (0*8?I)B2I:O*O(?I€l?I) into O*@?r so that (p8a)Ob-.p&ab. It follows that if we set (Lp; I a)b : I,pi& a;b, then this product of Xpr I ai and D is single-valued and coincides with the image of Xp;8 a;8 b in 6* I ?I. It is clear that the product,()p; I a)b Xpr I arb turns 6* I ?l into a right ?I-module. We yecall that we have the bilinear form (p, o): (hp,h,) on O* which defines the linear mapping of O* 816* into @ so that p I r - (p, c). If we combine this with the linear mapping of ?I @ ?I into ?I we obtain the linear i n to a 8 U :? I suchl that (p8a)8 mapping of ( 6*8U )8 (6 * 8 ? I) (r 8 e) -+ (p,c)ab. This defines a O-bilinear mapping of b*'8 ?I such that the value (p I a, r I D) : (p, r)ab. Since (p, o) is symmetric and ?I is commutative this is a symmetric bilinear form. Also if
VIII.
CHARACTERS OF THE IRREDUCIBLE MODULES
253
c e ? I , t h e n ( ( p 8 a ) c , r 8 D ) : ( p & a c , r 8 D ) : ( p , r ) a c ba n d ( p @ a , r8b)c :(p , r ) abc and s imi l a rl y (p & a ,(t8 D )c ) - (p 8 a,r& b)c w hi ch implies that (r, !), tc,y e 6* I ?1, is ?l-bilinear. Next we define a linear mapping, the gradient, of ?I into 6* I ?l: a -+ aG such that e(M)G : M I e(M) and a linear mapping, the La\lacian, of ?I into [: a --, ay' such that e(M)y' - (M, M)e(M)' We have
(e(M)e(M''o::rf*.^;f;#^M + M,): (M+ M,)o,e(M)e(M,) - M @e( MY (M ' ) + M' 8 e (M ' )e (M) - (e(M)G)e(M') + (e(Mt)G)e(M) . The linearity then imPlies that (29)
@b)G:(aG)b+(bG)a,
a,beW'
We have
(e(M)e(M'":,;l!n|rff ;'],,;:f.*r,Yi,"#,:,#;'::#,)e(M + (M', M')e(M)e(M') (e(MY)e(M') + 2(M & e(M), M' & e(M')) + (e(M')t)e(M) - (e(M)t)e(M') + Z(e(M)G, e(M')G) + (e(M')t)e(M) -
This implies that (30)
@b)t - (at)b * Z(aG,bG) + (bt)a .
We now return to the formulas which we developed for the multiplicity nx of the integral linear function M in a finite'dimensional irreducible module of highest weight A, T the element of O (rational number) determined by'the casimir operator. we consider again (1.9): Tltx : (M, M)nr + >
in**i'(M
+ ia, a) .
We multiply both sides by e(M) and sum on M. @
This gives
rx : xl + 4 AZno*ntM + ia, a)e(M), which we multiplY through bY
LIE ALGEBRAS
fl(e(a) - 1): d>0 Il@@) - 1)eil> O @? a) - 1): * 8' ,
a+0
bv (26). This gives (31) !rxQl+Q\Q'z
:
Fe,4
fIe@) - r)/tx+t,(M+ ja, a)(e(Mif o) - e(M)).
The coefficient of. e(M * aD in the right-hand side of 1(31) is
* ia,a) - fin"*o+ra(M+ kl + t)4, a)) fl(e(il - D(2""*r,(M / \i':o r=:-o
{+-,
: II @(il r- I)na(M, a) . F*a
Hence (31) can be written
(32)
in the form
*rxQ'+@4Q' - r)e(a)l,nn(M,a)e(M) : @@) Aff
l
- ( > a & e ( a ) f l ( e ( 9 ) - 1 ) ,> M 8 n * e ( M ) \ \dFo
{*7
x
/
: (=t Q'G,xG)- *Z((QG)Q,xG). Hence we have rxQ' - @4Q' : 2((QG)Q,xG) : 2(QG,xG)Q and canceling Q * 0 in the integral domain ?I, we obtain rxQ-QDQ-Z(QG,xG)
- (xQ)/ - (xt)Q - (Q4x, bv ($0). If we set / : XQ, as before, we obtain
(33)
rf : fl - (QOx.
SinceO: X"erdet S(e(dS)) and (dS,d'S): (d, d) we haveQy'-- (d, d)8. Since r - (A + d, I + d) - (d, d) (eq. below (21)) thesd substitutions convert (33) to the following fundamental equation norf (34)
fr':Qt+6,A+B)f.
.orr*"quently The element / - rO is an alternating element ".rd of the form this element is a linear combination of elementsj e(M)o. Moreover, w€ can limit the M which are rfeeded here by looking at the form of r and a. Thus we havel x -- I,n*e(M)
VIU.
CHARACTERS OF THE IRREDUCIBLE MODULES
where we now consider the summation as taken just over the -weights M of the representation. Also we have seen that Q linear com' xsew(dets)e(ds). If we multiply we obtain xQ as a binationof terms e(M+dS)whereM is a weight:f:Znu+asa(M+ds)' Maweight,SeW. Now e ( M * D S ) r ' -( M + 6 5 , M + 6 5 ) e ( M + d S ) : (MS-' + d, Ms*' + d)e(M+ DS) so that e(M * dS) is in the characteristic spaceof the characteristic root (MS-'+d, MS-'*d) of /. Since/ belongsto the characteristic root (l + d, I + d) f.or r' and characteristic spaces belonging to distinct roots are linearly independentit follows that f is a linear combinationof e(M * dS) such that (MS-' + d, MS-' + d) - (A + 6,1+ d) . By Lemma 3, (MS-' * d, MS-'+ d) : (l + B,I + d) for the weight Ms-'implies that MS-t : A. Hencewe see that f is a linear com' bination of the terms e(l + d)S. If we apply the projection operator (1.1w)o tolwe see that f - Ve(A* d)a: ?Es(dets)e(l + d')s. since d S < D i f S + L , ( l + d ' ) S < / l + d i f S + 1 . H e n c et h e c o e f f i c i e n t of e(1 + 6) in our expression for f is a. on the other hand, the coefficientof this term in ?(Qis na: I. Hencerl :1 and we have proved weyl,s Theorem. Let wt be the irreducible module for 8 with highestweight1. Then the characterTa: \nne(M) of 8 in Tlt is giaen by the formula det S(e(A+ d)S) ' r,r"E (det S)e(dS):) (35) where O - (U2)Xa>odt a a root. This theorem means that the expression on the right is divisible by 8 : Es (det s)e(ds) in ?t and the quotient is the character T,of the representation. It is easy to see that this result gives Weyl's original formula (1,) l, det S exP((l + d)S) (36)
X,1lh):
his result to obtain by a in the comprex case. *"r;t:-ployed limiting process the dimensionality of TJt 2n* - Zr(0). We proceed
256
LIE ALGEBRAS
to obtain the same result by a somewhat similar device. We introduce the algebra AQ) of formal power,series in an indeterminate / with coefficients in O. We recall thalt the mapping of power series into their constant terms is a homonhorphism ( of O(t) onto A. We can also define homomorphisms o{ tr into O b y em ploy ing e x p o n e n ti a l s :e x p e :1 * z + ( 2' z12!) + ... w hi ch i s defined for any z e O with zero constant term. We have the relation exp (e1* zz)- (exp zr)(exp zz); hence, if ), A, Q e g*, then exp (1, pX exp (p, p)t : €xp Q. * p, p)t. In particular, this holds for ], -- M, p: M', integral linear functions on 6, which implies, in view of. (2t), that we have a homomorphism (o of ?I iinto @(/) such that e(M)Cp : exp (M, p)t. Now consider 7-nC oC. Since Vo: ltrte(M) we have X,t|p:Zn*exp(M, p)t and, since the const{nt term of an exponential is 1, xnCK - 2n*: dim llt. We shall obtain the formula for dim!]t by applying CoCto (35),taking p: d - (1/2)>,a>&. Let o : Euer(det S)S as before and let M, M' be ilntegral linear functions. Then e(M)oCn, a
Hence (37) e(M)oCa- e(6)oCx- e(- d)C,.fI @@)(* - I)
(Lemma 4)
o>o
exp (- B,M)t fl (exR@, M)t - t)
I
o>0
: "!o(..oltr,
M)t- .*p+ (- a, irot),
since a - (1/2)Xa>ca. Applying this to (3b) we obtain
Now
(3e)
F,(e"o*
(a,M)t - exp |o(A * 6, a,)'lfl,ro(8, d)' . We write i -: D(hr'): 1' and 1 >since Then 0. Ei)(/rt tni 2 0, d 2ki4, h j : L ,::.,1 , d: X li. H e n c e w e re q u i re (\U n ; * 1)i ;, I,kp' )t : 2;,t(mi * l)ft;(tr, c1)' and Li,ik{},;, a.)t . Now a) - 2-()';'d) (l;, di)' - ?(l;' (at, (d,, a)
d,)
' \o" *'!
(a,,, a',)
- d;ttt)i
Hence (A + 6, a)' : 2(m; * L)w;ki and (d, a)' : I,w;ki and T
\(m; * r)w;k; dimTJt-II=*
(41)
t.rlt
)wtk;
where the product is taken over all the sequences (&,, kz, '' ', kt), ftr 2 0, such that Zhial is a root. We have seen that this set as well as the w; can be determined from the Dynkin diagram. G* H er e u) r : 3, I r z :1 a n d th e ro o ts ? ta a ,1d1z , d r * dz, dr* 2d2, ar * 3az, 2a, * 1ar. Then (41) gives (42)
dim IJt :
* l)(mz + L)(3mr+ mz + 4) hl(m, .(3mt * Zmz* S)(mt* mz + 2)(2m, * ms,+ 3)l '
LIE ALGEBRAS
if the highest weight of tn 1: lz we obtain, respectively, previously. Bu l22. H e r e u t ) r : 7 1 ) z :" ' are (43) d.ild;+r+...*ai, dr*dt+r+ di*...* These contribute formula:
(M)
is flhh * mzf,z. For I I - i' and 14 and 7, which we frad obtained : r,t)t-r:2,wt: 1rkrd: fuf where the h, ate non-negative integers and {a, B, "., p} is the set of positive roots. We have lP(0):1 and P(M) : 0 unlessM : Lm,a, m'inon-negativeinteger and al defined as before. Then e(M): ift . . . xft, xi : e(l). It is cdnvenient to replace the group algebra ?I by the field fr of power iseries of the form (x?' -. - xl\-'f where / is an infinite series with coefrcients in O in the elements rlr ... xlt, z; nor-r€gative integral. In fr we can consider the "generating function" )a.E SP(M)e(M ) which is defined since P(M) - 0 unless M : Zma r, //ti2 0. It i is clear that we have the identity
$ + e(a)* e(2a)+''') . "I*P(lM)e(M):,4 S i n c e( L - e ( a ) ) - r : 1 * e ( a ) * e ( 2 a ) + . " ,
we have the identity
(> P(M)e(Ml)ng-e(a))-1.
(47)
rue$
/ t)0
We re-write Weyl's formula (35) as
(Znne(u)))
(detS)e(dS):"e(detS)a(l+ d)S.
We replace M by - M in this and multiply the result through by e(d) to obtain (48)
2n*(-
M )) E (det S)e(d- dS)
By Lemma 4, Q : X (det S)e(dS)- e(B)fI (1.- e(-a)) . ,
VIII.
CHARACTERS OF THE IRREDUCIBLE MODULES
Hence
L (detS)e(d dS)- dII> 0 (1 e(a)).
sew
Hence if we multiply both sides of (a8)bV X'eg P(M)e(M), we obtain, by (47),that \nxe(-M):
(4e)
( L (dets)e(d- (A + D)s))(4",t
)e@))
:X-%., s)p(M)e(hI+6-(,l+d)s).
;:K
Comparison of coefficients of. e(- M) gives Kostant's formula for the multiplicity nN in the irreducible module with highest weight l, namely, (50)
nM:)
(detS)P((l + d)S- (M + d)) .
where 7.t;:'2nfi'e(M)is We now consider the formula for X,trX,t, the character of the irreducible module lltt with highest weight z/r. We have seen that X,rr1,,tr:\m,il,t, where mn is the multiplicity in !Il,8St, of the irreducible module with highest weight 1. The If we multiply summation is taken over all the dominant l. (det we get formula apply Weyl's and S)e(dS) through by X"err
/ '" y, (det?)e((1,+ d)")) l\ ,lt'€nSf r ' e ( Ml)/ ' ,I\ ,r e-w / :|AouZr*(det
S)e((l + d)S) .
The summation on the right-hand side can be taken for all I e 3 if we define ln,tr:O for non-dominant A. Applying the formula (50) f.or nfii we get (
/
\'l
+ d)s- (M+ a))e(M) X (aets)P((1, {t &>( /)f ) -g\ser
T)e((/, + d)")) ffi @et :
Hence
A
*^fi,(detS)e((l+ d)S).
LIE ALGEBRAS
r€s 8,r€w l€s If we put (12 + d)? * M:
sery
,,1+ d on the left we get
ST)P((A,* d)S+ (1, * d)" - (A + z|))e[,]+ a)
s , r €w ,>' es ,-E-Jdet
/
\
l€ci
_ \ sne€w S ( x+ 6 ) S : l + 6
/
/\ ( > (detS)ze(,,+d),s*r-6 > le(l + a'1 l€$ \sew / ,.te$ \seur
/
Hence
X (det ST)P((A'+ d)S* (A, + d)" - (1+ 2D))
s,r€w
s€w :
lrtA + >
(det S)zrz16rsi6 .
8€w 8*r
It is easy to see that if I is dominant, then (l + 0)S and hence (l + d)S - d is not dominant if S + 1. Hence if I is dominant then t/fiu+Et,s,-d:0 if S + 1 and so we obtain the formula
(51)
lnA: X (a"t S?)P((zt,+ d ) s + ( A , * d ) T - ( r + 2 d ) 8,TEW
for the 'multiplicity !lt, 8 !I,.
of the module with
highest weight i
in
Exerciees The notationsand conventionsare as in ChaptersVII andlVIII. 1. Let p eS f . Gi v ea d i re c tp ro o fth a t p l p Sfor everyS €W i f and o n l yi f p ( h r ) >0-, i : 1 , 2 , . - . , 1 . 2. (Seligman). Prove the uniqueness assertionin Theoreryr 2 by using the fact that there exists a finite-dimensional irreduciblemodulerlJlwith highest weight I satisfying: A(h,) are distinct and positive. Note that 1S : I if nS : S, so that 2(A, a)l(at, ut) : 2(15, aaS)/(a;S, aaS) : This leads to l(hd : tl(h) if alS: a1.
2(A, alS)l@r,S, aeS).
MODULES VIII. CHARACTERSOF THE IRREDUCTBLE
263
3. Call a weight .,1in 9Jt a frontier weight if for every root a * 0 either A + a or I - a is not a weight. Show that if lJl is finite-dimensional irreducible then any two frontier weights are conjugate under the Weyl group. 4. Let the base field be the field of real numbers. If a is a root let Po be the hyperplane in S defined by a(h) = 0. A clwmber is defined to be a connected component (maximal connected subset) of the complement of gryop, in F. Show that every chamber C is a convex set. A set of roots I is a d,ert.ning system for C if C is the set of elements ft,satisfying a(h) > 0 Defining systems which are minimal are called fundamental for all a € I. systems. Show that these are iust simpte systems of roots determined by the lexicographic orderings in S*. 5. Show that the group algebra :X (of the group S of integral functions on S) is a domain with unique factorization (into elements). 6. Prove that if Pe ?l is alternating,then P is divisible by Q = l5'(detS)e(DS). 7. Let 4 be the automorphismof !l such thate( l)n: s(-.1). Show that if X is the character of a finite-dimensionalmodule Sl then 1a is the character of the contragredient module lJl*. 8. Let 9Jl be a finite-dimensional irreducible module whose character satisfies Nn: t. Assume the base field algebraically closed. Show that the image !n under the representation I in 9Jt is a subalgebra of an orthogonal or a symplectic Lie algebra o{ linear transformations in !Jl. g. Let S be the split three-dimensionalsimple Lie algebra with canonical h. Show that the character of the (m + l)'dimensional irreducible basis e, .,1l, tu where n: e(l), t(h) = 1. Use module 9Jl-+r for fr is c- * nn-r + ... *r!Jl"nt. this to obtain the irreducible constituents of 9J1,,,11@ l0 (Dynkin). Let !J? and Il be finite-dimensional irreducible modules for g. prove that 9JlI It is irreducible if and only if for every I in any simple ideal of I either $ll : 0 or ltl : 0. Use Weyl's formula to show that the dimensionalities of the four ll. basic irreducible modules tor Ft are: 26, 52, 273, 1274. Use.Freudenthal's formula to obtain the character of the 26-dimensionalbasic module' 'Weyl's formula to prove dim lJt = 2I if lJt has highest weight lr 12. Use for Bt. lB (Steinberg). Let lJt be the finite-dimensional irreducible module with - MS is a sum of highest weight A. show that M is a weight of Dt if t positive roots for every S e W. 14 (Kostant). Prove the following recursion formula for the partition function P(M): (det S)P(M - (d - dS)). P(M) : -I 8€w
s+r
CHAPTER IX
Automorphisms In this chapter we shall study the groups of automorphisms of semi-simple Lie algebras over an algebraically closed field of charac' teristic 0. If e is an element of a Lie algebra B of characteristic 0 such that ad e is nilpotent, then we know that exp (ad e) is an automorphism. Products of automorphisms of this type will be called invariant automorphisms. These constitute a subgroup Go(8) of the group of automorphisms G(8) of 8. If r is any auto' morphism, then r-t(exp ad z)r: €XP ad.z" and ad e" is nilpotent. It is clear from this that Go is an invariant subgroup of G. The main problem we shall consider in this chapter is the de' termination of the index of Go in G for 8 finite dimensional simple over an algebraically closed field O of characteristic zero. We show first that if 8 is finite-dimensional over O (not necessarily simple), then Go acts transitively on the set of Cartan subalgebras, that is, if b, and 0z are Cartan subalgebras then there exists a o Q,Go such that Oi: Os. A conjugacy theorem of this type was first noted by Cartan in the case of 8 semi-simple over the field of complex numbers and it was applied by him to the study of the automor' phisms of these algebras. The extension and rigorous proof of the conjugacy theorem is due to Chevalley. This result reduces the study of the position of Go in G to the study of automorphisms which map a Cartan subalgebra into itself. If 8 is semi-simple, then the Weyl group plays an important role in our considerations. We shall require also some explicit calculations of invariant automorphisms due to Seligman. The final results we derive give the group of automorphisms for the Lie algebras At, Bt, Ct, Dt, I > 4, G, and tr'r. It is noteworthy that similar results can be obtained in the characteristic f case (see Jacobson [8] and Seligman [4]). As usual, for the sake of simplic ity we stick to the characteristic 0 case. In the next chapter we shall extend our final results for non-exceptional simple Lie algebras over algebraically closed fields to algebras of this type [265]
LIE ALGEBRAS
266 over arbitrary
base fields (of characteristic
I.
0).
Letnmas from algebraic geometry
Let sl be a finite-dimensional vector spacewith basis (ur,ttr,"',tt*) over an infinite field A. Any r has a unique repfesentation as and the Fi are the coordinates of r relative to the basis r:ZE;u; (u). Let f Qr .. ., f,*) be an element of the poltynomial ring AIl,,, .. ., A^l in the ln indeterminates )q with coefficients in O. Then .f (lr, . . ., f,^) and the basis (ze) define a mapping f of St into (In thi s O by t he r ule t h a t th e i m a g e f(x ) -f(E ):f(€ ,, " ' )€* ). chapter we shall often use the notation /(r) rather than xf ot xt .) We call f a Folynomial function on m. If (u'i, "', uk) is a second basis, then it is readily seen that the function / can be defined by another polynomial in O[i.r, .", ]^f with t'espect to (ul). In this sense the notion of a polynomial function is independent of the choice of the basis for St. The set of polynomial functions is an algebra 0[!m] relative to the usual composition$ of functions. We recall that if / and g are functions on lJl with vallues in 0, then (f + g)(x) : f(x) + g(x), (af)(x) - af(r) for a in {0, (fil@ : f(x)s(x). The mapping f (1,, . - -, f,^) -,f e Ottrtl is a homomorphism of for all o l J r , . . . , A - f i n t o @ [ f i J . S i n c e@ i s i n f i n i t e ,f ( { r , ' . ' , ' € ^ ) : 0 ., .. (1r, is an ,l-) the homonlorphism 0. Hence O implies in f 6c isomorphism. The isomorphism maps ,lr into the projection function zr such that ni(x) : tn. Hence it is clear that the zi generate the algebra O[St]. Let Tt be a second finite-dimensional space ovet O iwith the basis (ur, ur, '", an). A folynomial mapping P of Ut into Tt is a mapping where ri + b{E', "',€^), of the form x:ZTE*4ea ! :),irliar ., . . . .., is indepbndent of the Al)i, f,-7. This notion f,*) e Pi(lr, basis. The set of polynomial mappings of St intd Tt is a vector space under the usual addition and scalar multiplidation of mappings. The resultant of a polynomial mapping P of Ul to Tt and a polynomial mapping Q of Tt to S is a polynomial mapping PO of ![t into S. The notion of a polynomial functioll is the special case of a polynomial mapping in which the image space is the one-dimensional space @. Hence if. P is a polynoniial mapping of !ft into Tt and f is a polynomial function on Tt, thefr Pf is a polynomial frrnction on Sl. The mapping f - Pf is a fnapping of the algebra @[It] of polynomial functions on tt into @tt!ml. We now
IX. AUTOMORPHISMS
look at the form of this mapping. Thus we have P(x) - y where and x:28&6 !-Ztliui (1) tli : |i(Er, ' ' ', E^) , bi( l r, ..., f , - )e O l^b . . . , f ,^ 1 . Al s o w e h a v e f(y ) - f(n' , " ' , l n), f (pr, -.., pn)Q AIpr, ..., Fn]l,pi indeterminates. Hence P/ maps r into
(2)
fQ'G), br€), "', P"(E))
wh e re b i €) = f { 8, , €r , . ..,8 * ).
l f f, g e O [l t], th e n
- f QJ.il "'s P"GD (f + s)(f,(t),' ", P"(ED + s(b,(E), "', P"(E)) ' ' ', . .', : p"GD (f (p'(E), a(f af ,(f), P"(€)) ' '', p"GD. '", P"GD : f @'G),"', b"(0)s(fr(f), (fg)(f'G)s This shows that f -->P/ is a homomorphism op of o[Tt] into otml. Conversely, let a be an algebra homomorphism of @[Tt]into @[ill]. Consider the projection pi: X TJnt)n-rri. Suppose pi is the mapping a n d l e t P b e th e p o l y n o mi al mappi ng of 2E ;u ;-fi (E r , €r , ' ' ' , E ) Then Ppl where vi:fi(E). !m into Tl such that Z€ui-+Zttiai maps LErur into /r'(f) so that Ppi: p7. Since the pi generate O[It] it follows that o coincides with the homomorphism a' determined by P. Thus every homomorphism of @[yt] into OlUll (sending 1 into 1) is realized by a polynomial mapping of fi into Tt. It is easy to see that if P and Q are two such polynomial mappings, then the homomorphism dp: oq if. and only if P: Q. Hence we have a 1: 1 correspondence between the polynomial mappings of ![t into It and the homomorphisms of O[It] into O[!ft]. Of particular importance for us is the . set of algebra homo' morphisms of @[It] into the base field @. This can be obtained in a somewhat devious manner by identifying A with the algebra @tlnl of polynomial functions on fi : 0 and applying the foregoing result. However, it is more straightforward to look at this di' rectly. We note first that if y e It, then the mapping or: .f -, f (y), the specialization of .f tt !, is a homomorphism of OlUtl into O. Conversely, if o is any homomorphism of @[It] into 0, we let 7; : p7, as before. Then it is immediate that o: dy wh€tl y - }niui. It is clear also that if lr * lz in Tt, then or, I 6vz. Then we can define a Let f e OlVt) and let a : X a;u,i,€ fi|. linear function d"f on Sl by
LIE ALGEBRAS
zffi
(d"f)(YEu;):-tf (%\ Er. \o/t /xr:a,
( 3)
It is easy to see that d"f is independent of the basis used to define this mapping. The linear mapping d,.f is called the diferential ol f at a. We have the following properties d"(f + 9): d"f a d"9 d"(af) - a(d,f) d"(fg) - f (a)(d"g) + g(a)(d"f) .
(4)
If / is an indeterminate and,f is extended to 11116rin the obvious wny, then we have the following relation in the algebra Oltl, which is a consequence of Taylor's formula
(5)
f (a + tx) = f (a) + t(d"f)(r)
(mod/z) .
More generally, let P be a polynomial mapping x:}T|,ru;-t of St into It. Then we define a linear mapping doP, the }Tfif)ui diferenti.al of P at a by
(6)
(d,P)(x):,8 . $;W)^,_,,t0),,
Again, one can verify that this is independent of thetbases chosen in Ul and Il. Also one has the useful generalization of (5): (7 )
P(a * tx) = P(a) + t(d"P)(x)
(mod fz) .
A set of generators for the image space Vl(d"P) iis the set of vectors
(8)
(d,P)(ut): (K) . ^.=,*,i +
Hence d"P is surjective if and only if the Jacobian rgatrix (\ ev )'/
((%\ ),' \\ ard / t,*="*/
i:L,...,ffi,
j:!,2,...,n
has rank z. Lpuu.r 1. Assume A pe'rfect. Then if d"P is surjactiae for sonNe a, the homomorphism oe is an isomorphism of OWtl irtto A[sXl]. Proof: Our hypothesis is that one of the z-rowed fninors of the matrix (dbtldl) is not zero. If. or is not an isomorphi$m, then there exists a polynomial f(p,,. . ., p^) *0 such that f (pl€), pr(E),.- -, f*(E)) :
IX. AUTOMORPHISMS
0 for all fr in O. This implies that f (pr(,1),. . ., p"(l)): 0, that is, the polynomials fr(f,r, - . ., f,^), - . ., 2o(]r, . . ., f,^) are algebraically dependent. If this is the case, then we may assume that the polynomial f + 0 giving the dependenceis of least degree. The relation f (p,(l),. . ., y'"(t)) - 0 gives
(10)
O:K:
This contradicts the hypothesis on the Jacobian matrix unless (0f l0p)(0,(l), ...,2"(i)) :0. Since the degree of / is minimal for algebraic relations in the f '(l), ' ' ' , F^(]) we must have (0f i0p) - 0, j : 1, " ', n. This implies that f is a non'zero element of O-which is absurd-if the characteristic is 0. If the characteristic is p+ 0, we obtain that f is a polynomial in p!, pl, "', $f;. Since @ is perfect this implies that f : 9p, g a polynomial in the p's. Then g(pr(l), . . . , P^(A)): 0 which again contradicts the minimality of the degree of f. The main result we shall require for the conjugacy theorem for Cartan subalgebras is the following Tnnonpu 1. Let O be algebraically closed and let P be a foly' nornial mapping of silt into [t such that d"P i,s suriectiue for some a e fl|. If f i,s a non'zero polynomial function on !Il, then there erists a non-zero polynomi,al function g on Tl such that if y is any element of n safisfying g(y) + 0, then there erists an x in [n such that f(x) * 0 and P(x) - y. In geometric form this result has the following meaning: Given an "open" set in Ul defined to be the set of elements r such that f (r) + 0 for the non-zero polynomial f , then there exists an open set in !t defined by g(y) + A, g a non-zero polynomial which is com' pletely contained in the image under P of. the given open set in !m. (Suggestion: Draw a figure for this.) We shall see that Theorem 1 is an easy consequence of the following theorem on extensions of homomorphisms. Tnsonpu 2. Int O be an algebraically closed field and let f be an ertension fi.eld of O, \ a subalgebra of f and 2I' an extension algebra of \I of the form ?I' : Vlur, Ltzt. ' , u,7, h e T. Let .f be a non-zero element of A'. Then there eri.sts a non-zero element g in T such that if o is any homomorphism of 1l into O such that g' + 0,
27A
LIE ALGEBRAS
then o has an extension homomorfhisrn t of ZIt iilo
A such that
f'+0. Proof: Suppose first that r : l, so that ?[' : }LIul. CaseI: z is transcendental over the subfield 2 of f generated by ?1. We write Let g:f, an{ let a be a f :fo*fru + "'*f^tf , fr€Zl, f,+0. homomorphism of ?I into A such that g" + 0. Consider the polynomial f{ + f{ ^ + . .. + f:,^' in alll, .i an indeterminate. Since .f: * 0, this polynomial has at most m roots in @ so we can choose c in O so that ),6fic' + 0. Let r be the homomorphism of ?I' : ?I[z] into @ such that 2 anu'-' l, aici. This is an extension of o ' and f + 0 as required. Case II: u is algebraic over E. The canonical homomorphism of ?I[i], I an indeterminate, onto ?I/ : ?Itul (identity on ?I, ^ -, u) has a non-zero kernel $. Since tf e V[ul, f is algebraic over J also. Let bQ,), q(^) be non-zero polynomials in ?l[,1]of least degree such that f(u) - 0 and A(f) : 0. ' Then these polynomials are also of least degree in lli.) such lhat p1u1: g, qU):0. Hence they are irreducible in tt,il. Let gr be the leading coefficientof p(^) and gz:4(0) and choosl g : 9$2. We shall show that g has the required property for f . Thus suppose a is a homomorphism of ?I into O satisfying g" : gigi + 0 and supposer is any extension of 6 to a homomorphism of ?I' -II[u] into A. Since g (f) : 0 we s hal l h a v e g " (f' )--0 , w h i c h i m p l i es thal f' + 0 si nce g"(0) + 0. Thus we need to show only that the hompmorphism 6 can be extended to a homomorphism of ?I'. For this purpose let c be a root of P"(l) - 0 and consider the homomorphipm r' of ?I[,t] which coincides with a on ?I and maps .i into c. Lgt /t(i) be any element in the ideal $. Then the minimality of the {egree of..bQ) implies that there exists a non-negative integer & such that g\h(]') is divisible in ?Itil by 2(,i). Since fr"(c):0, (gi)oh"(c)= 0 and since g{ + Q, h"(c) : g. Hence h(},)" - g and so $ is mapped into 0 by t' . It follows that r/ induces a homomorphism r of ?I' :2llu) = ?lt,il/$ which is an extension of o. Now assume the result holds for r - l. Let E = 2llu,l so that ? I' : E f ur , . . . , t t , - t). T h e n th e re e x i s ts a n e l ement hE , E such that any homomorphism p of E such that hP + 0 has an extension r to 'there exists geA such By the caser:1, ?l'such thatf'+0. that any homomorphism d of ?I into 0 such that ,9" * 0 has an extension p to E - ?J[u,)such that hp + 0. Hence r is an extension of o such that f' * 0 as required. We now give the
IX. AUTOMORPHISMS
271
Proof of Theorem 7: The hypothesis on doP implies that or is an isomorphism of Olnl into O[!ft]. Let ?l : OlTtl"" s ?I/ : Olffil. If Er, - , ., r^ are the projection mappings of !m into @ we have ?I' : ofnr, "', fr^l: ?I[zrr,'", fr^). Since ?I' is an integral domain we may suppose it imbedded in its field of quotients /'. Hence we can apply Theorem 2 to ?l and ?lt. Let f be a non-zero element of ?I/ - O[g?]. Then there exists a non'zero element g e @[It] such that if y is an element of Tt such that g(y) * 0 then the homo' morphism o: h"P -, h(y) of 2I : Al\t\"' into O, which satisfies goPn: g(y) + 0, can be extended to a homomorphism r of. OIfltl into O satisfying f' + 0. We have seen that r has the form k -- h(x) where r is an element of Sl. Then f (x) + 0 and for every h e sll, -h,nP: hnPn. This means that h(P(r)) : h(y). Hence P(x) y and the theorem is proved. 2.
Coniugaea of Cartan subalgebraa
Let B be a finite-dimensional Lie algebra over an algebraically closed field of characteristic 0 and let S be a Cartan subalgebra of 8. Let (11)
8:0*I8"
be the decomposition of 8 into root spaces corresponding to the roots 0, d, B, ... of 0 acting in 8. If lz e 0 and eu e g,,, then there exists an integer r such that e*(ad h - a(h)I)' : g. This is equivalent to the condition that a(h) is the only characteristic root of the restriction of ad h to 8,r. The a are linear functions on 0. is a root and If reSp (p-0 or p+0) then lxerl-O or p*a : g * 2a is a 0 latter case or 8o**. In the either e llxerle'l lxe") root and ffre,le,) € 8o*rr. If we continue in this way and we take into account the fact that there are only a finite number of distinct roots we see that r(ad er)k :0 for suffi,cientty high ft. This implies that ad e, is nilpotent for every ed e 8e, d + 0. It follows that if nrr r Srrr " 'r €rk € 8r' dtr dzr " 'r dk non'zero roots, then (12)
? : exp (ad e",) exp (ad edz).. . exp (ad e"*)
is an invariant automorphism of 8. Now let (ht, hz, " ' , ht, €t+rt " ', 0o) be a basis for 8 such that (hr, hr, ., ., h) is a basis for Q and the elements e;11,. . . , €n are in root spaces 8r, a. * 0. Let ]r, . .., f,n be indeterminates, P -
LIE ALGEBRAS
I
o(f,r, -.., i*) and form the element /t
(13)
(\];h; \i
\
. . . exp (ad i"e") ) u*p (ad,)4+(t+') /
: *
b ;(f,,,..., f,o )h*; i ,p ,(l ,,
where the f; and fu are polynomials in the i's. a polynomial mapping
(14)
.. ., f,n)et These determine
r, i Eihi* il,n,-r ! b;G)h;+> f^nei
in 8. The product a.p ... p of the non-zero roots is a non-zero polynomial function. It follows that there exist Ei e 6 such that if ... p(ht) + 0. Then the characteristic 20 - \lfihi, then a(ho)B@o) roots of the restriction of ad ho to 8, * 8p + .. . * 8o are all different from 0 and so this restriction of ad &0 is non-singular. We shall now calculate the differential dnoP of. P at ho. For h:ZlErhr, e:2i*r€p1, let f be an t h i s p u r p o s ew e l e t r : h * e , indeterminate and we consider P(ha+4h+e)) -- (ho + th) exp (ad tE*qp.t) exp (ad tEr.ze*z) . . . exp (ad tE"e") = (ho + th)(L * ad tE+gt+r). . . (t * ad tE*e,)t : h0 + th + h0 adtE*p*r * .. . + h0 ad.tF^ea : ho * th * tErcrlhoer*rl+ ... * tEofh,e,l , : ho + th * tlhoel
(mod /') (mod f'z) (mod t'z) (mod fr) .
If we compare this with (7) we we see that dnoP istthe mapping (16)
h*e-+h*[hoel.
Since h-+h and e->lhoel are non'singular it follows that droP is surjective. We are therefore in a position to apply Theorem 1. Accordingly, we have the following result: If ,f is a polynomial function * 0 on 8, then there exists a polynomial function g + 0 on 8 such that if y e 8 and g(y) * 0, then there is an r in 8 such that P(r) - y and J@) + 0. 'We recall the definition of a regular element a of 8 as an ele'
IX. AUTOMORPHISMS
273
ment such that ad a has the minimum number l' of 0 characteristic roots. We recall also that if c is regular, then the set of vectors Z belonging to the characteristic root 0 of ad a is a Cartan subalgebra. It follows from this that if 0 is a Cartan subalgebra and 0 contains a regular element a, then e is just the collection of elements h e 8, such that h(ad a)' :0 for some integer r. We shall need also the characterization given in g 3.1 of regular elements. For this we take the element u - }Ir)';h; * I,i*rlfit in 8r, P A(lr, - .., f,n)and we consider the characteristic polynomial
(17)
f "(A): det (il - ad u) : f , n- r { A;)} " -t + ... + (-L )" -' ' ro -,,(),r)A "
Then ro-1'(A1,..., f,n)is a non-zero homogeneous polynomial of degree n - l' in the I's and if x - Zt;h; * \Eiet, then r-+ r,,-1,(r) : t,-y(Eu . . . , E) is a polynomial function on 8. The element r is regular in 8 if and only if rn-y(x) * 0. (It will be a consequence of the theorem we are going to prove that l' : l.) We now consider again the Cartan subalgebra S and the basis ( hr, ' '', h , , et + r t ' ' ' , €n) fo r 8 . We a p p l y T h e o re m I to the pol ynomial function .f : rilr,which is + 0 since tn-r, * 0 and op is an isomorphism. Accordingly we see that there is a non-zero polynomial function .q on 8 such that every y e 8 satisfying g(y) + 0 has the form P(r) where .f (x) - r,_t,(P(x)) : ro_t,(y) + 0. Hence every y such that .q(y) + 0 is regular and if x - I,,l:g,h; * Li*rEiei, then y - P(r): (> E,h;)(expadEn&n) ... (exp adi*e*) - hn where h:28;h and T is an invariant automorphism. Thus y is the image of an elemerlt h e 0 under an invariant automorphism. It follows that h: yn-t is regular. It is now easy to prove the conjugacy theorem for Cartan subalgebras. Tsponnu 3. If Q, and bz are Cartan subalgebras of a finitedimensional Lie algebra ouer an algebraically closed field of characteristic 0, then there exists an inuariant automorphism q such that
0l : 0'.
Proof : There exists a non-zero polynomial functioo g; such that if y is an element satisfying gi(y) + 0, then y - h'lt, h; a regular element in 0; and h an invariant automorphism. Since g$z * 0 we can choose y so that gr(y) + 0 and gz}) + 0. Then y : hl' : hzz, h; a regular element of Qr, I; ?n invariant automorphism. Then hz: hl., T : rlfitzt. Since hi is regular and is contained in ,br it
LIE ALGEBRAS
274
follows that 0e - Fl. Remarks. It is a consequence of the theorem that levery Cartan subalgebra contains regular elements and that all Cartan sub' algebras have the saine dimensionality / which is thd same as the number // indicated above. It is easy to see also thalt if the notations are as before, then the regular elements of 8 belonging to Q are just the elements fto such that a(ho)P(h') "' p(h') + 0. 3.
Non-isomorphism of the split aimple Lie al'gebras
We shall apply the conjugacy theorem for Cartafi subalgebras first to settle a point which has been left open hit\erto, namely, that the split simple Lie algebras which were listed in $$4.5-4.6 are distinct in the sense of isomorphism. We recall that these , , andE. w e r e :A u l Z l , B t , l > 2 , C t , 1 2 3 , D t , l > 4 , G r , F r , ' , D a E given in following the are of these algebras dimensionalities The table: di,mensi,onality ty0e t(t+ 2) At t(zt+ L) Bt t(zt+ r) Ct, t(zt- r) Dr T4 G, 52 Fr 78 Ea 133 Et 2'48 Ea For the classical types and for Gr, Fn and Eu this was derived in 'W'e have proved the existence of Et and .Earin S7.5. The $ 4.6. dimensionalities of these Lie algebras can be derived by determining the positive roots directly from the Cartan matrices. We shall not carry this out but we shall assume the result lfor these two Lie algebras. To prove that no two of the Lie algebras we have listed are isomorphic it suffices to assume the base field algebfaically closed. This is clear since 8, = 8, implies 8,r zgrn for any extension P of the base field. We therefore assume 0 algebraically closed. The subscript / in the designation Xt (e.g., At, E) fdr our Lie alge' bras is the dimensionality of a Cartan subalgebra. lThe conjugacy theorem shows that this is an invariant. Hence npcessary condi-
IX. AUTOMORPHISMS
275
tions for isomorphism of Xt and,Yy are I : l' and. dim Xr : dim Yr,. A glance at the list of dimensionalities shows that the only possible isomorphisms which we may have are between Bt and Ct, I > 3 and between Bo and Eu and between Cs and 86. The latter two have been ruled out in an exercise (Exercise 7.6). It therefore remaini to show that Br 7L Ct, I > 3. Since Cr has an irreducible module in a 2l'dimensional space it will suffice to show that if ![t is an irreducible module f.or Bt such W e c o u l d e s t abl i shthi s, as t ha t ![tBt* 0, t hen dim g ft > 2 1 + 1 . in a similar discussion f.or G, ($ 7.6), by using the fact that the set of weights is invariant under the Weyl group. However, w€ can now obtain the result more quickly by using Weyl's dimensionality formula. We observe first that (8.41) shows that if llt is an irreducible module of least dimension satisfying ![tBr * 0, then !ft is a basic module (cf. also Exercise 7.13). The dimensionalities a n d 2 t ( c f . ( 8 . 4 5 )a n d E x e r c i s e o f t h e s ea r e ( " f ' ) , k : I , 2 , . . . , 1 - L 8.12). Since I > 3, these numbers exceed 2/. This proves our assertion and completes the proof that Br and Cr ?r€ not isomorphic if , = 3. 4.
Automorphisme of semi-simple Lie algebras oaer en algebraicalla clased, field
Let 8 be a finite-dimensional semi-simple Lie algebra over an algebraically closed field of characteristic 0, 0 a Cartan subalgebra, f r : {d r,...,d t } z s im p l e s y s te m o f ro o ts re l a ti v e to €), ei ,f;,h;, i: I,2, '..,1, d set of canonical generators for 8 determined by z. Thus the h form a basis for Q, 0r € 86n,ft, € 8,-"n and we have the following relations: [h;hil: 0 leJil - 6.iihi
(18)
le;hil:
Ai;€; - - Arrf,
Ifthl (Ad is the Cartan matrix determined by z. where Let r be an automorphism. Then 0" is a second Cartan subalgebra. Hence there exists an invariant automorphism a such that 0' : 0". Then the automorphism r' : 16-' maps 0 into itself. We now consider an automorphism r(:c') which maps the Cartan subalgebra 0 into itself. lf e, e I, we have le"h) - a(h)e". Hence
LIE ALGEBRAS
276
leLh'\ - a(h)eL. It follows that 8l : 8p where B is a root' In this wayweobtain a mappingd--rp of thesetof roots. Sincee,'e 8B ' we have leLh")- p(h")el. Hence we see that (19) a(h) -* P(h') Let rx d,enotethe transposein Q* of the restriction qf r to'Q. By Then (19),implies that definition,if E e,b*, then t"(h):f(h'). followingl g the have : we Hence d,",-'. 0,* d ot of I such that $' - p 1. Let t be an automorphi,sm PnoposrrroN a is any ro'ot of b in 8, if Then S. o/ for a Cartan subalgebra0 8|:86(r+1-l
(20)
where r* is the transpose in b* of the restriction of c; to Q. We note next that if fr : {dr, .. ., a,r}and Fi d\'i')-r, then z' : {Fr, .- ., B,} is a simple system of roots. Thus ever}r root has the form + E k&i where the &i are non-negative integerd. If we apply This ("*)-t we see that every root also has the form *. Z kih prove next guarantees that z' is a simple system. We PnoposttloN 2. Let n and n' be si,mple systems of roots. Then there existsan inuariant automorphism o such that Q" - $ and ota*t-t Tt' .
Then we know Proof: Let r: {dr,dzt' ',dr}, r' : {Fr, "',0t}. '" $oir for suitable Weyl (Theorems 8.1, 8.2) that nt : TcSar,,S"r, reflections S,,. It is therefore clear that it suffice$ to prove the zS"a. For this purpose we introduce the invariant result for r'automorphisms expad Efi and expad €et, E e A. In lour calculation we shall use the formulas for the irreducible representations of the three-dimensional split simple Lie algebra given in (36) of $ 3.8. Wenotethat the matrices of the restrictions of ad/6, adei to 8; Aei * Ahi * Ofi, using the basis (er,hi,lh;f;l) are, respectively,
*''-'(+ j, s) s )' '';\, B,(E': (22)A{E): ( ^'r i (i i r)
(2r) "or-,(l
Hence for exp adEfi, expad Ee; in 8,i we have the matrices
\o
o
tl
\2f' -zE
IX. AUTOMORPHISMS It follows that the matrix of the restriction to 8; of o;(E)= expad Ef;expadE-'erexpadti
(23) is
--( 3 j, A;(E)8,(E-')a,(t)
(24)
f,;)
\28_, o
N e x t l e th e b s a ti s fy a u(h)-_0. Then I n p a rti c ular , hii( o - - h r ttE): h. Since 0 is the direct lherl: 0 : lhfil and consequently lf vectors /l such that a;(h) : 0 it Ah the subspace of and sum of of ar(F) to 0 is Moreover, restriction the that clear is Q';'g'= b. the reflection determined by h;: h -->h - lZ(h,h;)l&t, h;)lh; h h l2a,(h)loti(h,t))h,r lf p e $J*, then [2(h,h,)l(h,1, h,)lhat
o(n \
21u,(h), - 'o\l"r),a{h) : p(h)- ?(l'o'),a{h) . : p(.h) /,",) '" d;) d{h')
d{h")
/
\di,
This shows that the transpose inverse of the restriction to .b of or(f) is the Weyl reflection Soi-' in 0*. Hence the invariant automorphism al(f) satisfieszrc;{€)*) - vSot: vt , as required. Proposition 2 and the considerations preceding it show that if r is an automorphism such that 0' : b, then there exists an in' and if r' : ra-', then variant automorphism 6 such that b":0 o((r')*)-' : r for the simple system ft. We simplify our notation again by writing r for r' . Then we have a permutation i -' i' of. i -- I,2 , . . . , / s uc h t ha t o \t' )-r * d i ,. Al s o w e h a ve l e' rhi l - Zei , - 2 f : , t e i f f l : h i . S i n c ee l e 8 , , , , ,f i e 8 , - " , , w e h a v e e i : tf:hil-Iti€i,,fi : v;f;,. Then l4l p;viht,. Since le;,hr,'l:2er and [eihi): Z ei so that plv ; ley h; , \ : Z p & r,, w e h a v e v i : y t;' a n d hi : h.i ,. S i nce le i h 'i l : A i$' r , le; , hi, ) : Ai ;i :r,. H e n c e w e h a v e (25) i ,j : 1 , 2 , " ' , 1 . A n , j ,- - A o j , The subgroup of the symmetric group on 1, 2, - . . ,l of. the permutations i -, i' satisfying (25) will be called the group of automorphisms If we recall the definition of the Dynkin of the Cartan matrix (A). diagram of the Cartan matrix, it is clear that if 4,, dzt .. ., d7 zty the vertices of the diagram, then any element i -t i' of the group of automorphisms of (At) defines an automorphism of the Dynkin diagram, that is, a 1 : I mapping d; -+ e,;, such that (dr.,at) : (ar,,ar) and for any i, j, the number of lines connecting d4 to di
LIE ALGEBRAS
is the same as that connecting dr and,ai,(cf. $4.il. The argument used to show that the Dynkin diagram determines the Cartan matrix shows also that the converse holds, that is, if di-'+ 4r, is an automorphism of the Dynkin diagram, then i -- i' is an auto' morphism of the Cartan matrix. Now suppose that r is an automorphism such that 81,t: &rt, i: !,2, ...,1. Then we have the identity mapping f -i,' : i' in the foregoing argument. This shows that ei : F&r f: - po'fo, hi : hr. Hence r acts as the identity in 0. Convetsely, Prop. 1 shows that if the restriction of r to 0 is the identity,'then gL: 8, for every d so ef, : pier f', -- trtt f r For these autortrorphisms we have the following PnoroslrtoN 3. If c as an automorhhism such that h' -- h for eaery h of a Cartan subalgebra O of 8, then r ls an inaariant automorphism. Proof: We have seen that ef, -- p&; and,f{ - piL.fi. Let o;(6) be the invariant automorphism defined by (23) and let ou(E)- ot(E)oi(I) We have seen that hli(E' - -hi, hai(Et- h if at(h) - 0. It follows that h-irit : h for all h. The matrix relative to (ei,h;,lhJtl) of the restriction of. ot(E) to 8; is the product of the matrix in (2+1 by the matrix obtained from this by taking 6 : 1. The result is
(26)
(n)
diag{t2,l,t-'}.
Hence we have (28)
. e?i'et: E'er: Enooe,
We wish to calculate next e?t'E' for j + i. We have 1fr;1: 0 and -A,iifi. Hence fi generates an irreducible inodule for 8i. lfihr): 1,2 or 3. If. A;t: -2 the There are four possibilities -A;t:0, module is equivalent to 8; and the argument just used shows that fe;Gt g-eufi and this implies that e?i''' : (-F)tti€i. lf Aii - Q, - et: (-E)t;le;. Next let [,fr8r] : 0 and this implies that elict -L. : acting the matrices e; in the module generAu Then of f;, ated by f5 are, respectively
(n)
lo
\o
1\
o) '
/0
\-t
It follows that the matrix of a;(f) is
0\
o) '
IX. AUTOMORPHISMS
(30)
(_?, il
Hence we have f}'G' : -E.fi and that of ar;(f) is diag {-8, -5-'}. -3. Then the -E-tet: : (-E)oiiei. let A;i: Finally, znd e'iiGr to to (36) of $ 3.8: are, according and e; forf; matrices representing
1
l0
0
\
l0
0/
I\o0
0
01
0
o 1 ol. [ -^s o, I -4 o o o rl'
(31) \--l fl
\\o0
0
0
ll o/-
_3
0
0t
It follows that the matrix of. oa({) as defined in (23) is
0
0
t'l6t o -rE ol. I o I o zE-'o ol t0
(s2)
\-6F-'0
0
0l
Consequently, the matrix of ornE) in the module generated by ft Hence f f'u' : -E'f i and solitet is diag {-t" , -8, -E-' , -F-t}. -- (-E)^njei. Thus in all cases we have ?E)-'ei seit€t- (-E)^niei
(33) (i:ior
j+i).
(34)
Q = Q ( E ' , F I ,. . ., F ,) :
Nowset t' t {-E ) .
c o (-€ r)o n (-F r)" '
Then (33) implies that (35)
e l - g l ' , t E fi . - . € f i e i ,
i : 1,2, --.,1 .
We recall that the matrix (Ad is non-singular and its determinant is clearly an integer d. Hence we have an integral matrix (Brr') : dI. Let i be fixed and set Eo: (p'fd)"i*, such that (Ad(Bi h : l, "',1. Then (36)
(E d1a tA I i+. " + Bi. a t i € _ o0 ei : e' 1..... 1t 7p2rrt i
errl \r, e i :
g?{Er...,ett: e"rrlo)"irA|i+"'+Bit^,trn: e,
if
/ qe j i + j
.
It is clear that a suitable product of the invariant automorphisms defined here for j : I,2, . . . ,l coincides with the given auto' morphism r in its action on the ei,f;, h;. Since these are generators it is clear that r is an invariant automorphism. It is now easy to see that the index of the invariant subgroup
280
LIE ALGEBRAS
Go in G does not exceed the order of the group of adtomorphisms of the Cartan matrix. In fact, we can display a sublroup K of. G which is isomorphic to the group of automorphisms qf the Cartan matrix such that every r e G is congruent modulo Go,to a te e K. Let P: i-ri' be an automorphism of the Cartan matrix. Then we have the relations -0 , lh, i ,,h i ,l fe o ,fi ,l : B i l h y fet'hi'l: Aitet' , [f;'hi'f : - Ainfn' , Hence the isomorphism theorem for split semi-simplel Lie algebras (Theorem 4.3) implies that there exists a (unique) dutomorphism tp of 8 such that eI": er, .f{':f;,. The set ofr these automorphisms is evidently a subgroup Kof G isomorphictto the group of automorphisms of the Cartan matrix. Now let r be any automorphism of 8. We have seen that r is congruent modulo Go to an automorphism r, such that 6r" : €). Also we know that rr is congruent modulo Go to an automorphism rz such that Q"z: Q and rc(t;t-r : n for the simple sy$tem of roots T : { d r , d z t - . . , d t } r e l a t i v et o 0 . W e h a v e 8 L 1 : & r n , , ' ,:i 1 , 2 , . . . , 1 and i -r i' is an automorphism P of. the Cartan matri*. Let r.r" b€ the corresponding element of K. Then 6 : tztFt satisfies 8Zr: 8r* ga-or. 9-rn, i : 1,2, . . .,1. Hence o is an invariant automorphism, by Prop. 3. Thus r is congruent modulo Go to rr. It can be shown that no element of K is in Go, urhich means that G is the semi-direct product of. K and Go. This is equivalent to showing that the index of Go in G is the order of the group of automorphisms of the Cartan matrix. A proof of this will be indicated in the exercises. We shall now restrict our attention to 8 simple. The result stated will follow for the ll-ie algebras Ar, Bt, Ct, Dt, I + 4, Gz and Fn from the explicit determination of the groups of automorphisms for these Lie algebrps which we shall give in the next section. The result will alsq be clear for Et and. Ee. We now examine the groups of automorphisms of the connected Dynkin diagrams. We recall that these are the ones which correspond to the simple Lie algebras 8. The types are tAt, I 2_7, Br, 122, C,l>3, D t , 1 2 4 , G r , F r , E a , E z , E a .I f w e l o o k a t t h e s e d i agr am s as giv e n i n $ 4 .5 w e s e e th a t fo r A r ,B t,C r,Gr,Ft,E , and Ee the group of automorphisms of the diagram is lthe identity.
281
IX. AUTOMORPHISMS
1111 Fof ,4r, I > 2: a-o. dr
. . s-e-
dz
dt-r
we have in addition to the dt
identity mapping the automorphism dt-, dt+rt.
For
1 Dr:
I
I
d1
d2
a-;...01
I -zodt dt_z
\9
o,_, I
we have the identity automorphism and the mapping dr'-+ di i < I - 2, dFr -'+dtt dt -+ dt-t which is an automorphism. These are the only automorphisms if / > 5. For /: 4 the diagram Dr i
o'
o-o1'" d1
dz
\o
dg
has the group of automorphisms which permute dt, ds, dl and leave d2 fixed. This is isomorphic to the symmetric group on three elements. For
liae
Ea,1-1_!1-l_: d1
d2
ds,
dq,
d5
the group of automorphisms consists of the identity mapping and the mapping a6--+dat d;-) a6-;, i 35. It is now clear that we have the following Tuponpu 4. Let 8, be fi.nite-dimensional simple ouer an algebrai,cally closed field of characteristi,c 0, G the group of automorphisms of 8, Go the inuari,ant subgroup of inuariant automorphisms. Then G : Go unless 8 is of one of the following types: At, I > t, Dt or E6. In all of these ca,sesexcept Da, the inder [G : GoJ< 2 and for D', [G : Gu] 5 6. Remarh. The group G is an algebraic linear group (cf. Chevalley tzl). It is easy to see that Go is the algebraic component of the identity element of G. If @ is the field of complex numbers, then G is a topological group and Go is the connected component of 1 in G. 5.
Explieit
determination of the automorphisms for the simple Lie algebras
Let 8 be a semi-simple subalgebra of @r,,0 the algebra of linear
LIE ALGEBRAS
transformations of a finite-dimensional vector space tover o. Let z fu an element of B such that ad Z is nilpotent. since the algebra of linear transformations adg t3 is semi-simple it tfollows from Theorem 3.17 that there exists an element H e I such that [zH] z. This implies (Lemma 4, $ 2.5) that z is nilpotent. we have ad Z - Zn - Zz (Zn: X - XZ, Zz: X -- ZX). Hence exp ad Z : Exp (2" - Z") : exp Zn exp (.- Z) , : since [Z*2"]: 0. Then
(37)
exp ad 7 - (exgZ)nGxp(- Z))" - (expZ)nbxp Z);' .
If we set A : exp z, then the automorphism exp ad z of. 8 is the mapping (38)
x -+ A-txA
.
we now consider the simple Lie algebras of types iAt, Bucu Dt, Gz, and Ft. At. Here 8 is the Lie algebra of linear transformattions of trace zero in an (/ + l)-dimensional vector space. we can identify g with the Lie algebra of matrices of trace 0 in Ot+r. lt A is a nonsingular matrix, then X-+ A-|XA is an automorphistn of 8. Since the only matrices which commute with all matrices bf trace 0 are the scalar matrices it is clear that the automorphism$ X-- A-'xA, X -. B-'XB are identical if and only if B : pA, p e A. Besides the automorphisms X--+ A-IXA we have the automorphfism X-- -X, of 8 and, more generally, we have the set of atrtomorphisms X ---+- A-'X'A. Suppose the automorphism X -, -X' coincides with one of the automorphisms X --, A-'XA. Then we have (3 9 )
A-' XA :
-X'
for all X of. trace 0. This implies that -X - A' X'(A')-r - - A, A-'XA(A')'| so that A(A')-' commutes with all X. It follows tthat A' : pAi hence At : *.A. The condition (39)can be rewrittenr as A-'X'A : -X. The set of X's satisfying this condition is either the symplectic Lie algebra or the orthogonal Lie algebra. Accordingly the dimensionality is either (l + t)(l + 2)tZ (odd / only) or I(l + t)lZ (/ even or odd). Since the dimensionality of the space of (l * l) x (/ + t)
IX. AUTOMORPHISMS
matrices of trace 0 is 12+ 21, we must have either l' + 2l : ( / + l X / + 2 ) l Z o r l ' * 2 1 : l ( l + l ) 1 2 . T h e o n l v s o l u t i o ni s l : L in which case/' * 2l : (/ + lX/ + 2)lZ. Thus we see that X-'-X' coincideswith an automorphismof the form X- A-'4A only if
f ;
wehave-x' : A-'xA forA: (j, l) "'u
result obtained at the beginning of this section shows that The "it,j;t;.,n.. every invariant automorphism of ,8has the form X-, A-|XA where A is a product of exponentials of nilpotent matrices. For the Lie algebra A, this is the complete automorphism group. For At, I > l, we have the automorphism X -t -X' which is not invariant. Hence lG: Gol- 2 and the automorphisms are the mappings X -' A-|XA and X '- - A-'X'A. Tnponpu 5. The grouf of automorphisms of the Lie algebra of The 2x2 matricesof trace 0 is the set of ma\bings x-- A-'xA. g:roup of automorphisms of the Lie atgebra of n x n matrices of - A-'X' A' iroru 0, n > 2, is the set of mafpings X-+ A-tXA and X-' Q.) characteristic closed of (The basef.eld a is algebraically Bt, Ct, Dt. These algebras are the sets of matrices X satisfying -S for Ct' S-tX'S - -X where S: 1 for Bt and Dt and S' : The siie of the matrices is 2t for Ct and Dr and 2l + 1 for Br' We take t>2for Bt, />3 for Cr and l>4 f'or Dt. Let O be a matrix such that (40)
O'SO : PS
where p + O is in @. Then O is non-singular and if X e 8, then Y: O-'XO satisfies (4 1 )
S - t y , S : S - t ( O-tX O ),S= S -rO ,X,(O,)-IS : - S - ' O ' SX S-' (O ' )-' S : -p O-' X(p -' O) : -O-'XO : -Y
.
-Hence Y e 8 and consequentlyX -r Y O-'XO is an automorphism of 8. Since the base field is algebraically closed we can replace - Y, OiSO' : S. If we O by p-LO : O, and we obtain Or'XO, D,, O is an orthogonal and for Br write O f.or Or, then we see that It is easy to (S: matrix. a symplectic O is for C,, 1) and matrix verify that the enveloping associativealgebra of 8 is the complete matrix algebra. We leave it to the reader to prove this. It follows
284
LIE ALGEBRAS
that the only matrices which commute with all the dlements of 8 are the scalars. Hence if Or'XOr: O;'XO, for all X e 8, then Oz--oOy p€A. Let Z be an element of 8 such that ad Z is nilpoterft. Then we have seen that Z is nilpotent and the automorphismt exp ad Z has the form X-+ A-'XA where A:expZ. The nilpotqnce of Z implies that exp Z is unimodular (Exercise 5.4). Also, We have S-'(exp Z)'S - S-'(exp Z')S
- expS-'Z'S: €xFen
- (expZ)-' .
Hence A : €xp Z is orthogonal for Bt and Dr and symplectic for Ct. For ,Br and Ct every automorphism is an invariant automorphism. Hence in these cases the automorphisms of 8 have the form X-'O-'XO where O is unimodular and satisdes O'SO: S. For Bt this states that O is a proper orthogonal dnatrix (corresponding to a rotation). For Cr it is known that if Ol is symplectic (O'SO : S) then O is necessarily unimodular (see Artfin [1], p. 139, or Exercise 12 below). Hence this condition can be dropped. Now consider Dt. In this case there exist orthogonal matrices O of determinant -1 and we cannot have O : pO, where O, is a proper orthogonal matrix. Thus it O : QOr, then p: +I and in either case det O : det pOt : det Ot : 1. This contradictiort implies that the automorphism X--'O-|XO, where O is improper brthogonal, is not invariant and we see that G = Go. If I > 4, then the index of Go in G is 5 2; hence this index is 2 and every hutomorphism of. Dt has the form X-rO-'XO where O is orthogonal. We therefore have the following Tnsonnu 6. Let O be algebraically closed of characteristic 0 and let 8, be the Lie algebra of skew matrices or the "symplectic" Lie algebra of rnatrices X such that S-tX'S - -X where S': -S. Assume the number of roLt)s n 2 5 in the odd-di,mensional skew case, n ):6 in the symplectic case and n Z l0 in the eaen-dimensional skew case. Then the groups of automorphism, of 8, in the shew cases consist of the mappings X-->O-|XO wherg O is orthogonal. In the odd-di.mensionalcase one can add the condition that O is brofuer. In the symflectic case the group of automorphisms is where O'SO - S. the set of mappings X-tO-'XO The case of. D, is not covered by this theorem. It can be shown that in this case the group GlGo is isomorphic to [he symmetric I
IX. AUTOMORPHISMS
285
group on three letters and the group G can be determined. This will be indicated in some exercises below. We consider next the Lie algebra Gz. Here we use the representa' tion of 8 as the algebra of derivations of a Cayley algebra 6 ($ 4.6). Now, in general, if O is a non-associative algebra, D a derivation of G, A an automorphism of 6, then A-'DA is a derivation. It follows that the mapping X--+ A-'XA determined by an automorphism of 6 is an automorphism of the derivation algebra 8 - E(6). In the case of O, the Cayley algebra over an algebraically closed field of characteristic 0, we know that every automorphism of B is invariant and so it is a product of mappings of the form X-' A-'XA where A: axPZ, Z in 8 and Z a nilpotent linear transformation in 0. Since Z e 8, Z is a derivation of 0. 'It follows that A : €xP Z is an automorphism of 6. We therefore see that every automorphism of 8 has the form X -, A-'XA where A is an automorphism in 6. The same reasoning applies to the Lie algebra Ft. Here we represent 8 as the derivation algebra of the exceptional Jordan alge' bra Mi and we obtain the result that the group of automorphisms of 8 is the set of mappings X -- A-'XA where A is an automorphism of. Mi. Tnponpu 7. Let 8, be the Lie algebra of deriuations of the Cayley algebraE'orofthelordanalgebraMfoueranalgebraicallyclosed rt.eM of characteri,stic 0. Then the automorphisrns of t haug the 'form X--+ A-,XA where A is an automorphism of 6' or of Mf .
Exercises The base field in all of these exercises will be of characteristic 0 and all spaces are finite-dimensional. l. Show that any non-singular linear transformation A can be written in + N, N nilpotent, the form AoAc where Aw is unipotent, that is, Au:l and At is semi-simple and A, and At are polynomials in 14. Prove that if A: BtBo and Bo and Bs commute where Bo iS unipotent and 8c is semi' simple then Au : Bw and Ar - BE. (Hintz Use Theorem 3.16.) 2, Let r be an automorphism of a non-associative algebra ?[ and let r = rrrrr 3s ir 1. Prove that rr and cu are automorphisms. (Hint: Assume the base field is algebraically closed and use Exercise 2.5. Extend the field to obtain the result in the general case.) 3. Let I be a finite-dimensional simple Lie algebra over an algebraically
286
LIE ALGEBRAS
closed field and let Go be the group of invariant automorphisnis of 8. Prove that 8 is irreducible relative to Go. 4. Let G be the split Cayley algebra over an algebraically closed field, Go the subspace of elements of trace 0 in G: a * d : 0. We ftrave seen that the derivation algebra t(:6r; acts irreducibly in 0o ($7.6). Use this result to prove that Go is irreducible relative to the group of automorphisms G of G. 5. Same as 4. with G replaced bv lff (cf. g 4.6). Exercises 6 through 9 are designed to prove that if I is semi-simple over an algebraically closed field, then G/Go is isomorphic to the lgroup of automorphisms of the Cartan matrix or Dynkin diagram determintd by a Cartan subalgebra 0 of 8. In all of these exercises 8, 0, G, Gd', etc., are as indicated in the text. 6. If a€8 and Eo:[zllzal:}l, then dimGo ] l, the dimensionalityof a Cartan subalgebra of 8. Slcetch of proof: Note that dirn Gc : dim 8 rank (ad o). Show that if o is a regular element, that Go : S a Cartan suba l g e b r as o d i m G a : l . l f . ( u , . . . , x t n ) i s a b a s i sf o r 8 o v e r @ , a n d( { r , . . . , f n ) are indeterminates, then c : 2 €rut is a .regular element of 8p, P : @(tt) so rank(adr):n-1. If 6:\alur t h e s p e c i a l i z a t i o nh : F t shows that rank (ad a) < n - I. Hence dim Go 2 l. 7. If r is an automorphism in I let I, be the set of fixed points of r: D" : D. Prove that if r is invariant, then dim I" 21. Strcetchof prmfz \il'e have r = €XF Zr exp Zz - . . exp Z, where Zt : ad zt is nilpotent and we have toshowthatrank (r-1) < n-L L e t f i , . . . , f " b e i n d e t e r m i n a t easn d let P be the field of formal power series in the 6c, that is, the quotient field of the algebra Q < tu. . ., f" ) of formal power series in the fo with coefficients in @. Then r(f) = exgtrZrexglzZz... expt Z, ii an invariant automorphism of 8r. The matrix of r({) relative to the basisr(uu . . . , un) of I has entries which are polynomials in the {a and the specialization fe: 1 gives the matrix of r relative to this basis. Hence if dim lrrer 2 l, a specialization argurnent will show that dim I" > | (semi-simplicity anrd the rank are unchanged in passing from I to 8p). Now the exponential formula can be usedto showthat {f) = expErZr...exptrZr:expZ where Z:ad?, ze 8p (Exercise s.LZ). Then I'rer f Gz so dim frrel Z I by 6. 8. Let r be an invariant automorphism such that 0" = 6 And ,n(e*)-t- r, for a simple system of roots t. Then h, : h for every h e 0. Sketch of proof : (r'*)-t induces a permutation of the set of roots which we can write as a product of cycles (9r, . .., B)(Tu ..., Ts).. .. Since n(er)-r : n, (r*)-t leaves the set of positive roots invariant; hence the 8c in a cgcle (Bu ...,9r) are either all positive or all negative. 8pr * ... + 8B, is invhriant under r. If ap, is a baseelement for 8pn, then we have el, : vtaB2toEr:),rza1s,. . . , e'Fr: ,treFt so that the characteristic polynomial of the restrlction of r to 8 f t + . . . + 8 B " i s 1 7 - v r . . . v 7 . I f v r . . . y r * L t h e n t - 1 i s i n o t a f a c t o ro f this and consequently (8p, + . .. * 8Br) fl I, = 0. If this holds for every
IX. AUTOMORPHISMS
287
c y c l e ,t h e n . I , c 0 s i n c e I = S @ ( 8 p r @ " ' @ 8 p , ) O ( 8 v r O " ' O 8 r , ) + " " = h for all h e 0. Since dim.[, ] t by 7. it will follow that .It S and h" = h , h € b a n d Q o=; p f t 1 * 0 , N o w l e t o b e t h e a u t o m o r p h i s m i n S s u c thh a t l r t that o is invariant' generators)' shown have We : (et, canonical pi'ft fi ft,h, : = If a is a positive root then o : I kst and tZ p!'pt' "' u!',' and eo-a c o n d i t i o ns s a m e t h e s a t i s f i e s o r r ' = a u t o m o r p h i s m u l k r h o k r . ...r 't' r k a, e - r . T h et ' I - v l " u l p i t" ' P i l i f t h e ' l w h e r e " ' u ! as r and e'p'r=r'req.,"',a-Pr='traB1 s t n o n ' n e g a t i v ei n t e g e r s , * B,= Iisiar*0 Btare all iositive'and Fr*'" . . . a l l n e g a t i v ae n d B r * ; . . * 8 r = a r e i f t h e h t ' t 8r andvr... rl : vr... vlFi" - Lirrnr. SinEe e, * 0 for sotnei we can choosethe p's so that u{ " ' ti + 1 =h, h€8' forlvery cycle. Then the argumentused before shows that h:' H e n c el L r= h , h e S . of the 9. Prove that G/Go is isomorphic to the group of automorphisms Cartan matrix. form 10. Let I = (Di-r, I + 1 : 2r and let r be an automorphism of' the that such exist r there that and I, dim that zr Show 8. in A-LX'A f,-r dim ft : r. let I/ be ll. Let I be semi-simpleover an algebraically closed field and of Go of I such the subgroup of Go of 7 such that S4 g F, K the subgroup thathr=hforallft.€s.ShowthatKisaninvariantsubgroupoflland Weyl group') that I{/K = W. (This gives a conceptual description of the matrix symplectic a is if 12. Use the proof of Theorem 6 to prove that O : l' det O then 0, with entries in a field of characteristic if a= al *oo, 1 3 . L e t O b e a s p l i t c a y l e y a l g e b r aa n d l e t d : a I - Q ' o N t o ) - f V t U ) l 'V e r i f y o o € G o . S e t N ( o ) : q , d : d a a n d( o , b ) : l l N t o + b ) witt that (o, b) is a non-degeneratesymmetric bilinear form of maximal : show index. Prove that N(ac, b) : N(o, bZ) and N(co' b) N1o' 6b)' Hence p,:l(cr' +cn) are in the that if c€Go, then ca (t+rc), ct'lr+cr) and to the form (o' b)' Show relative G (Dr) space of the orthogonal Lie algebra form Bt + I [8.,Fa11 where has the algebra Lie this of element every that c , c i ,d t € G o . l{. Use the alternative law (eq. (4.79))to prove the following identity in G c(ab)* (ab)c: (ca)b+ a\bc) * a(bcn). Use this to prove the principle of local triality: or Z(ab)Rc- \a,cr,)b A in G which is skew relative to (o, b) there transformation For every linear (8, linear transformationssuch that skew of pair c) unique a exist (ab)A : 1o.B)b* otbC) . 15. Show that the mappings A - B and A -* C determinedin 14 are auto' morphisms of the orthogonal Lie algebra. Prove that every automorphism of this Lie algebra is of the form X -. O- tXO where O is orthogonal or is the product of one of the automorphismsdefined in 14. by an automorphism X - O-|XO.
288
LIE ALGEBRAS
16. Let 8r and 8s be two subalgebras of Dr isomorphic tb Ds. prove that there exists an automorphism of. Dt mapping 5J1onto fJ2. 17. Show that the automorphism in At, I ) l, such that fu - ft, ft - et (cf. p.127) is not invariant. Show that for Du I24, this automorphism is invariant if and only if I is even. 18. (Steinberg). If 6 is a Cartan subalgebra let Go(b) be group of automorphisms generated by au exp ad e where e belongs to a rdot space of g relative to 6 (cf. Chevalley [7]). Let &rr be a second Cartan Subalgebra and let Go(6r) be the corresponding group of automorphisms. Show that there exists ? € Go(S), ?r € Go(br) such that S? : S?t. Then This S : i0ltr-t. implies that Go(0) :4r7r'Go(fir)TrT-, so that Go(6) : 19.
Prove that Go(6) :
ti-tGo(p)n :
TlLtGo($)71 :
Go(Or) .
l
Go, the group of invariant automorpfrisms.
CHAPTER X
Simple Lie Algebras over an ArbitrarY
Field
In this chapter we study the problem of classifying the finitedimensional simple Lie algebras over an arbitrary field of characteristic 0. The known methods for handling this involve reductions to the problem treated in Chapter IV of classifying the simple Lie algebras over an algebraically closed field of characteristic 0. One first defines a certain extension field /' called the centroid of the simple Lie algebra which has the property that 8 can be considered as an algebra over I' and (8 over /-)o : J7-s.r8 is simple for every extension field I of. T. It is natural to replace the given base field @ by the field /'. In this way the classification problem is reduced to the special case of classifying the Lie algebras I such that 8o is simple for every extension field g of the base field. If 8 is of this type and J? is the algebraic closure, then the possibilities for ,8o are known (A1, Bt, Cr, etc.) and one now has the problem of determining all the 8' such that 8o is one of the known simple Lie algebras over the algebraically closed field .?. This problem can be transferred to the analogous one in which the algebraically closed field I is replaced by a finite'dimensional Galois extension P of. T and 8r is one of the split simple Lie algebras over P. This is equivalent to the problem of determining the finite groups of automorphisms of 8p considered as an algebra over A which are semi-linear transformations in )3p over P. We shall consider this problem in detail for 8r one of the classical types At - D except Dr. Our results will not give complete classifications even in these cases but will amount to a reduction of the problem to fairly standard questions on associative algebras. For certain base fields (e.g., the field of real numbers) complete solutions of these problems are known, so in these cases the classification problem can be solved. The classification problem for the field of real numbers is quite old. A complete solution was given by Cartan in 1914. Simplifications of the treatment are due to Lardy and to Gantmacher. For [28e]
290
LIE ALGEBRAS
an arbitrary base field of characteristic 0 the results for the classical types are due to Landherr and to the present aluthor, for Gz to the author and for F. to Tomber. It is worth notilng that most of the results carry over also to the characteristid p + 0 case. This has been shown by the author. References to the literature can be found in the bibliography. 7.
Multiplication algebra and, centroid of a non- awoeiative algebra
Let ?I be an arbitrary non-associative algebra over la field o (cf. S 1.1). lf. a e [, the right (left) multiplication aa (a")iis the linear mapping tr --+tra (r -+ ax). we define the multiplication atgebra T(Dt) to be the enveloping algebra of all the an and at, s e A. Thus E(?r) is the algebra (associative with l) generated hy the ar, an. If ?I is a Lie algebra, then !(?I) is the enveloping algebra"rrd, of the Lie algebra ad ?r. we define the centroid r(?I) of ?r to be the centralizer of E(?I) in the algebra 0(?r) of all linear transformations in ?I. Thus the elements of /- : /'(2I) are the linear transformations for all A e !(?I). Evidently,'y e T if and 7 such that [r, A]:0 only if lvanJ: 0 : [rar.] f.or all a e 21, and these conditions can be written in the form
(1)
(ab)r:@fib-a(br)
a , be ? I
Lnuur 1. If A' : T, then I- i,s commutatiue. Proof: Let y,6eI', a,beTI. Then (ab)rt-((ar\b)d-(a:)e|) and (ab)r8 - (a(brDd : (a|)(bi. If we interchange T and d we obtain (ab)dr : @6)(bi - @)(bS). Hence (ab)(rd - dr) : 0. Since ?It : ?[, any element c of ?l has the form c - D a;hr It follows that c(rd - dr) : 0 for all c and f" is commutative. I A non-associative algebra ?I is simple if tr has rlo (two-sided) ideals + 0, + [ and ?l' + 0. since ?12 is an ideal it follows that
?l' : 2I for ?r simple. Hencethe lemma implies that /' r is commutative. The ideals of a non'associative algebra ?r are just the subspaces which are invariant relative to the right and left multiplications. These are the same as the subspaces which are invariant relative to the multiplication algebra E = t(?l). It follows that ?r is simple if and only if ! is an irreducible algebra of linear tralnsformations. lf. r e 2I, then the smallest E-invariant subspacecontdining r is rE.
X. SIMPLE LIE ALGEBRASOVER AN ARBITRARY FIELD
29I
The converse is Hence if. x+0 and U is simple, then rT.:?J. - ?I for every x + 0, then tr is easily seen also: if ?I'z* 0 and rT simple. We recall the well-known lemma of Schur: If ! is an irreducible algebra of linear transformations, then the'centralizer of A is a division algebra (for proof see, for example, Jacobson - E(?l) for ?l simple [2], vol. lI, p. 27L). In the special case of ! this and Lemma 1 give Tnponou L. The centroid f of a simple non'associatiue algebra is a field. Since the centroid I' is a field we can consider ?l as a (left) vector space over l'by setting Ta:&T, & €?I, 7 € l'. Then condition (1) can be re-written as
( 1')
r@b)- (ya)b- a(rb) ,
which is just the condition that ?I as vector space over associative algebra over /' relative to the product ab over (0. A non-associative algebra ?I will be called central if /' coincides with the base field. If ?I is simple with then ?l is central simple over l.; for we have
/- be a nondefined in 2t its centroid centroid /',
Tnponnru 2. Let \ be a sirnple non-asnciatiue algebra ouer a rteld O and let I'(= O) be the centroid. Consider ?l as algebra ouer I' by d.efi,ningTa: aT, a € ?I, 7 € l'. ThenTI is simfle and central ouer I' and. the multiptication algebra of V, oaer T is the sarne set of trans' formations as the multiplication algebra of il ouer A. Proof: Since f =- A it is clear that a /'-ideal of ?I is a O-ideal so ?I is f-simple^: Similarly, the centroid f of ?I over f is contained in /'; hence F : T and ?I is central. Let t denote the multipli-- f\. the set cation algebra of ?l over l'. Then it is clear that i of /'-linear combinations of the elements of !. Now let Eo be the subset of E of elements A such that rA e t for al| r in /'. It is clear that to is a @'subalgebraof !. lf a, x € U, then r@x) - a(Tx) : Qa)r which means that Tar,: atT : (ra)r.. Hence at e' To and similarly, an e To. Thus ts contains all the left and the right multiplications and consequently to : a. W'e therefore have t : /'! : !' We consider next the question of extension of the base field of a non-associative simple algebra. The fundamental result in this connection is the following
292
LIE ALGEBRAS
THponpnr 3. If \ is a non-associatiuecentral simple a'l,gebraoaer o and P is any extension field of A, then ?Ie is central sdmble ouer P. Next let w be an arbitrary non-associatiaealgebra ouerla, let y' be a subfi,eld (ouer A) of the centroid and suppose the r'-algehra T 8z?l fs siatple. Then il is simple ouer O and y' - l. Proof: For the proof of the first assertion we shall need a wellknown density theorem on irreducible algebras of lirlear transformations. (See,for example, Jacobson [2], vol. II, p. ZT2.) A special case of this restrlt states that if t is a non-zero irredubible algebra of linear transformations in a vector space ?l ovef a and the centralizer of ! in @(U) is the set @ of scalar multiplijcations, then t is a dense algebra of linear transformations in ?I ever O. This means that if {xr, xr, ..., )tn} is an ordered finite set of linearly independent elements of ?I and !r, !2, .. . t ln ?ro n lrbitrary elements of ?I, then there exists a T e T. such that k.iT : y.;, i I,2, ..-,n. The density theorem is applicable to the rhultiplication algebra t of a central simple non-associativealgebrar Now let P be an extension field of a and consider the extension algebra ?Ip, for which we choose a basis {urle e I} consisting r of elements u, e 2I. Let x and y be any elements of ?Ip with x + O. Then we can write r : ZiFttci, y : Zinrx, where {xr, xr, . .., xn} is a suitable subset of the basis {u"} and the f's and 7's are in p. We may assume also that E, + 0. The extension to 2I" of anf element of t is in the multiplication algebra t of ?lr ov€r P. i Hence there existsan A;ei such that xlAi:x; and )ciAt-$iif i*I,i: 1,2, ".,n. Then trA;: Efir and A:ZT:rV$tLAi € t an6 satisfies xA -- t. Thus xt e y and since y is arbitrary, rt. : ilr. This implies that ?Iie is simple by the criterion we noted before. Let C be a linear transformation in ?Ip such that LCAI : 0 flor all a e t. Let tio be one of the base elements and apply the previous consi d e r at ionst o r : uo : : 1' 1, x l t y : u d ,c . L e t Are E sati sfy )c1A 1 r ; A 1 : 0 , i + I . T h e n u , C : x 1 C: x r A r C : r r C A r : g A r : TJC.: vJfla. Since Zr depends on uta,we write 4r: Qo, and so we have urC : Quu4ot 0, Q P. Next we note that if u, and uB 6re any two base elements, then there exists 2 BoF e t such that urBaF: uF. This is clear because of the density of t. Then ppup: uBC: utBrBC : uoCBap: gocrrBry: pu6p. Hence Qa: 0F: p and C is the scalar multiplication by the element p. This shows that the centroid is P, so ?Ir is central over P. This completes the proof of the first assertion.
X. SIMPLE LIE ALGEBRASOVER AN ARBITRARY FIELD
293
Now suppose ?I is a non-associative algebra ovet A such that f &o?l is simple over y', I a subfield containing A of. the centroid /- of ?I. We can consider ?I as algebra over / (6a : a6, a e il, d e /) and we show first that ?l is simple over y'. Thus let E be a l' ideal in ?I. Then the subset of elements of the form X r;8 bi, Tt e T, D, e E is a l-ideaL in /" 8, ?I. The properties of tensor products over fields imply that if E + 0, ?I, then the ideal indicated Hence E: 0 or E - ?I' is a proper non-zero r'-ideal in T &oII' Also ?I' + 0 since (r8r?I)'z+ 0. This shows that ?I is simple over l. The proof of T'heorem 2 shows that the multiplication algebra E of ?l over 0 is the same set as the multiplication algebra of ?l it is now clear that over y'. Since we have xT.- ?I for all r*0 Consider ?I again as /' is a field. 0 and also ?l is simple over mapping that the checks one l. Then over algebra
(2)
2rr8
ao-.}rrar(= \at;)
onto ?I' T;Q T, or€V is a l'algebra homomorphism of I8r?I r' -i ndependent. 1 , th a t = l let T a re and f Su p p o se 7 € f, $lso Thenarelationt&ar+18a2:0,at€'2L,impliesthattheat:o' the Then r&ar+18a2+0and' az:-Ta. C h o o s ea 1 : a * 0 1 r', f + Hence if. image of this element under (2) is Ta 7a:0. This image. then (2) has a non-zero kernel as well as a non'zero y'. Hence contradicts the assumption that /'8, U is simple over y' and the proof is comPlete. f : It is immediate that a dense algebra E of linear transformations g(U). in a finite-dimensional vector space is the complete algebra .., . transformation, linear any xn) is a basis and A is Thus if (xr, If ?l HenceA:TeT" t h e n t c o n t a i n s? s u c h t h a t x t T : t c t A . that seen we have then /-, with centroid simple is non-associative transforof linear algebra ! dense is a algebra the multiplication mations in ?I considered as a vector space over f. Hence if ?l is finite-dimensional over 0 and consequently over /., then ! is the complete algebra of linear transformations in ?I over l-. We therefore have the following THponpru 4. Let ?1 be a fi,nite'dimensional simple algebra with centroid T and. multiptication algebra 7.. Then T" is the complete set of linear transformations in ?I considered as a aector space ouer T. Let the dimensionality [?l : O] of ?I ovet O be n and let lT : Ol:
294
LIE ALGEBRAS
r, Ia: rJ - nx- Then it is well known that n - rm,and we now see that Again let ?I be arbitrary and let a -, at be an isomorphism of ?t onto a second non-associativealgebra ?I-. Then d is a 1 :1 linear mapping of ?I onto 2I-and (ab)e: a0b0,which implies that (3) a70 : o(at), , b*0 : 0(bo)n. Thus ( 3')
o-tazo - (ao)",
o-rb*o : (be)* .
This implies that the mapping x-+0-tx0 is an isdmorphismof the multiplication algebra !(u) onto !([). It is clear ilso that T -) T0: 0-'T0 is an isomorphism of the centroid r([) onto r(D. lf aeDl we have (ar)o ._qaee-try: a0T0. In particular, if ?l is simple and ?I is consideredas a vector spaceover 1., then we have (4) (ra)t : Toao Next let D be a derivation in ?I. (aD)b * a(bD), which gives
Then D is linear I and (ab)D _
(5)
Ia", Dl: (aD), , fb*, Dl: (bD)* This implies that the inner derivation x-+ [x, Dl in o(ll) maps t(?I) into itself. consequently,this inducesa derivation T'-+Ti,= _ lrDl in r(tr). By definitionof Tu we have (ar)D_(aD)r*a7d so that for simple ?l we have (6) ja)D: T"a + r@D) . we can state the results we have just noted in the following convenient form. Tsponpu 5. Let ?r and E be centrat simble non-associatiaealgebras ouer a rt,eM T and tet A b_ea subfietd.of f . Then any isomorphisrn 0 of \ ouer a onto E oaer o is a semi-liineartrinsformation of a ouer T onto E oaer l. Any deriaation,D in T ouer o satisfies(ra)D: Tda+ r@D), T e T, a ea where d i,s a deriaation in T as a fi,eld oaer O. we add onefurther remark to this discussion.suppgsE(ur,...,ilo) is a basisfor ?I over /- and that u;tr: : Z T;irur, Iiir G /', gives the lrultiplication table. A I : 1 semi-rineartransformatiOnd of ?I onto ? I -m a p s t h e u r i n t o t h e b a s i su ! r ,i : 1 , . . . , " tiiL,o*r-r. * a is a @-isomorphism we have utol: Zrtiouf; where d acting ol Tit*
X. SIMPLE LIE ALGEBRAS OVER AN ARBITRARY FIELD
2%
is the automorphism in /' associated with the semi-linear mapping 0 of. It. In many cases which are of interest one can choosea basis so that the rtir e @, that is, ?l : ?Iop where ?lo is a non'associative algebra over o. Then we have ulul : 2 rrtoutowhich implies that ?I and I are isomorphic as algebras over /-. 2.
Isomorphiern of extension algebras
Let tr be a non-associative algebra over O and let P be a finitedimensional Galois extension field of A, G : {1, s, . . . , u} the Galois group of. P over O. In this section we shall obtain a survey of the isomorphism classes of non-associative algebras E such that 8.. : ?Ip. We recalt that if ?l is a vector space over @ then I can be identified with a subset of ?I". This subset is a O-subspaceof ?Ip which generates ?Ip as space over P and has the property that any set of elements of ?I which is @-independent is necessarily P-independent. Moreover, these properties are characteristic. If (Qr,Qz,. . ., Q*) is a basis for P over @, then any element of ?Ir has a unique representation in the form 2 p;)c;,tr € ?I' If s e G, then s defines a semi-linear transformation U, in ?I" by the rule
(7 )
(\ oux)U,: 2 pln .
It is easy to check that (J" is independent of the choice of the basis in P over O, that the automorphism in P associated with U' is s (that is, (tr) U": E (xU"), E e P, x e Ur) and (8)
U"UI:U*,
Ut:t-
Hence the U, form a group isomorphic to G. If we take pt - 1, then the elements of ?I have the form Prxr,xr e ?I and it is clear from (7) that these elements are fixed points for every U", s e G. It is easy to show directly that ?I is just the set of fixed points relative to the U". This will follow also from the following basic lemma: Lpuu.c, 2. Let tl be a aector space ouer a fi.eld P which is a f.nite-d.imensional Galoi,s extension of a fi.eld O. Suppose that for each s in the Galois group G of P oaer O there is associated a semi-li,near transformation U" in ?I with associated autornorphism s in P such that (Jr: L, (J,(lt: U*. Let 2I be the set oL fixed (J,, s e G. Then V is a O'subspaceof ?I such \oints_relatiue to the that 2I ?Il".
LIE ALGEBRAS
Prgof: One verifies directly that lI is a @-subspdceof [. If rce ?I and p e P, then 1,66p"(x(J,) e ?I since (A"p'(x(J,))U, : 2"p't(xU,(Jr) : Z,p"'x(J,, - )"p'(x(I"). Let (pr, Qz,. . ., An)be a basis f.or P over o. Then it is well known that the n x n matrix whose rows are (pi, pi, .. ., pL), s € G, is non-singular. It follows from this that r is a P-linear combination of the n elements 2,pi@(t,) which belong to ?I. Thus the p-space m spanned, by ?I is fr. Next let frr, ' '', fr, be elements of 2I which are @lindependent. Assume there exist E; € P, not all 0, such that Zi|iqi : 0. Then we may take r minimal for such relations and *" Ftruy suppose t' : 1. Evidently we have r ) L and we may assumd that Er 6, o. Then 0 : (Xif 6)(J, : lfi]xi and we can choose s iin G so that Ei + gr. we can therefore obtain a relation 0 - xlF.r h - zi€lxr : l,i-r(Er - Ei)x' and this is non-trivial and shorter than the relation This contradiction shows that the rr.'s are p-inde}ilixi :0. p e ndent . Henc e fr:? 1 .. lf x : Z E f i; , € t € P , 1 1 e ? I, th e n r(J ,:Z €' txt. ffhus the (J, are the transformations we constructed before for trn. we return to the situation considered before in which rve were given [l : lI. and we defined the u" bv (z). we saw that ll s g, the @-subspace of fixed elements relative to the t/". On the other hAnd, *. hu.r. fl: ?r" - Ep. Let {e,) be a basis for ?l over o alrrd,let b e E. T h e n 6 : X E $ t E r e .P , x , i e { e , } . S i n c e t h e r r a n d D i a r e i n E t h i s relation implies that these elements are o-dependent and since the x; zte O-independentwe have D: X d.tx;, d,;e (0. Hence De?I and E : 2l so that lt is just the set of fixed elements relative to the U,. Our results establish the following Tnnonou 6. Let fr be a aector space oaer p, a fi.nite-dimensional Galois ertension of the fietd o. be a:(finite) set of Jut {u,ls€G} semi'linear transformations in ?r ouer p such that:i (1) (Jr: L, u,u1 : u", (2) the automorphism in P associatedwith u, is s. Let 2I be the A-spaceof fixed elements relati.ue to the U,. ,,Then fr : U" and the correspondence{t/,}-?I is a bijection af thq set of finite sets of semi-linear transformations satisfying (r) and (p) and the set of A-subspaces2I of Lsailsfying fr: 2I.. Now assume that ?r is a non-associative algebra over o. Then it is easy to check that the u" defined by (T) are alrtomorphisms of ?I - ?Ip considered as an algebra over A. Convprsely, if the group { U"} of semi-linear transformations in fr is given and every
X. SIMPLE LIE ALGEBRAS OVER AN ARBITRARY FIELD
297
U, is an automorphism of fr over O then the set ?I of fixed elements is an algebra over A. Hence the correspondence of Theorem 6 itduces a bijection of the set of groups {U"}, U, za automorphism of ?I over @ and the set {?I} of O'subalgebras ?l such that ?Ir -fr. Let ?I and E be two of the t['algebras such that 2I,e: Ep and suppose A is an isomorphism of ?I onto E. Let {%} be the group of semi-linear transformations associated with A, {U,} that with 2I. The isomorphism A has a unique extension to an automorphism A of. fr over P. The mapping A-'(J"A is a semi-linear transfor' mation in fr with associated automorphism s in P. Also A-'U;A: The set of fixed points rela' A-'U"A. 1 and (A-'U,A)(A-'U.A): V", s e G. tive to the A-t(J,A is the space E. Hence A-'U,A: Conversely, let {U,}, {V"} be groups of semi'linear mappings as in Theorem 6 such that the (J,, V, are automorphisms of ?I as nonassociative algebra over O and assume there exists a P'auto' morphism A of. fr such that
(9)
V,:A-'U"A,
sec.
Then 2IA: E for the associatednon-associative algebras 2I and E. Hence ?I and E are isomorphic as algebras over O. We therefore have the following THnonsM 7. Let fr be a non-associatiae algebra ouer a fi'nite dimensi.onal Galois extension P of a fi.eld O. Then the corre' spondence of Theorem 6 between sets of semi'linear transformati,ons (2) and O'subspaces V such that 2Ie : lfJ,j in E satisfyins $) and E induces a bijection of the set of {U") such that eaery U" is a O-algebra isomorphism and. the W which are A'subalgehras of 8. The corresponding O-subalgebras_areisomorphic if and only if there erists an autornorphism A of V oaer P such that (9) holds for the associated groups. If E is a non-associative algebra over O such that Ep : ?Ie, then E can be identified with its image in fr : ?I". In this way we see that Theorem 7 gives a survey of the isomorphism classes of algebras E such that Ep =ei, via certain similarity classes of groups of automorphisms of fr over A. This is the type of result we wished to establish. In the sequel we shall be concerned with finite'dimensional nonassociative algebras over a field O of. characteristic 0 and we shall be concerned with the question of equality Vo : Eo for two such
2W3
LIE ALGEBRAS
algebras, g being the algebraic closure of the base field o. Let (er, "',€*), (fr, -..,.f^) be bases for ?I over o and E over o respectively. Then the equality ?Ip: Eo implies thal fi: Z p;iei, p;te Q, i,i --1, "',ffi. Let J be the subfierd over oigeneratedby the p;1. Since €v€r}' p11 is algebraic over o and thebe are only a finite number of these, , is a finite-dimensional extension of t0. since J? is algebraic and of characteristic zero, J isl contained in a subfield P of' J? which is a finite-dirnensional Galois extension of O. Since the prie P it is clear that ZTpfrtrZ1pei and the converse inequality holds since et : Z p,;tfi, ei) : @i)-r. Also it is clear that the f; and the e; both form linearly independent sets over P. Hence \Pe;-?Ip and ) p.fo: Er, so ?Io:,Eo for ;7 the algebraic closure of the base field A implies ?[p = Er for p a suitable finite-dimensional Galois extension of. 0. 3.
Simple Lie algebral of types A-D t
we now take up the main problem of this chapter: the classification of the finite-dimensional simple Lie algebras over any field, o of characteristic 0. If 8 is such an argebra and /. is its centroid, then /' is a finite-dimensional field extension of @ arrd g is finitedimensional central simple over r. converscly, if g is finitedirnensional central simple over any finite-dimensional extension field /- of a, then 8 is finite-dimensional simple overia. If gr and 8z two isomorphic simple Lie algebras over @,thdn the respective"r€centroids f, and Tz isomorphic so both algebras *"y U" "te considered as central simple over the same field | -TrzTz. Moreover, Theo'em 5 shows that we have a semi-linear transformation d of ,8r over /. onto 8z over /. such that d is a O-isomorphism of 8r onto 82 as non-associative algebras over o. These results reduce the classification problem for simple Lie algpbras over O to the following problems: (l) the classification of the finite-dimensional field extensions /. of o, (z) the classification of the finitedimensional central simple Lie algebras over the f in (l), and (3) determination of conditions for the existence of a semii-linear transformation d of two central simple Lie algebras ovef /- such that d is a tD-algebra.isomorphism. To make this concrete we consider the important special case: o the field of real numbers. Here any algebraic extension field f of o is either @ itself or is isomorphic to the field of complex numbers. If. f is the field of complex
X. SIMPLE LIE ALGEBRASOVER AN ARBITRARY FIELD
2W
numbers, then /- is algebraically closed and we know the classification of the finite-dimensional simple Lie algebras over l-. We recall that the algebras in this classification (Ar, Br etc.) all have the form 80",80 an algebra over O (Theorem 4.2). It follows from the remark following Theorem 5 that the algebras in our list (which are not l--isomorphic) are not isomorphic over @. Thus, to complete the classification over the reals it remains to classify the central simple algebras over this field. Now suppose 8 is a finite-dimensional central simple Lie algebra g is the algebraic over /' which is any field of characteristic 0. If closure of /a, then 8o is simple. Conversely, if 8 is finite-dimensional over /'and 8o is siniple, then 8 is evidently simple so its centroid f' is a finite-dimensionat field extension of l'. This can be considered as a subfield of g. Then 8", iS simple since (8r')o: 8o is simple' Hence 8 is centrat by Theorem 3. Thus 8 is central simple over l' if and only if ,8o is simPle over .o. Since g is algebraically closed of characteristic 0 we know the possibilities for 8o . They are the Lie algebras 4r, I > L, Bu I 2 2, C,, l2 3, D1, I > 4, Gr, Fr,Eu, Er, E in the Killing Cartan list' If 8o is the Lie algebra X in this list then we shall say that 8 is of usually the subscript / will be dropped and we shall tyfe x. speak simply of g of. type A, type B, etc. For each type X we shall choose a fixed Lie algebra 8o of this type. For example, we can take 8o to be the split Lie algebra over /' of type X. Then our problem is to classify the Lie algebras 8 such that 8o: 8oo' For a particular 8 there exists a finite-dimensional Galois extension P such that 8p : 8oe and we have seen that to determine the 8 which satisfy this condition for a particular P, then we have to look at the automorphisms of 80p over /'. We shall study the cases of 8o of types, At, Bt, Ct, Dt, I > 4. In this section we shall give some constructions of Lie algebras of types A to D. In the next section we give the conditions for isomorphism of these Lie dlgebras and in $ 5 we shall prove that every Lie algebra of type At, Bt, Ct, Dt, I > 4 can be obtained in the manner given here. The starting point of our constructions is the fundamental Wedderburn structure theorem on simple associative algebras: Any finite-dimensional simple associative algebra U is isomorphic to an algebra @ of all the linear transformations of a finite'dimensional .ruito, space St over a finite'dimensional division algebta r'. An equivalent formulation is that ?I : y'n the algebra of n x z matrices
3OO
LIE ALGEBRAS
over a finite-dimensional division algebra l. (For I a proof see: Jacobson, structure of Rings, p. gg, or Artin, Nesbitt, and rhrall, Rings with Minimum condition, p.gz.) If the base field t- is alge. braically closed then the only finite-dimensional division algebra over f is f itself , so in this case w: Tn for some zi If the base field is the field of real numbers, then it is known that there are just three possibilities for y': y' : T, / - T(i) the field of complex numbers or y' the division algebra of quaternions. i (Theorem of Frobenius, cf. Dickson [1], p. 62, or pontrjagin I1l, t p. 1ZS.) The center o of a finite-dimensional simpre associative algsbra is a field and the centroid consists of the mappings cr: cn, c e G. This can be identified with the center. If ?I is central (G - f)l and g is the algebraic closure of T, then ?Io is finite-dimensional simple over g. Hence ?Io= 9n for some n. Since [?I : f] : [Uo : gl = lg*: e) : 1xz this shows that the dimensionality of any fi.nite-dimerhsional central simple associatiue algebra is a square. Let 2I be a finite'dimensional central simple assocfative algebra over /-. Consider the derived algebra g _ ?Ii of the Lie altebra wb If g is the algebraic closure of l', then go: (fli)o : 1ul;i = g'nu since ?Io* 9o. On the other hand, we know that if n _ I * l, I > L, then gLr, is the Lie algebra of (/ + 1) x (/ * r.) matrices of trace 0 and this is the simple Lie algebra At over g. Since go = gLt it follows that 8 - ?IL is a central simple Lie algebrb of type Ar. Next let ?I be a finite-dimensional simple associative algebra with an inuolution J. By definition, / is an anti-automorpfrism of period two in ?I. Hence -"/ is an automorphism of ?Ir. Tfie set g(U, /) of /-skew elements (a" : -a) is the subset of fixed blements relative to the automorphism -/. Hence this is a subhlgebra of ?rz. The anti'automorphism / induces an automorphism i in the cenrer 6 of E which is either the identity or is of period two. In the first case./ is of first kind and in the second / is an involution of second kind. we assume first that / is of second kind. More precisely, w€ assume that G : P: r(il a quadratic extension of tlie base field f an_dthat q" : -q. Let 0(2I, D be the space of /-symrrfetric elements (o'-o). A n y a e ? I h a s t h ef o r m a _ b * c , b_LAll+a')iO c : L @ - a ' 1 e @ . I f D e b , q b e 6 a n d i f c e 6 t h e n E c e g . H e" n" ac e the mapping r -' qx is a 1 : 1 linear mapping of 6i onto 6 so we have the dimensionality relation: [€ : f] - [6: TI. Since 6 n 6 _ 0 and ?I-6+O, ?I-6Sg and [?I:r] _2[g:r]. Thus [?I:p]_
X. SIMPLE LIE ALGEBRAS OVER AN ARBITRARY FIELD
3OI
We recalt also that [?I:P] is a square z', so +tlt, Tl:[6:f]. ( a r , . . . , a n z )b e a b a s i s f o r 6 o v e r / ' . T h e n e v e r y : n 2 . L e t ig,fl the form \iza,ra; + [i'|n(qar) : \ p;ail P;: ?I has .t"tn.nt of It follows that the a; form a basis for ?I (or ?Iz) T. d;, e d; * F;Q, F; - 2Ir over P. Let I be the algebraic over P. This implies that 6p closure of T which we may assume to be an extension of the field P. Then 6o: (6p)o : (U, over P)o -- Qnr.since (?I over P)a = 9"' Let 8 be the derived algebra @' : e(A, n' . Then 8a = QLz so 121,8o lf n-l+t, [ 8 p : g ] : n 2 - 1 a n d [ g ( U ,h ' : f ] : n ' - I . of type 8 simple is central Hence is the simple Lie algebra At. At. We have now given two constructions of Lie algebras of type A' We summarize our results in the following Tnponpru 8. Izt 2I be a finite-d,imensional central simple associatiae algebra ouer T, 2I + f . Then the deriaed algebra DIL is a central sllmpu Lie atgebra of type At, I > r. Int 2I be a fi.nite-dimensional the simple associ,itiaealgebra with center P a quadratic extension of kind. second of possesses inaolution an ?I J Oasi fieU T and, tuppott g(U,,D be the Lie algebra of shew Supiose also that Il + P. I-et algebra @(V, I)' is a central simfle the deriaed, etiments of W. Then l. I At, tYPe > of Lie algebra Before pro.""ding to the discussion of the Lie algebras 9(?I, /), involutions in / of first kind we quote some well'known results on (cf. Jacobson [3], pp. 80-83). algebras of linear transformations Let ![t be a finite-dimensional vector space over a division algebra y'' I and let G be the algebra of linear transformations in Sl over -' Ar if and only if Then it is known that G has an involution A two. If the d of d-'-/ has an anti-automorphism -period one or : l- is _com' that implies drd, d and ffir: period is one, then d y' d = d' take can one then if is commutative, mutative. Conversely he-rmitian --+d non-degenerate a define can one given, then is lf. d, y' relative to d -'+d. Such or skew hermitian form (x, y) in lIt ovet a form is defined bY the conditions (10)
(r, + xz,!) -- (trr,y) -t (xz,!) , (dr, y) : d,(x,y) ,
(x, y' * yt) : (x'y') * (r' y') (x, dY) - (x, Y)d
for x, )Ct,JCz, !, !r, lz e IIl, d e /,
(11)
(r,:y) : (f,x)
or
(x,y) - -T, x) ,
302
LIE ALGEBRAS
according as the form is hermitian or skew-hermitian, and nondegeneracy means that (r, z) : 0 for all r implies z .+0. If tt = d, then we obtain a symmetric or skew bilinear form. lf',(ut,ctz,.. .,14o) i s a b a s i sf o r ! f t o v e r y ' , t h e n ( x , y ) : Z E { , ; for x*ZE;r4;, !: Znrui defines a non-degenerate hermitian form and (r, y) : ) E;prr; is a non'degenerate skew hermitian form if. V - - p + 0. rf. tl-= d, then non'degenerate skew bilinear forms exist for sl if and only if !m is of even dimensionality over /'. Let (x,y)r be any nondegenerate hermitian or skew hermitian form associated wittr d -- tt in l. lf. A € 6 we let Ar denote the adjointof. ,A,rel4tive to (r,y), that is, Ar is the linear transformation in !ft such that (r4, y) _ (r , t A ' ) f or x , - y e St. T h e n i t i s e a s i l y s e en that' A -l " ' i -s an involution in @. Moreover, one has the fundamentalr theorem that every involution t of.0 is obtained in this way. In particular, supposey' : /- is an algebraically closed field. Since y' : T, the only anti-automorphism of. / as algebra over /' is the identity mapping. Hence (x, y) is either a non-degenerate symmetric bilinear form or is a non-degenerate skew bilinea, form. The latter can occur only in the even-dimensional case. The Lie algebra g(@, determined by the form is the Lier algebra Bt if. "I) the form is symmetric and dim ![t :21 * I. The Lie algebra g(@,,/) is cr if the form is skew and dim m - 2l and it i. D, if the form is symmetric and dim Sl :2,. Now let ?I be a finite-dimensional central simple associative algebra over /" which has an involution / of first kind. If g is the algebraic closure of f , then ?Ioz !)o. The extension ! of. J to a linear transformation in ?Io is an involution in ?Io and 6(?I o, _ D @(21,I)o. Since 2Io= Qn and / is an involution in g, the result above shows that 6(?Io,./) is one of the Lie algebraS Bt, Ct or Dt. We assume now that l >- 2, I2 3 or l > 4 according as 6(?Ie, is /)'see Bt, ct or Du Then the algebras @(?Io,J) are simlle and we that (?I, "/) is central simple over A of. types Bu Ct or D1. We recall that n is the dimensionality of the space lll considered before and that n' : fQo: Qf - [?I : r]. For Br we hdve n - zl * r and [g(?I, I): I']: [O(?Io,D: A : t(Zl + 1). For Ct, n : 2l and I@: fl: K2l + l) and for Dt, n : Zl and [@: f] : l(Zt - 1). We can now state the following Tnponpu 9. Let ?I be a finite-dimensional central sirnple associatiue algebra ouer I of dimensionality nt and suppose 2r hus an inuolution
X. SIMPLE LIE ALGEBRAS OVER AN ARBITRARY FIELD
303
q(}I, D be the Lie algebra of J'skew elernents of J of fi.rst ki,nd. Let fu. If n : 2l * L, then [g(u, I): rl : l(21* r) and 6 es central : l(21 * t) or simple of tyfe Bt for t > 2. If n :21, then [@ : /'] simple central is Inthefornrcr case a3surne,= 3. Then @ tef-D. I > 4. : assulne we then l(21 L) of type ct. If n -2t and t6: rl Then @ is central simPle of tYPe D4.
Conditions for isomorPhiem
Let 8, and lJz be finite-dimensional central simple Lie algebras f over the field /- of characteristic 0- Suppose O is a subfield of A:8r:o8r. and that 8r and 8z are isomorphic as algebras over Then we know that a @'isomorphism 0 of ,8, onto 8, is a semithe linear transformation of 8, over /' onto 8z over /'. We denote : (ra)' have associated automorphism in l' by d also so that we / ' w i t h o v e r T t o t i f . T e r , a e g r . L e t ( a t , . , . , a * ) b e a b a s i s f o r S , ' ' ', ot^) is a the multiplication table la;ai7: Z Tiixa*; Then @1, algebraic the : I be Let basis for 8z over l- and laqtall X rlioa?,. closure of. T. Then it is a well'known result of Galois theory that d the automorphism 0 in T has an extension to an automorphism g basis a form a0' the and 8ro over for basis in 9. The ar, form a to; € 9, ) ortna'r, for 8zo over g. It follows that the mapping \a&r+ other the =o$zo. On $qo Thus 8zo. 8rp onto of is a @-isomorphism g' Hence field closed algebraically the over hand, 8ro is simple it has a basis over J? whose multiplication coefficients are in the prime field and so are in O. This implies (remark following Theo' remS)that8ro4ogzo.Wehavethereforeprovedthefollowing Lpuu.o,3. I-et \ and' gz be finite'dimensional central simple Lie algebras ouer a. rt,eU.f of characteristic zero- Izt 0 be a subfi'eld ol T and' 9 the algebraic closure of r' Then 8' 3o 8' implies 8ro *p8zo' This result implies that the only @-isomorphismswhich can exist for the Lie algebras of Theorems 8 and 9 are those between the algebras defined in Theorem 8 and between algebras of the same type (8,, Ct, D) of Theorem 9. We shall call the Lie algebras of tite torm ?I! of Theorem 8 Lie algebras of tyfe Ar, those of the form g(?I, I)' , I of second kind, Lie algebras of type An' For the latter class we assume / ) 1 if the type is Ar. This amounts to assumingthatt?I:Pl:n'>4'Weshallsupposealsofromnow on that t > 4 for the algebras of type Dr. We consider next the
304
LIE ALGEBRAS
enveloping associative algebras of the Lie algebras of Theorems 8 , 9. LsuMA,4. Let o, T, e be as in Lemrna s. (r) Let tr oe a finitedimensional central simple associatiaealgebra ouer I of Eimensionality n" > I. Then the enueloping associatiuealgebra of g 12IL ouer A is ?I. (2) Let 2I be a finite'dimensional simple associatiubalgebra with center P a quadratic extension of T and with an inuolution I o/ secondkind. Assume [?I: P] : n, > 4. Then the enueloping associatiue algebra of 8 : 6(?I, I)' oaer o is rr. (3) I*t rr bq finite-dimensional central simple ouer T with an inuorution ! of first kind and Then the enueloping associati.uealgebra of g -[?I: rJ : n' > l. Proof: we note first that in all cases the enveloping associative algebra of 8 over @ is the same as that of g over its centroid /.. T h u s s i n c e S i s a v e c t o r s p a c eo v e r / . , y l e g i f 1 6 / . a n d / e g . Hence the two enveloping associative algebras indiQated coincide with the set of sums of products 1,,1,.. . lt, lr e g. This remark shows that we may as well assume the base field b - T and we shall now do this. In the cases 1 and 3 we introduce tire algebra ?Io: !)n, fl ) 1, and we consider 8o which is a subalgebru of !2*". We know that 8o is the Lie algebra of matrices dl, 8,, C, or Dt as defined in g 4.6. In all cases an elementary dirdct calculation with the bases given in g 4.6 shows that the g-subalgdbra generated by 8o is Qn, that is, (8o)* - lJn where the * denotes fhe enveloping associative algebra. If 8* denotes the enveloping aspociativ. utg.bra over a of 8 (in ?I), then it clear that the o-subspace of. e* spanned by 8* is (t3o)*: gn. Hence 8* contains a Uasis for gn over 9. Since the elements of this basis are conitained in ?I ii follows that they constitute a basis for ?I. Thus we have g* : lI. The argument just used cannot be applied readily, to the case Z (Lie algebras of type Au) since in this case 2lo:,gnD!)n. We therefore proceed in a somewhat different mannbr. Let p _ 0(q) where qt : -q, as before. Then q e 6(til,jD and gt g.(U, n + qg@, Hence it suffices to show that I) : g(?I, ,/)*. 8*= 6(?I,). Since0-T a n d ,B - @ , w e h a v e L 6 : t \ ) : n z a n d [8: @]: n' - l. Hence, it suffices to show that g* contains an element a e @, e@'. Assume the contrary that g* n g : 6,. Let bi, i:1,2,3, be skew elements. Then it is immediate that {brbz6r}= brbzbs + bsbzbtis skew. If the bt e @', then {6,Dr6r}€ @' : g* n-6.
X. SIMPLELIE ALGEBRASOVERAN ARBITRARYFIELD
305
We now consider the algebra (?I over P)o: 9,. We have seen that this algebra has a basis consisting of elements of 6 and that g'*r. has a basis of elements in 8 : 6'. The multilinear character of. {brbzbs}now implies that {b,brbr)e g'u for all br € 9',r,- If we take bi: b this implies that Ds€ 9'*r, if b e g'"r;. Thus we must have tr D' : 0 for all b satisfying tr b : 0. This is impossible since n t 2. For example, we can take b : Zer,- €zz- €ssso that tr b : 0, tr b' :6 + 0. This contradiction shows that 8* = O and 8* : ?I. We are now ready to prove our main isomorphism theorems. In all of these A, I' and I are as in the foregoing discussion. Tnponsu 10. Let 2I and E be fi.nite'dimensionalcentral simfle associatiuealgebras oaer I' and. let 0 be an isomorphism of ?Jl ou€r O onto E!. ou€r 0. Assume [?I: f] : nz ) | and [E: /-] > 1. Then if n :2, 0 can be extended in' one and only one way to an isom o r p h i s m o f? J o u e r o o n t o E o u e r o a n d i f n > 2 t h e n 0 c a n b e extended in one and only one way to either an isomorphism or the negatiue of an anti-isomorphism of W ouer CI onto E ouer O. Proof: We have 2lo= Q*andEna g^. By Lemma 3, !J'6 =oE!,oi hence Q'*, = 9'*r. and m : n. Thus we may assume that ?Ia: 9n: Eo so that Q* has a basis (ar, . . . , ant) such that the at form a basis We may assulrlealso that (ar, "' , anz-t)is a basis for ?I over f. for W', over /'. Now d is a semi-linear mapping of. 2I'r.over I' onto Si over f whose automorphism in I' we denote by 0. Then @ 1 ,.'-, at r - , ) is a bas i s fo r El o v e r /' . If 0 i s e xtended to the is an automorphism automorphism 0 in 9, then Ll'-'r2tar->rtno! 0' in 9'*r, over @ which is a semi-linear transformation with asso ci a te d aut om or phis m 0 i n 9 . L e t (e i ), i ,i :I, " ' ,fl , be the usual matrix units f.or Qo. Then the mapping \ at;i€;i-r !, ofti€ii, o;; € g, is an automorphism 0" of the associative algebra Q* over 0 whose associated automorphism in A is 0. d" induces an auto' morphism 0" in the Lie algebra 9'*r. and since 0' and d" have the same associatedautomorphism in Q, T : (0")-td' is an automorphism in Q',t over 9. By Theorem 9.5 this has the form X -' M-'XM if. i f' n:2 a nd it eit her ha s th i s fo rm o r th e fo rm f,- + -M-' X ' M !)" of an is automorphism n > 2. Since the mapping X-+ M-'XM and X -, M-'X'M is an anti-automorphismof Qn, 7l can be realized by an automorphism of. !l* over Q if. n - 2 and either by an automorphism or by the negative of an anti-automorphism of. Qo over 9 f.or n > 2. Then 0' : 0"T can be extended to an automorphism
LIE ALGEBRAS
C of 8" over o if n - 2, or to an automorphism ( og the negative of an anti-automorphism C of g, over A if n > Z. Since d, Is the mapping 2 a&r+ ) ,tal it is clear that 0tcoincides with the given 0 on LrL. Since the enveloping associative algebra over o of ?Il is ?I and that of al is E it follows that c maps r[ onto E and consequently 0 can be extended to an isomorphism or the negative of an anti'isomorphism of ?I over o and E over @. sinc0 ?Il ienerates ?I over @ it is clear that the extension is unique. This result shows that if ?Il and El are isomorphib as algebras over o, then ?I and E are either isomorphic or anti.tisomorphic as o'algebras. The converse is clear since any isombrphism of ?I onto E induces an isomorphism of ?Ii into Ei and if d is an antiisomorphism of ?r onto E then -d induces an isomorphism of ?ii onto El. our result also gives a description of the group of automorphisms of ?Il. If n - 2 or if ?I has no anti-automprphism, then the group of automorphisms of III over O can be itdentified with the group of automorphisms of 2I over a. u n)z and ?I has an anti'automorphism /, then it is easy to prove by a feld extension argument that the automorphism a -, -at in ?Il is not of the form a - eo, 0 an automorphii sm of ?I. It follows that :the group of automorphisms of ?I over o is isomorphic to a subgfroup or ina"* two of the group of automorphisms of ?Il over 0. If we take o : l, then it is a known result of the associative theory that every automorphism of ?I over o is inner. This could also be deducedifrom the form of the automorphisms of. ll,*" u, field extension argument. Then it follows thai the automorphisms" ot ?I1 over A are of the form x -, m-,trln or of the form x --, -m-r)c"m where J is a fixed anti-automorphism in ?I. Tnponpu 11. lzt ?Ii, i:1,2, be a finite-dimensional simple asalgebra ouer I with center a quadratic fietd 'itti*tp; - r(i) and
yfoliy!
t?I':';;t n?>4 and let a be a subfietdof r. Then any o-isoworphism0 of 6(2[', J)t onto 6(?Ir,J)' can beextendedin one and onry one way to a 0-isomorphismof 21,onto 2Iz. The Lie algebra @(?[,, Jr), is-not isomorphicto any Lie algebra EL of type Ar. Proof: We may choosea basis (ar, . .., e,,z)for ?It over P, so that (ar, . .. t anz-r)is a basis for 6(?11, /,)/ over I. and fof ?I{z over Pr. If .? is the algebraic closure of. T (chosen to contdin P, and Pr), then (?Ir over Pr)o: !)n, so 6(2t,, fr)o: gnr" and 6(?I,, f)' : O'^r".
X. SIMPLE LIE ALGEBRAS OVER AN ARBITRARY FIELD
307
Similarty, w€ have 6(Ur, h)b = Q'*rr. Assume there exists a a' Then we know that isomorphism of @(fl',,I')' onto Q(9Ir,h)'. g'*r"=oQt*rz so nr: tlz: n andwe may assume that ?Ir and ?Iz are @-subalgebrasof g,nsuch that any Pr-basis for ?Ii is a basis for 9". Let 0 be a @-isomorphism of 8r : 6(U,, /r)t onto 8z : @(21r,Jr)'. Then d is semi'linear in 8, over /' onto 8z over l' with associated automorphism 0 in T. If. (au ' ' ' , anz-r) is a basis for 8r ov€r /", then (al, . .., at*r-r) is a basis for 8, over l' and both of these are bases for Qt*z over g. If the automorphism d in /" is extended to an automorphism d in g, then the mapping 2l'-'tror-'-2co1la0; is an automorphism in Q'*t over A whose associated automorphism in g is 0. The proof of Theorem 10 shows that this can be extended to an automorphism or the negative of an anti-automorphism of the associative algebra gn over o. Since the enveloping associative algebra of @(?[, ])' over @is 2Ir it follows that the isomorphism d on 8r over O can be extended to an isomorphism or the negative of an anti-isomorphism of U, ovet O onto ?Ie over O. If the second possibility holds let ( denote the anti-isomorphism. Then /'C is an isomorphism of ?1, onto ?Iz and if a e 6(?I,, /')' then a0 : -a€ -- qrr( since arl : -a for a e @(VL,I)| . Hence /t( is an isomorphism of ?Ir which coincides with d on 6(?I b ft)' . Thus in every case we can extend d to an isomorphism of ?It onto ?Iz. This extension is unique since ?Ir is the enveloping associative algebra of O(U,, ,I,)'. This proves our first assertion. Next let 8r : @(?I,,./,)' and suppose we have a O-isomorphism d of 8r onto 8z : Et where A is central simple associative over l'. The argument just used for ?1, and ?Iz shows that 0 can be extended to an isomorphism d of ?I, onto E. Since the centroids of the Lie algebras 6(?I',.I,)' and Ei consist of the multiplications (x-t yr) in these Lie algebras by the elements T e f it fotlows that 0 maps l" into itself. Hence Pr: l@r) is mapped into a subfield of E which properly contains f . On the other hand, Pt is the center of ?I, so this is mapped into the center f of E. This contradiction proves that 8r: @(Sr, "/r)' cannot be isomorphic to El. We consider next the Lie algebras of types B, C and D which we defined in Theorem 9. The main isomorphism result on these is the following THoonsu 12. I*t Vi, i,: L,2, be a finite-dirnensional central simlle associatiae algebra oaer T such that Vi has an inuolution t; of f.rst
LIE ALGEBRAS
hind. If [\: r] : n?, then [O(?I,,J): I,] : l;(Zh * L) V nn: Zl,i,* L and [@QI;, J): T]: I;(2h* t) or t;(21;- t) if ni:21;. the respec-
tiae cases indicated we assulne that l;2 Z, h 2 3 , l r ) t 4"In . Then if A is a subfield of f , any O-isomorphism 0 of g(U,, l) ont7 @(?lr,!) can be extended in one and only one way to a O-isomorphism of V1 onto
u,2.
Proof: If g is the algebraic closure of O, then W";sv gn6 and CI(2[r,l)o is the..lil.algebra Bt, ctn or Dlnaccording zs ?tt : zrt * r, -t) rti : 2lt and [6(?I;, J) :/-] : tt(2t; * o, nr : Zliand [E(?I;, f;) :/-] : l;,(21;- 7). It follows that if g(?I,, Jr) =r@l(Wr,lr) then nr!' 1z: lt and we may suppose that ?Iro: go - plzoand g(?[r, Jt), - @(Vr,Ir)o is either the Lie algebra of skew symmetric matrices in p" (ivpes B or D) or the Lie algebra of matrices satisfying @-'A,e:'_A, I a skew symmetric matrix with entries in trre prirne field (type c). Let (at,...,a*) (m:l(zt+l) or t(zl r)) be a basis for @(u', /r) over /. and hence for 6a over o. If d is a o-isomorphism of 6(?I', /r) onto @(V2,-J2), 0 is semi-linear with associated automorphism 0 in f and (aer,.. ., ol) is a basis for 6(?Iz, /2) over /. and for @o over 9- If d is extended to the automorphism d in o then Zaia;+ ! ,ulal, otr € g, is an automorphism 0' in 6o ov€r o with d as its associated automorphism in e. If (e;r.) is a usual matrix basis for g* over g then X at;i€ii-r ) aeiieli is an aut5morphism d// of 9o over J? whose associated automorphism in g is id. Moreover, since I has entries in A, d,, maps 6o into itself and so it induces an automorphism 0" in 6o over @. It follows that 0t70rr1-ris an automorphism of @o over a. In all cases this rhas the form x-+ 114-txM and so it can be extended to an inner automorphism of !2n over g. It follows that 0' can be extended to an automorphism ( of 9, over Q. Since the restriction of ( rto 6(1I,,,I,) is the given d and since the enveroping associative alge$ra gifr,, /,1 .Ui, is c maps ?I, isomorphically on rlz and coiti.ides "r with' 0' on 6(u', "/'). Thus 0 can be extended to a o-isomorphism of ?I, onto ?I2. Since O(2t,, /,) generates llr this extension is unique. 5.
Completeness theoreml
Let 8 be a finite-dimensional central simple Lie aflgebra of type A over r. Then there exists a finite-dimensional Galois extension
field P of. I such that 8p = P1,", n 2 2. Thus we m4y suppose that S i s a l'-subalgebra of Plz such that the p-space spanned by g is
X. SIMPLE LIE ALGEBRASOVER AN ARBITRARY FIELD
309
P|, and elements of 8 which are /'-independent are P-independent. We have seen also that for each s in the Galois group G of. P over x there corresponds an automorphism lJ, of Pl,,t oYet I' such that (px)U, - p'(x(J"). If (ar, " ', Q'oz-r)is a basis {or 8 over /-, hence We ior P!,, over P, then (J, is the mapping >l'-'P;a;--+2pia;. (J,(Jt: (J, elernents set of is the and 8 which U* \76ait I, have T;e T, is the set of fixed elements for the U,, seG. We have seen in the last section that U" has an extension U' in the enveloping associative algebr? P* of. Plt, such that U' is an automorphism of Po over l' if n:2 and U, is either an automorphism of Po over T or the negative of an anti-automorphism of Po over T E v er y y €, P * i s a s u m o f p ro d u c ts x fi z " ' )G; x;€ P ' nr,, if n )2 . p e P and (J" is an automorphism of Po, then (prr "' x,)U,: and if (prrU,)(rzu,) ' " (x,U,) - p"(xr(J,)"' (x,U,) : p'((r, "' x,)U,).' If U" is the negative of an anti-automorphism, then (px, ''' x,)U,: (-L)'(r,U") . '- (rzlJ,)(pxJJ"): (-1)'p'(x,U")" '(rrU,): p'((x' " ' x,)U"). Hence in either case U" is semi'linear with automorphism s in P. since (J,(Jt and (I* have the same effect on the set of generators is valid in Pn. Plz of P" we have (J"(Jr: (J"t. Similarly Ut:1 lf. (J, and (Jt are the negatives of anti-automorphisms then Uss-r : (J,[1;' is an automorphism. It follows immediately from this that the subset H of. elements s e G such that U" is an automorphism of Pn over l' is a subgroup of index one or two in G. Case I. H: G. The subset of Po of. fixed elements under the U, is a l--subalgebra U of P* such that }le: P* (Theorem 7). Hence ?I is finite-dimensional central simple over l-. Evidently 1 and 8gU and so 8':8 =VL. On the other hand, tS:rl:n28s type ,4r of L i e a a l g e b ra i s Hen c e 8 :? Il L VL :rl :n' - 1. defined on p. 303. Case II. H + G. Then I/ has index two in G. We know also that n > 2 in this case. The subset of. P of elements F such that E" : E, s e F/, is a quadratic sub4eld l'(q) over l.. It is clear from the form of the U, that x : 212-'p;ar satisfies *(f , : 1, s e F/, if Thus 8rror, the set of f(q)-linear and only if all the fie/'(q). is the set of elements of. Pl," which are combinations of the a;, fixed for all the U,, s €, H, and I/ is the Galois group of P over f@). It follows from case I that Srror:Ul where ?I is central simple over f(q) and is the enveloping associative algebra of 8rror. Now let / € G, 4 H. Since f/ is of index two in G, an element Now Zi'-'p;a; of 8",0r is in 8 if and only if (2p;ai)(L:2i;ai.
310
LIE ALGEBRAS
,
ut: -/ where / is an anti-automorphism of p" over 1.. since (2 pP)Ur: 2 F;a; for pi e f@) and p the conjugate of p under the automorphism + L of. f(q) over I', J- -(Jt maps B;101into itself. Hence / induces an anti-automorphism in the enveloping algebra ?I of 8.,0,. If pe T(q), -ye?I, then (py), - p!". Herrcelirr[u.", the automorphism p + p in I'(q), so / is of second kind. since aut:a for aG,8, at: -aandaeg(?I,/). Thus gLg(u,/) and 8' : 8 s @(?r,/)'. comparison of dimensionalities oirer /- shows that 8 - 6(2r, r)'. Hence 8 is a simple Lie algebra of type Ar. We have therefore proved the following Tnponprrr 13. Any central simfle Lie algebra of type Ar. is i,sornorphic either to a Lie algebra V,n, 2I a fi.nite-dimensional central simple associatiue algebra or to an algebra @(a, I), whete 2r is f.nitedimensional simple associatiue with an inaolution I of qecond kind. we consider next the Lie algebras of types B, c and D in the following Tnuonsu 14. Izt 8, be a centrar simple Lie algebra of type 81, I > 2, Ct, 12 3 or Du 14 5. Then g is isomorhhic to a Lie- ilgebra g(u, /) where { ,s a fi.nite-dimensional central simfle assoiatiue algebra, J an inuolution of fi.rst kind. in 21. Proof: There exists a finite-dimensional Galois extension p of T such that 8e is the Lie algebra @(.pn,J) of /-skew ntatrices in po where/is the involution x-x': the transpose of x[n p,,, or the involution x--Q-'x'Q -e and the entriies of where e': e are in the prirne field. Also we have n>E if n-21 *L, nz6 if. n:21 and the involution is x-re-'x'e, and z > 10 in the remaining case. For each s in the Galois group G of p over /. we have the automorphism (1, of. 8.p over f : ZTpra,;+ .X plar where (ar, . .., a*) is a basis for 8 over f and for Bp oyer p. g is the subset of 8p of elements which are fixed relative to the u,. The conditions on n insure that (J" can be extended to an automorphism u" of the enveloping associative algehra p* of. ge (Theorem 12). The extension (/, is semi-linear in po wilth associated automorphism s, U, - 1 and (J,U: (J* hold in p". Hence the subset of P* of elements which are fixed relative to the (J", s € G, is a subalgebra { of. P^ such that ?Ip - po. Hence ?I is finitedimensional central simple over l-. lf. X € 6.e, then Xr : _X and XU, e 6r. Hence XrfI, - -X(J, _ (X[J)r. Thus f(Jt: U,/ holds
X. SIMPLE LIE ALGEBRASOVER AN ARBITRARY FIELD
311
in 6ip. Since P" is the enveloping associative algebra of @p: U"J in Pn also. This implies that I @(Pn,D it follows that fU": hence maps ?I into itself and / induces an involution in ?I over lwhich is of first kind since l. is the center of U. lf. a e I then On the other hand, &t : -a so a e @(W,I) and 8 s @({, /). g(?I, hp = g(P*, I) : Br. Hence 8 : 6({, /). This completes the proof. 6.
A claser look at the isomorphism
conditions
We have seen in $ 4, that if Jr, i : L,2, is an involution (either kind) in a finite-dimensional simple associative algebra 2L ov€f O, then 6({ r, Jr)' =o@(2I2,/')/ implies that ?Ir and ?Iz are isomorphic. It therefore suffices to consider one algebra ?I - ?I' ?Iz and cong(?I, g(?I, K)' where sider the condition for isomorphism of /)/ and isomorphism any I and K are involutions in ?I. We have seen that d of 6(?I, /)/ onto g(?I, K)/ can be realized by an automorphism - dK holds of ?I. If. ae g(?I, J)t then ato : -a0: aoK. Thus l0 in 6(tr, /)'. Since the enveloping A-algebrasof 6(fl, /)' and g(21,K)l We shall call the inin ?I or K:0-'J0. are ?I we have I0:0K volutions / and K of II cogredient if. there exists an automorphism We have seen that cogredience is a d of D such that K:0-'f0. necessary condition for isomorphism of @(V, n' and g(?I, K)'. '/d where d is Conversely, if. J and K are cogredient and K: 0' g(U, g(?I, K) and @(2I,I)' an automorphism then d maps /) onto onto 6(?I, K)'. Hence d induces an isomorphism of the Lie algebra g(?I, /)/ onto g(?I, K)'. Thus g(?I, J)' =rg(U, K)' if. and only if / and K are cogredient. We see also that the group of automorphisms of the Lie algebra 6(?I, I)' can be identified with the subgroup of the group of automorphisms d of the algebra ?I such that 0J: J0 . Now let ?I : @ the algebra of linear transformations in the finitedimensional vector space !ft over the finite dimensional division algebra y'. Let a--+d be an involution in / and let (x,y) be a hermitian or skew hermitian form corresponding to this involution. Then the mapping A-, A*, A e 8,, A* the adjoint of A, is an involution in @. We recall that A* is the unique linear transforand we have (y,x) - eTrJ) mation such that (xA,l):(x,lA*) -1 : according as the form is hermitian or skew where e 1 or is a second involution in I and (r,t), hermitian. Suppose d--d'
3T2
LIE ALGEBRAS
,
a second hermitian or skew hermitian form correspodding to this so we have (t, x)r: er(tr,yXwhere er : tl. Suppose tbe involution determined by this form is the same as that given by (x, y). Thus i f. ( x , y A *) : ( x A , y ) th e n (x ,y A* )r:(x A,y )r. Let u and o be arbi trary vectors in $t. Then r->(x,u)a is a linear tran$formation in ul and one checks that its adjoint relative to (x, y) is x -+ e(r, a)u. Hence we have ((r, u)a,!)r: (r, u)(u,!)r:
(r, e(y, a)u), e(r, u)r(y, u)l
Since x, !, 14,a are arbitrary this shows that (x, y)r:)(x, t)p, p * 0 in /. lf. a e y', (x, at), : (x, !)rd' gives (x, y)dp _ (*, y)pa,; hence d' : p-tdp. Also we have (y, x)p: er((f,,y)p), * eet(r, !), : erp-tp1fi1l: eqp-tp(y, x)p. Hence p: e*,D. Conver$ely, let p be any element of r' satisfying F : + p.+ 0. Then a direct verification shows that a-rd' : p-tdp is an involution inl and (k,y)r=(r,y)p is hermitian or skew hermitian relative to this involution . lf. (x, y) is skew hermitian and p : -p, then (r,y)p is hermitiin. Hence if I contains a skew element + 0, then a skew hermitiihn form can be replaced by a hermitian one which gives the sanre involution A-.A* in G. If. / contains no such elements theni p: p for all p e / and this implies that / is a field. Hence we may restrict our attention to hermitian forms and to alternate forrns (/ a field). Two such forms (x, t), and (.r,y) give the same involution in @ if and only if. (x,J)r : (r,y)p, F: p if a+a is the involution of (r,J). It is known that any automorphism of G has the forrn A -+ s-rAS where s is a semi-linear transformation in lut over y';(Jacobson[3], p. 45). If d is the automorphism in y' associated with s, then one checks that'(r,y)r: (rs,Js)o-t is hermitian or alterirate with involution a--7a0ye-'. Moreover, if. A € G then (xAS,yS)o-' - (rSS-'AS, yg)e-t * (rS, rS(S-t,451*;e-t : (rS, y(S(S-'eS)*S-r;5;e-1 . Thus (rA, t)r: (fr,y(S(S-r.4S)*S-'), and the involutlon in @ determined by (r,J), is A-'S(S-'AS)*S-'. Thus if we, call A- A*, l, and A -, S-'AS, d, then the new involution is K - 0f0-t which is cogredient to J. Elecause of this relation it is natural to extend the usual notion of equivalence of forms in the following manner:
X. SIMPLE LIE ALGEBRASOVER AN ARBITRARY FIELD
3I3
Two hermitian forms (r, y) and (r, y), are said to be S'equivalent if there exists a 1: 1 semi-linear transformation S with associated isomorphism d such that (r, J), : (rS, yS)0-'. It is well known that any two non-degenerate alternate forms are equivalent in the ordinary sense. Hence the involutions in O determined by any two such forms are cogredient. Moreover, these are not cogredient to any involution determined by a hermitian form. Our results imply also that the non-degenerate hermitian forms (x, y) and (r, y)r define cogredient involutions in 0 if and only (rS,JS)o-'p where S is semi-linear with automorphism if. (r,!)t: p 0 and is symmetric relative to a'-rd'=@)t-' 7.
Central simple real Lie algebras
We shalt now apply our results and known results on associative algebras to classify the central simple Lie algebras of types A-D (except D.) over the field O of rcal numbers. By Frobenius' theorem, the finite-dimensional division algebras ovet A are: O; the complex field P - AQ), i' : -l; the quaternion division algebra r' with basis L, i, i , & such that
(12)
i':i':k2:-L, jk:-hj:i,
ij--ji==k, hi:-ik-i.
a: d + pi + ri + Bk -> d : involution I has the standard 8k. Since the automorphisms in / are all inner, d rj Pi in y' is either standard or it has the form a + q-'dq involution every The dimensionality of the space of skew elements where 4: -q. under the standard involution is three; under a-+q-tAq the dimen'We denote the automorphism ;t 1 in P over O bV sionality is one. p+F. By the Wedderburn theorem, the finite-dimensional simple associative algebras over O are the full matrix algebras Oo, P" and y'o. These can be identified with the algebras @(O,n), @(P,n) and @(/,n) of linear transformations in lUl over O, Pand y' respectively. The algebras @, =E(O,n) and y'n?@(l,n) are central and have only inner automorphisms. The center of @(P,n) = P* is P. In addition to inner automorphisms, E(P, n) has the outer automorphisms X- S-tXS where S is a semi-linear transformation with associated automorphism p+V in P. All our algebras have involutions and 6(P, n) has involutions of
314
LIE ALGEBRAS
second kind. The involutions of o(@,z) have the fbrm x-- x* where x* is the adjoint of,X relative to a non-degenerafe symmetric or skew bilinear form in ![l over o. Involutions ddtermined by skew forms are not cogredient to any determined by ta symmetric form. Any two non-degenerateskew bilinear forms a;e equivalent so these give a single cogredience class of involutions. If (r, y) and (r, !)r ate non'degenerate the criterion of the last $ection shows that the involution determined by (r, y) is cogredieht to that of (r, y), if and only if. (x, y) is equivalent to a multiple of (r, y)r. Since (r, y) is equivalent to any positive multiple of (x,ly) it follows that the involution given by (r,y) is cogredient to that of (r, t), if and only if. (x, y) is equivalent to !(r, y)r. If k, g is nondegenerate symmetric it is well known that there exists a basis (ur, ur, . . . , un) for Ul such that
(13)
(u;,u+): 1 , (ut,u)--1 , (u;,u):0,
L
0. Arch. Math. 5 (1954)' pp. 274-281' field of characteristic p > 0, t4l Lie groups and Lie hyperalgebras over a I-vI. I: Comment. Math. Helv. 28 (1954),pp.87-118. II: Am. J. Math. 77 (1955), pp. 2|8-244. III: Math. z. 62 (1955), pp. 53-75. IV: Am. J. pp. Math. 77 (1955), pp. 429-452. V: Bull. soc. Math. France 84 (1956)' n7-239. VI: Am. J. Math. 79 (7957), pp. 331-388' groups. Mathematika. 2 (1955)' pp. tsl witt groups and hyperexponential
2L-3r.
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Stolr{gBnc, R. lU Prime power representations of finite linear groups, I and II. I: Can. J. Math. 8 (1956),pp. 580-591. II: Can. J. Math. I (1957),pp. 347-351. I2l Variations on a theme of Chevalley. Pacific J. Math. I (1959), pp. 875891. l3l The simplicity of certain groups. Pacific J. Math. 10 (1960), pp. 10391041. I4l Automorphisms of classical Lie algebras. Pacific J. Math. 11 (1961). Uber eine Beziehung zwischen geschlossenenLieschen Gruppen und diskontinuierlichen Bewegungsgruppen Euklidischer Raume und ihre Anwendung auf die Aufzahlung der einfachen Lieschen Gruppen. Comment. Math. Helv. 14 (1941-42),pp. 350-380. I2l Kristallographische Bestimmung der Charaktere der geschlossenenLieschen Gruppen. Comment. Math. Helv. 17 (1944-45),pp. 165-200. lU
Ttrs, J. tU Sur les analogues alg6briques des groupes semi-simples complexes. Colloque d'AlgBbre. Brussels, (Dec. 1956). tzl Les groupes de Lie exceptionnels et leur interpretation g6om6trique. Bull. Soc. Math. Belg. 8 (1956),pp. 48-81. l3l Sur la g6om6trie des 8-espaces. J. Math. Pures Appl. 36 (1957), pp. 17-38. t4l Les "formes r6elles" des groupesde type E'e. S6minaireBourbaki. (1958),
t5l t6l
Sur la trialit6 et certains groupesqui s'en deduisent. Publs. Math. Inst. des Hautes-Etudes No. 2 (1959), pp. 14-60. Sur la classification des groupes alg6briques semi-simples. Comptes Rendus 249 (1959), pp. 1438-1440.
328
LIE ALGEBRAS
ToMBER,M. L. tu Lie algebras of type F.
proc. Am. Math. Soc. 4 (1gs3), ,pp. zs9-26g.
Wntl,
H. lU The Classical Groups. princeton Univ. press, 1939. I2l Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch IineareTransformationen. I, II and tII. I: Math. z. zg (Lgz6),pp. 221_309. II: Math. z. 24 (1920), pp. BZB-J?6.III: Math. z. 24 (1926),pp. 3zz-39b. WItt,
E.
Treue Darstellung Liescher Ringe. J. Reine Angew. MAth. LTT (LIST), pp. 152-160. tzl spiegelungsgruppen und Aufziihlung halbeinfacher Liescher Ringe. Abhandl. Math. Sem. Univ. Hamburg. t4 (1941),pp. 289_b32. t3l rreue Darstellungen beliebiger Liescher Ringe. collect. Math. 6 (1953), pp. 107-114. lU
t4l Die Unterringeder freien Lieschen 2t6.
Math. Z. M (1956),pp. 195-
Z.nssnunlus,H. tU I2l
t3l t4l 15]
Uber Liesche Ringe mit Primzahlcharakteristik. .Abhandl. Math. Sern. Univ. Hamburg. 13 (1939),pp. l-100. Ein verfahren jeder endlichen p-Gruppe einen Lie-Ring mit der charakteristik p zuzuordnen. Abhandl. Math. sem. univ. Hamburg. lg (1939), pp. 2N-207. Darstellungstheorie nilpotenter Lie-Ringe bei charakteristik p > 0. J. Reine Angew. Math. t$z (lg4}), pp. 150_155. Uber die Darstellungen der Lie-Algebren bei Charakteristik 0. Comment. Math. Helv. 26 (L}SZ), pp. ZSZ-214. The representations of Lie algebras of prime characteristic. proc. Glasgow Math. Assoc. Z (L954), pp. l-86.
INDEX Abelian Lie algebra, 10 Adjoint mapping, 9 Ado's theorem, 202 Algebra alternative algebra, 143 associative algebra, 3 associativealgebra of linear transformations, 5 constants of multiplication of an algebra, 2 convention on algebra terminology, 15 of differential operators, 175 Jordan algebra, 144 Lie algebra, 3 Lie algebra of linear transformations, 6 non-associativealgebra, 2 Almost algebraic Lie algebras, 98 Automorphisms of Au Bt,, Ct,,Dt, Gz Fu 28I-285 Basic irreducible modules, p. 226 Campbell-Hausdorff formula, 175 Canonical generators, 126 Cartan matrix, 121 Cartan's criteria, 68, 69 Cartan subalgebra, 57 conjugacy of Cartan subalgebras, 273 splitting Cartan subalgebra, 108 Casimir operator, 78 Cayley algebra, L42, 317 Center, 10
Characteristic polynomial, 60 Characters, 239, 249 primitive character, 259 Cohomology groups, 93-95, 174-785 Compact form, 147 Complete Lie algebra, 11 Complete reducibility, 46,75-83, 96103 Composition series, 48 Derivation, 7, 73-75, 79 anti-derivation, 179 inner derivation, 9 (sr, sz)-derivation, 177 Derivation algebra, 8 Derived series, 23 Direct sum, 18 Dynkin diagram, 128 Engel's theorems, 36 Enveloping algebra, 32 Exceptional simple Lie algebras, 142 Extension of base field, 26-28 Extension of a Lie algebra, 18, 88 Exterior algebra, 178 Factor set, 89 Filtered algebra, 164 Fitting decomposition, 37, 39, 57 Free associative algebra, 167 Free Lie algebra, 167 Freudenthal's formula, 247 Friedrich's theorem, 170 Goldie-Ore theorem, 165
Central non-associativealgebra, 29L Centralizer, 28
Graded algebra, 163
Centroid, 291
Holomorph, 18
Chamber, 263
t32el
330
INDEX
Ideal, 10, 31 Indecomposability(for a set of linear transformations), 46 Invariant form. 69 Irreducibility (for a set of linear transformations), 46 absolute irrepucibility, 223 Irreducible modules for At, Bt and Gz, 226-235 Iwasawa's theorem. 204 Jacobi identity, 3 Killing form, 69 Kostant's formula, 261 Levi's theorem, 91 Lie element, 168 Lie's theorems, 50, 52 Linear property, 146 Lower central series, 23 Malcev-Harish-Chandratheorem, 92 Mqdule, 14 cyclic module, 272 contragredient module, 22 a-extreme cyclic module, 272 tensor product of modules, 21 Morozov's lemma, 98 Mostow-Taft theorem, 105 Multiplication algebra, 290 Nil radical, 26 Nilpotent Lie algebra, 25 Normalizer, 28, 57 Orthogonal Lie algebra, 7 Poincar6- Birkhoff - Witt theorem, 159 Polynomial mapping, 266 polynomial differential of a mapping, 268 Primary components, 37-43
Qnaternion
algebra, 316
Radical of associative algebfa, 26 of Lie algebra, 24
Rank, 58 Real simple Lie algebras, 313-316 Reductive Lie algebraf 104, 105 Regular element, 58 Representation adjoint representation,16 for associative algebra, 14 for Lie algebra, 16 i regular representatipn, 16 of split three dimehsional simple Lie algebra, 83-186 tensor product of rtepresentations, 2l Restricted Lie algebra of characteristic p, 187 Abelian restricted Lie algebra, 192194 Roots, 64
Schenkman's derivation tower theorem, 56 Semi-simpleand nilpotent components of a linear transformation, 98 Semi-simpleassociatirlealgebra, 26 Semi-simple Lie alge\ra, 24 structure theorem {or semi-simple Lie algebras, 7[ Simple system of roofs, 120 indecomposablesimple system, 127 Solvable Lie algebra, 24 Specht-Wever theoreni, 169 Spin representation, ?29 Split Lie algebra of linear transformations, 42, 50 Split semi-simple Lie, algebra, 108 existence theorem nor, 220 isomorphism theorein tor 127, 221
INDEX Split simple Lie algebras At, Bt. Ct, D t , G z , F + , 8 0 , 1 3 5 - 1 4 6 ;E z a n d Ea, 228 Split three-dimensional simple Lie algebra, 14 Splittable Lie algebra (in the sense of Malcev), 98 Steinberg's fotmula, 262 Subinvariant subalgebra, 29 Symplectic Lie algebra, 7 Trace form, 69 Triality, 287 Triangular Lie algebra, 7 u-algebra of a restricted Lie algebra,
r92
331
Unipotent linear transformation, 285 Universal enveloping algebra, 152 Upper central series, 29 Weakly closed subset of an associa' tive algebra, 3l Weight, 43 weight space, 43, 61 Weyl group, lI9, 240-243 Weyl's character formula, 255 Weyl's dimensionality formula, 257 Whitehead's lemmas 77, 89 Witt's formula, 194 Witt's theorem, 168
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