Indecomposable Representations of Graphs and Algebras

Memoirs of the American Mathematical Society Number 173

Vlastimil Dlab and Claus Michael Ringel Indecomposable representations of graphs and algebras

Published by the

AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island VOLUME 6· NUMBER 173 (end of volume)· JULY 1916

AMS (MOS) subject classifications (1970). Primary 16A64: Secondary 16A43, 17899,18EIO. Key words and phrases. Indecomposable representations of finite dimensional algebras, species, quivers, valued graphs, regular representations. homogeneous representations; Dynkin and extended Dynkin diagrams, Coxeter functors and transformations. Weyl group, root system, defect.

Library of Congress Cataloging in Publication Data

Dlab, Vlastimil. Indecomposable representations of graphs and algebras. (Memoirs of the American Mathematical Society no. 173) Bibliography: p. 1. Associative algebras. 2. Representations of algebras. 3. Representations of graphs. I. Ringel, Claus Michael, joint author. II. Titleo III. Series: American Mathematical Society. Memoirs ; no. 173. QA3.A57 no. 173 [QA251.5] 512'.24 76-18784 ISBN 0-8218-1873-2

ii

Abstract I. N. Bernstein, I. M. Gelfand and V. A. Ponomarev have recently shown that the bijec-

tion, first observed by P. Gabriel, between the indecomposable representations of graphs ("quivers") with a positive definite quadratic form and the positive roots of this form can be proved directly. Appropriate functors produce all indecomposable representations from the simple ones in the same way as the canonical generators of the Weyl group produce all positive roots from the simple ones. This method is extended in two directions. In order to deal with all Dynkin diagrams rather than with those having single edges only, we consider valued graphs ("species"). In addition, we consider valued graphs with positive semi-definite quadratic form, i. e. extended Dynkin diagrams. The main result of the paper describes all indecomposable representations up to the homogeneous ones, of a valued graph with positive semi-definite quadratic form. These indecomposable representations are of two types: those of discrete dimension type, and those of continuous dimension type. The indecomposable representations of discrete dimension type are determined by their dimension vectors: these are precisely the positive roots of the corresponding quadratic form. The continuous dimension vectors are the positive integral vectors in the radical space of the quadratic form and are thus the positive multiples of a fixed dimension vector. The full subcategory of all images of maps between direct sums of indecomposable representations of continuous dimension type is an abelian exact subcategory, which is called the category of all regular representations. It is the product of two categories U and H, where H is the largest direct factor containing only representations of continuous dimension type. The representations in H are called homogeneous and their behaviour depends very strongly on the particular modulation of the valued graph. One can reduce the study of the category H to the study of the homogeneous representations of a simpler valued graph, namely of a bimodule. On the other hand, the structure of the category U can be determined completely: it is the direct product of at most three indecomposable categories, each of which has only a finite number of simple objects, is serial, and has global dimension 1. The indecomposable representations which are non-regular can be described in the following way: there are two endofunctors C+ and c- on the category of all representations, called the Coxeter functors, such that the list of all representations of the form C- r P and C+rQ, where P is an indecomposable projective representation and Q is an indecomposable injective representation, is a complete list of all non-regular indecomposable representations. Also there is a numerical invariant, called the defect, which measures the behaviour of the indecomposable representations, and depends only on the dimension type. The defect of a representation is negative, zero, or positive, if and only if it is of the form C-rp, regular, or of the form C+ r Q, respectively. The paper concludes with tables of all valued graphs with positive semi-definite quadratic form. The tables provide, in condensed form, most of the information which is available about the representation theory of these valued graphs.

iii

Acknowledgement

The authors wish to acknowledge support of the National Research Council of Canada under Grant No. A-7257, of the Canada Council under Award No. W740599, of the Technische Hochschule Darmstadt, of the Universite de Paris and of the Sonderforschungsbereich Theoretische Mathematik, Universitat Bonn in preparation of this paper. They also wish to express their gratitude to Carleton University for the financial assistance in printing this paper as Memoirs.

Introduction

In a recent paper, I. N. Bernstein, I. M. Gelfand and V. A. Ponomarev [2] have shown that the bijection between the indecomposable representations of graphs ("quivers") with a positive definite quadratic form and the positive roots of this form observed by P. Gabriel [8] earlier, can be proved directly. They have introduced certain functors which allow to construct all the indecomposable representations from the one-dimensional ones in the same way as the canonic generators of the Weyl group produce all positive roots from the basic ones. In this paper, we are going to extend this result in two directions. On one hand, we shall consider valued graphs (and therefore "species" of [9] , [4]) instead of graphs in order to deal with all Dynkin diagrams rather than with those having single edges only; in this way, we recover previous results of ours [4]. And, on the other hand, we shall consider also valued graphs with positive semidefinite form (i. e. the extended Dynkin diagrams) and describe, up to the homogeneous ones, all their indecomposable representations. In the case of extended Dynkin diagrams with single edges, this yields the previous results of L. A. Nazarova [15] and P. Donovan and M. R. Freislich [7]. A valued graph (f, d) is a finite set f (of vertices) together with non-negative integers d jj for all pairs i, j E f such that d jj = 0 and subject to the condition that there exist (non-zero) natural numbers I j satisfying 1 djj/j

= djJj

for all i, j E f.

In addition, we shall always assume that the valued graph (f, d) is connected in the sense that, for every k, IE f, there is a sequence k, ... , i, j, ... , I of vertices of f such that d jj 0 for each pair of subsequent vertices i, j. Note that d jj may differ from d jj , but that d jj 0 if and only if d jj 0; let us call such pairs U, j} the edges of (f, d), and the ver-

*" *"

*"

(d jj , d jj )

tices i, j neighbours. In notation, we shall use the symbol:I : for the edges of J (f, d); if d jj = 1 = d jj , we write simply i - j- Let us remark that one can prove easily that every tree (graph without circuits) can be turned into a valued graph by choosing pairs (d jj , djJ of arbitrary natural numbers 0) for all edges of that tree. An orientation n of a valued graph (f, d) is given by prescribing, for each edge {i, j} of (f, d), an order (indicated by an arrow: i ---+ j)' Given an orientation n and a vertex kEf, define a new orientation skn of (f, d) by reversing the direction of arrows along all edges containing k. A vertex kEf is said to be a sink (or a source) with respect to n if k ~ i (or k ---+ i) for all neighbours i E f of k. And, an orientation n of (f, d) is said to be admissible if there is an ordering k l , k 2 , . . . , k n of f such that each vertex k t is a sink

(*"

Received by the editors December 13, 1974. 1 Observe that there is a one-to-one correspondence between valued graphs and symmetrizable Cartan matrices (see (12).

2

VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL

with respect to the orientation

sk

t-l

••• Sk Sk

2

1

£2 for all I ,;;;; t ,;;;;

n;

such an ordering is

called an admissible ordering for £2. It is easy to see that an orientation £2 of the valued graph (r, d) is admissible if and only if there is no circuit with orientation : - : - : 'I

.•• :

't-l

-

'2

'3

: _. ; therefore, in particular, every orientation of a tree is admissible.

' t -'1

A modulationlJJ/ of a valued graph (r, d) is a set of division rings F j , i E r, together with an F(Frbimodule ilj and an FrFj-bimodule jM j for all edges {i. (i) there are FrFj-bimodule isomorphisms jMj

"'"

HomF/jMj • F j )

"'"

11

of (r, d) such that

HomFj(Mj • F j )

and (ii) dim (Mj)Fj =

dij'

A realization (\))1, £2) of a valued graph (r, d) is a modulation \))1 of (r, d) together with

an admissible orientation £2. A repre!ientation X = (X j , j'fJj) of a realization (1))1, £2) of

(r, d) is a set of finite-dimensional right F(spaces Xj' i

E

r, together with Frlinear map-

pings j'fJj: X j ®F j jMj -

Xj

for all oriented edges 2 ; - j- A morphism a: X - X' from a representation X = (X j • to X' = (X;, j'fJ) is defined as a set a = (<X j) of Fj-linear mappings <X j : X j - X;, i E r, satisfying j'fJ;

(<X j ® 1)

= <Xjj'fJj

for all edges; -

j'fJ)

j-

One can see easily that the representations of (\))1, £2) form an abelian category which we shall always denote by LWI, £2). Given a valued graph (r, d), denote by Or the vector space of all x = (x)jEr over the rational numbers. In particular, for each k E r, k E Or denotes the vector with x k = 1 and x j = 0 otherwise. Also, for each k E by SkX = y, where Yj = x j for i -=1= k and Yk =

r,

define the linear transformation

-x k +

L

jEr

sk

:Or _Or

djkx j .

The symbol W = Wr will always denote the Weyl group. i. e. the group of all linear transformations of Or generated by the reflexions sk' k E r. A vector x E Or satisfying wx = x for all w E W will be called stable. A vector x E Or is called a root (of (r, d)) if there exists k E r and w E W such that x = wk. A root x is said to be positive (or negative) if x j ;;;. 0 (or x j ,;;;; 0) for all i E r. Given a representation X = (X j , j"P) of a realization (\))1, £2), we may define the mapping dim: L(\))I, £2) _ by dim X = (x), where x j = dim (X)F' for all i E I sion type of the representation X.

Or

r.

The vector dim X is called the dimen-

The main result of this paper is the following THEOREM.

Let (r, d) be a valued graph. Let (\))/, £2) be a realization of (r, d).

2 We shall show later that j'Pj corresponds (bijectively) to an Frlinear mapping j:;i(Xj --. X/~Fj jM j , and thus, our definition of a representation coincides with that of P. Gabriel in 19 J. Of course, realizations are called species in [9 J .

REPRESENTATIONS OF GRAPHS AND ALGEBRAS

(a) Then L('JJL Q) is of finite type if and only if (r, d) is a Dynkin diagram, i. e. a valued graph of one of the forms

.--.-- ..... - - .., · (1,2)._ ..... _ . ', · (2,1)._ ..... _ . ',

.-----. --------

..... -

-

..

._._1_._.. '

._._!_._'._ .. i

'

· --. --. --. --. --. - - . ', (1,2) • - - . - - . - - . ;or · (1,3) . Moreover, the mapping dim: L('JJl, Q) ---* Qr induces a bijection between the isomorphism classes of indecomposable representations of ('JJl, Q) and the positive roots of (r, d). (b) If (r, d) is an extended Dynkin diagram, i. e. a valued graph of one of the forms

· (1,4) .. · (2,2) . ,.'

....----- --. .------. .-----. . ..-----;

· (1,2). _

_ . (2,1) ..

· (2,1). _

..... _ . (1,2) ..

, ,

(1,2) (1,2) . - - . - - . ... e - - . - _ . ·,

eon

.-----

..______.

-

..... - . -(2,1) .;

(1,2) ~n.>-· .... - - . - - .., On

E6

> -- ..... _-. 0, and

therefore x k

Now, 2B(x, k)IB(k, k) sequently,

;;;,

1.

< 1 + fjlfk , and Sk X =

therefore, if fj f k , we

= dji . Note that = y, where

we may

Or by ~x

---+

for each i E f.

We are going to show that ~Sk

= s~~

(where s~ denotes the involutions of Or'). Indeed, for i (s~~x)i

'* k,

= (~x)i = fiXi = (~Skx\;

and,

(S~~X)k

=-

f kx k

= f k (- x k

+ LdiJixi i

+ Ldikx i) = (~Skx)k· i

Now, x = wj, where w is a product of suitable involutions Sk; consider the vector x' where w' is the corresponding product of the involutions s~. Thus x' =

w,-.l

f..j fj/

= w'j,

= 1..- w'~j = l-~x fj

fj

is a positive root in Or'. Since

Qr,(x') = Qr,(j) =

A< ~

= Qr,(k),

we see immediately that, according to the first part of the proof, either x'

=k

or s~ x' is positive.

However, if x' = k, then ~x = fjk implies that x = (fjlfk)k, a contradiction. If s~ x' is positive, then

fjS~X' = fjs~ ). ~x = ~skx, /

and therefore also Skx is positive. The proof is completed. PROPOSITION

the group finite.

1.5. Let (f, d) be a valued graph with a positive quadratic form. Then

W of all linear transformations of Or IN induced by

the transformations of W is

Proof Let f be a natural number with f i ~ f for all i E f. Let M be the set of all integral vectors x E Or such that Q(x) ~ f Let P be a regular matrix which transforms Q into a diagonal form:

11

REPRESENTATIONS OF GRAPHS AND ALGEBRAS t

Q(x)

=L

;=1

c;yl,

where y

= xP

and 0

< c; E

Q for 1 '!( i

'!(

t

'!( n.

Thus, NP consists of all vectors y = (y;) with y; = 0 for 1 '!( i '!( t. Let h be the common denominator of the entries in P; hence, hP is an integral matrix. Consequently, if x EM, then hy is an integral vector and, moreover Iy; I '!( J flc; for 1 '!( i '!( t. Therefore, under the transformation P, M is mapped into a set which is modulo NP finite. Thus, also the set M = M + NIN is finite. Now, the group Wof the automorphisms on Or IN induced by W transforms if into itself. And, since if contains a basis of Or IN, Wcan be embedded into the symmetric group on M and thus is finite, as required. Let (r, d) be an extended Dynkin diagram. Proposition 1.5 allows us to introduce a very important concept, that of the defect 0cx of a vector x E Or with respect to a Coxeter transformation c E W (or, what is the same, with respect to an admissible orientation of (r, d»: If E Wis of order m, then, given x E Or,

c

cmx

In this fashion, 0c : Or (oci) E Or, we have

---+

= x +(ocx)n,

where 0cx E Q

0 defines a linear form and thus, for the defect vector 0c 0c x

=

= L(oci)x;. ;

Notice that, since one of the integral components of n equals 1, all 0ci are integral. Also, one can see easily that 0c(cx) = 0c x . 1.6. Let x ;> 0 be a positive root of (r, d) whose quadratic form is positive .•• sk sk be a Coxeter transformation of Or. Then , . n 2 I (i) cx;j> 0 if and only if x = Pk for a suitable 1 '!( t ;;;;; n, where LEMMA

and let c = sk

t

Pk t

= ski sk2

••• Skt_l k t ·

(ii) c- 1 x ;j> 0 if and only if x = ~ for a suitable 1 '!( t t

q k =s k n Skn-l t

• ..

'!(

n, where

skt+l k t·

Proof This is an immediate consequence of Lemma 1.4.

1.7. Let x be a positive root of (r, d) whose quadratic form is positive and let c = sk ••. sk2sk 1 be a Coxeter transformation of Or. Then either there exists an inten ger r such that crx ;j> 0, or (r, d) is an extended Dynkin diagram and 0cx = o. LEMMA

Proof If (r, d) is a Dynkin diagram, then c is of finite order, say m, and y = "2;~=1 c!'x satisfies cy = y. Consequently, y = 0 and thus there exists 1 ;;;;; r;;;;; m - 1 such that crx;j>O.

If (r, d) is an extended Dynkin diagram, then by Proposition 1.5, the order of is finite; denote it again by m. Now, if

°

eX =1=

0, then

t!m x = x + s(ocx)n for every integer s.

cE

W

12

VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL

From here it follows that, for a suitable r r

'*

= sm (such

that seacX) is negative and large in ab-

solu te value), c x O. The following more general results cover the remaining case of a root of defect zero (see R. V. Moody[12]). LEMMA 1.8. Let (r, d) be an extended Dynkin diagram. Then there exists a natural number g, 1";;; g .,;;; 3, (called the tier number) such that the positive roots x of (r, d) are just the vectors of the form x = X o + rgn

with a non-negative integer r and a positive root Xo of (r, d) satisfying Xo ,,;;; gn. The tables in Chapter 6 provide the values of g and, for a particular admissible orientation which will be used throughout the paper, the value of a c ' For each oriented extended Dynkin diagram, there exist roots x* .,;;; gn with acx* = 0 such that the other roots Xo .,;;; gn with acXo = 0 are just the vectors of the form

L

r';;t';;r+s

ctx*, r, s;;. 0,

which do not belong to the radical space N (and which satisfy the inequality";;; gn). These vectors x* can be found in the tables (Chapter 6): They are the dimension types of the first indecomposable representation in each orbit listed there. Thus, we may summarize the preceding results and list all positive roots of a valued graph with a positive quadratic form. PROPOSITION 1.9. Let (r, d) be a valued graph and let c be a Coxeter transformation ofOr. (a) If (r, d) is a Dynkin diagram and m is the order of c, let, for each 1 ,,;;; t .,;;; n, at be the largest integers such that all C-rPk with 0";;; r .,;;; at are positive. Then the vectors t

x

=

c-rPk t' 0 .,;;; r .,;;; at' 1";;; t .,;;; n

are just all positive roots of(r, d). Similarly, if b t is the largest integer such that all Crqk t with 0 .,;;; r .,;;; b t are positive, then the vectors x = cr qk' 0";;; r .,;;; b t , 1";;; t ,,;;; n, t

are just all positive roots of (r, d). (b) If (r, d) is an extended Dynkin diagram, then (1) the vectors x = C-rPk ' 0";;; r, 1";;; t .,;;; n, are just all positive roots of (r, d) of t negative defect with respect to c; (2) the vectors x = c r qk' 0";;; r, 1";;; t .,;;; n, are just all positive roots of (r, d) of t positive defect with respect to c; (3) the vectors x = Xo + rgn, r;;' 0, where Xo ,,;;; gn with acx o = 0 can be derived from the tables of Chapter 6, are just all positive roots of defect zero with respect to c. Proof (a) follows immediately from Lemma 1.7 (with r chosen so that Irl is minimal) and Lemma 1.6. (b) is a consequence of Lemmas 1.6, 1.7 and 1.8.

REPRESENTATIONS OF GRAPHS AND ALGEBRAS REMARK.

13

Using the obvious relation cp k -- - q k t

t

for all 1

~ t ~

n.

and the fact that the number of roots of a Dynkin diagram is mn, (see, e. g. [3]), we deduce that the set

is the set of all roots and thus that the sets defined in Proposition 1.9 (a) have 1h mn elements. Let us conclude this chapter with a remark giving another characterization of positive roots of an extended Dynkin diagram of defect zero. REMARK. Given a valued graph (r, d) with a positive form, and a Coxeter transformation c. define in Or a partial order ~c as follows: x ~c y if and only if ctx ~ cty for all integers t. Obviously, this order is trivial if and only if (r, d) is a Dynkin diagram. In case of an extended Dynkin diagram, we can speak about c-positive roots: these are those roots x for which

x~c O. Minimal c-positive roots will be called simple c-positive roots. (a) For a root x of an extended Dynkin diagram (r, d), the following assertions are

equivalent:

(i) x is c-positive; (ii) 0c x = 0; (iii) c-orbit of x is finite. This follows immediately from Lemma 1.7. From the tables in Chapter 6, one can see that in each case, there are at most three orbits of simple c-positive roots. The simple c-positive roots are labelled there dim E}t). They are precisely the roots denoted in the remark preceding Proposition 1.9 by ctx*, t ~ O. That remark can be reformulated as follows.

(b) Every c-positive root is a (uniquely determined) sum of simple c-positive roots from the same orbit. 2. REALIZATION OF VALUED GRAPHS: THE COXETER FUNCTORS Let F j• F j • F k be fields. An F(Frbimodule jMj is said to have a dual bimodule if the FrFj -bimodules Hompj (;Mj , F j ) and Homp/jMj • F j ) are isomorphic. For example, if K is a common central subfield of F j and Fj such that K operates centrally on jMj , and if dim K jMj is finite, then jMj has a dual bimodule. Note that the FrFrbimodule jMj is a dual bimodule to jMj if and only if there exist nondegenerate bilinear forms

e{ : jMj

r:. jM

j

I

-+

Fj

,

eJ:

jMj ~ jMj -+ F j • I

Given jMj , the dual bimodule jMj (if it exists) is unique (up to an isomorphism), whereas the bilinear forms e{ and eJ may vary. 3 3 Although they are used in the construction of the Coxeter functors, the category L(ill, 0), as well as all results are, of course, independent of a particular choice of these forms.

14

VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL

CMi ) and p (iMk) have a dual bimodule, then so has the FfPi Pi i Pk Fk-bimodule p.cM)p. 0 p.(iMk)Pk' 4 For, given iMi' E{, EJ and kMi' Ef, E~, one considers If both bimodules I

~i

I

I

0 iMi together with the mappings 10 iMi 0 iMk 0 kMi 0 iMi

1 01) iMi 0

~

Fi 0 iMi ~ iMi 0 iMi ---+ F i

and E~ (l 0 E} 0 1), which are obviously nondegenerate. Now, let iMi be an Fi-Fj-bimodule with the dual bimodule iMi' If (X)Pi and (Xi)Pi are vector spaces, then there is a natural isomorphism Homp/Xi 0 iMi' Xi) ~ Homp/Xi , Xi 0 iMi)'

For, there is the well-known isomorphism Homp·C~·, ~.) ~ Xi 0 Homp·CMi ,

Pi

I

I

F,.)

~ Xi 0 iMi;

hence, the adjointness of 0 and Hom yields Homp/Xi 0 iMi' Xi) ~ Homp/Xi , HompiCMi , Xi)) ~ Homp/Xi , Xi 0 iMJ

Thus, for each Frlinear mapping ilfJi : Xi 0 iMi ---+ Xi' we have attached canonically an Fi-linear mapping i~i : Xi ---+ Xi 0 iMi; conversely, for i!/J i : Xi ---+ Xi 0 iMi' there correspond a unique i~i: Xi 0 iMi ---+ Xi' and we have iifli = ilfJi and i~i = i!/Ji' This notation will be used throughout this paper. REMARK. Let G be a subfield of F, and assume that the bimodule e F p has a dual bimodule. Note that this dual bimodule has to be pFe' Denote by pKp the kernel of the multiplication mapping pFe 0 eFp 4 pFp . We claim that also eFe' pFe 0 eFp, e(F!G)e and pKp have dual bimodules. This is obvious for eFe = eFp 0 pFe , and pFe 0 e Fp; both are self dual. There exists a nondegenerate bilinear form e Fp 0 pFe ---+ eGe' but this is just a nonzero map E: eFe ---+ eGe' Let eLe be the kernel of E. The nondegenerate bilinear form for e Fe is Ell: e F e 0 e F e ---+ eGe'

and since EIl(eGe 0 eLe) = 0 = EIl(eLe 0 eGe)' we conclude that Ell induces a nondegenerate bilinear form on e(F!G)e 0 eLe' and on eLe 0 e(F!G)e' Now, for pKp , we consider the nondegenerate bilinear form 1l(1 0

E

0 1): pFe 0 eFp 0 pFe 0 eFp ---+ pFp .

Let w = 1 : Fp ---+ Fe 0 e Fp be the mapping canonically attached to the identity 1 : Fp 0 pFe ---+ Fe' with respect to E; thus, by construction, the map Fe ~ Fp 0 pFe

w01 ---+

10E Fe 0 eFp 0 pFe ~ Fe 0 e Ge ~ Fe

is the identity. Note that w is an F-F-bimodule mapping. We claim that 4 In dealing with tensor products of the form we shall usually omit the letter

Fj.

iMi ® iMk' Pi

15

REPRESENTATIONS OF GRAPHS AND ALGEBRAS

p(1 ®

€

® 1)(w(1) ® xi ® Yi) = p[(1 ®

€)(W

® 1)(1 ® x;) ® Yi] = p(x i ® Yi) = XiYi

and therefore L>(1 ®

€

® l)(w(1) ® Xi ® Yi) =LXiYi = O.

i

i

As a consequence, p(1 ®

€

® 1) defines a nondegenerate bilinear form on (pFe ® eFp)/w(pFp) ® pKp ,

as well as on pKp ® (pFe ® eFp)/w(pFp). If dime F = dim Fe = 2, e Fp has a dual bimodule if and only if there exists IE F\ G such that either all the commutators [f, g] = Ig - gf, g E G, belong to G or all [f, g], g E G, belong to IG (i. e. IG = G/). For, a nondegenerate bilinear form

€ :

e Fp ® pFe

-+

e Ge is

simply a nonzero bimodule mapping €: eFe -+ e Ge · Now,either e(F/G)e ~ eGe which is equivalent to saying that there is IE F\ G with [f, g] s:; G, or e Fe = e Ge E9 e H e for some complement H, and this means that H = GI = IG for some IE F\ G. Now let (g,)l, D.) be a realization of a given valued graph (r, d). Let X = (Xi' j O.

De-

c on OriN by m, we have, for ar-

bitrary r,

dim(C+mrX')

= cmr(dim X') = dim X' + r(aX')n,

which can be arbitrarily large. On the other hand, since C+ preserves monomorphisms (for,

si;

involves only construction of a certain kernel), we have

dim(C+mrX') ~ dim(C+mrX) = dim X for all r, a contradiction. Thus (i) implies (ii). Conversely, (ii) implies (i). For, all summands in a direct decomposition of X have defect ~ 0, and the total sum of their defects is O. Hence, all are of zero defect. A dual argument yields the equivalence of (i) and (iii). DEFINITION. The representations in L('.D~, D) satisfying the properties described in Lemma 3.1 will be called regular.

Thus, we have PROPOSITION

3.2. Let

em, Q) be a realization of an extended Dynkin diagram.

Let

R(,)J/, D) be the full subcategory of all regular representations in L('.J)l, D). Then R('1ll, D) is an abelian exact subcategory of L('J)l, Q), closed under extensions. Proof. Let V, W E R('.D~, Q) and let a: V -+ W. Let a = alia' with an epimorphism a': V -+ X and a monomorphism a": X -+ W. Then, by Lemma 3.1 (ii), ax ~ 0, and by (iii), ax ~ 0; hence ax

= O.

Since subobjects of X are also subobjects of W, X is, in view of

(ii), regular. Thus, images of morphisms between regular representations are regular. On the other hand, if O-+V-+X-+W-+O is exact, then ax

= av + aw

and therefore, if any two of the representations are of zero

23

REPRESENTATIONS OF GRAPHS AND ALGEBRAS

defect, so is the third one. In fact, if any two of the representations in the above sequence are regular, so is the third one; for, subobjects of V are subobjects of X, quotients of Ware quotients of X and subobjects of X are extensions of subobjects of V by subobjects of W. This completes the proof. A simple object X of R(~l, .Q) is said to be homogeneous if dim X E N. An arbitrary regular representation is called homogeneous if all its simple composition factors (in the category R('JR, .Q)) are homogeneous. The full subcategory of R('JJL .Q) of all homogeneous representations will be denoted by H(~,)l, .Q). Now, consider the dual vector space Or. of all linear forms X: Or ---+ Q Write X = (X;) with respect to the dual basis,i.e. Xx = LiErXixi for x = (xi) E Or. For a representation X = (Xi' j'fJi) E L(9Jl, .Q), we write XX = x(dim X). For each wE W, define w* : Or. ---+ Or. by (aw*)(x)

For k E

r, sk

is an involution and, writing

= -Xk

f,k

For c

= sk n

= a (wx),

•" s

s

k2 k l'

and f,i

xsZ

=

EOI'.

X

t

we obtain

= Xi + dijXk

we have c* = s* s*k kl

{xlxc*

2

•.• s*

k n'

for

i

=I=-

k.

and

= X} = ll c

is a one-dimensional space generated by the defect vector d c . (One can verify that it is a complement of the image of Or under the mapping Or ---+ Or. defined by x ~ Br(x, -), whose kernel is obviously N.) Also {x I xc*(x) = X(x) for all X E Or*} = {x Icx = x} = N. DEFINITION.

A representation E E L(~l, .Q) is said to possess an equation if there

exists Tl E Or. such that

(i) TlE > 0; (ii) if TlX > 0 for some regular representation X, then E c.... X; (iii) if TlX < 0 for some regular representation X, then X ---- C+E. In what follows, Er = C+rE (with Eo = E) for a given E E L(~l, .Q). 3.3. Let E E L('1Jl, .Q) possess an equation Tl and assume that, under the action of the Coxeter transformation c, dim E has a finite orbit containing I ;;;. 2 elements. Then (1) C+1E ~ E and all LEMMA

Er ,

0

~ r

< I,

are (mutually nonisomorphic) simple regular nonhomogeneous representations.

(3)

'* E '* E X '* E

= 0 for all simple regular representations X,* Eo, X Ext (Er , X) = 0 for all simple regular representations X

(2) TlX

1

O~r 0, there is a direct summand V0 of R(X) of dimension type (1, 0) and T' (V0) ~ Eo· If 11X < 0, then, for the same reason as in n' there is a summand VI of type (1, 1) and T"(V 1 ) ~ E 1 · Ben" {E = Fz } is a generating set. Again, consider the contraction to 8 2 :

Zn _ 1

Zn_1

(1,2) ----4

T'(V o)

B

n-1

.

b. As before, If 11X

= Eo; and, if 11X < 0,

Bon.

{E = F z

n-2

> 0, there IS. a summand Vo of R(X) of type (1, 0) and

there is a summand VI of type (1, 1) and T"(V 1 ) ~ E 1 • ' E' = FO F F ••• F F G} is a generating set. First, consider the

30

VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL

(2,1~

b. If71X

> 0,

there is a direct summand V o of R(X) of dimension type (1, 0) and T' (V0) "'" Eo' If 71X < 0, then a summand VI of type (1, 2) contraction to B2 :Zn_2

occurs and T"(V 1) "'" E 1 . Second, consider the contraction to the diagram a2 summand

(2 ,1

~

b. If 71'X

> 0,

then a direct

of R(X) of dimension type (1, 1) occurs and T' (V~) "'" E~. If 71'X

Yo

< 0, then

there is a summand V'l of type (0, 1) in the decomposition and we get T" (V ~) "'" E'l .

COn'

{E

= Fzn-2 ' E' =. OF F F'" F F F} is a generating set. ~ b.

traction to the diagram zn_2

First, consider the con-

> 0, there is a direct summand

As before, if 71X

°

Vo

of R(X) of type (1, 0) and T"(V o) "'" Eo. And, 71X < implies the existence of VI of type (1,1) and T"(V 1 ) "'" E 1 . Second, consider the contraction to a2 ~ b. If 71'X > 0, there is a direct summand V~ of type (2, 1) and T'(V~) "'" E~. And, if 71'X < 0, then a summand V~ of type (0, 1) yields T"(V~) "'" {E = F z n _ 3' E' = ~ F F ••• F F~, E" = ~ F F ••. F F ~ } is a generating set.

E; .

i\.

First, consider the contraction to the diagram

A 3 : zn_3

/ ' b1 - b2

In the decomposition of R(X), the dimension types of the direct summands are

°?' b, ? 1

1

~; and

or 1

~.

1

and 1

~.

Now, if 71X

T'(V o) "'" Eo. If 71X

>

°

°b,

1

~,

there must be a direct summand V 0 of type

< 0, there must be a direct summand of type

°b, °?' or

But the first two cases lead to a quotient of X of negative defect - 1. Hence, there

is a summand VI of type 1

~,and

T"(V 1 )

E1 .

"'"

Second, consider the contraction to a 2

--

b 1 . Then a summand V~ of type (1, 0)

yields the subobject T'(V~) "'" E~ in case that 71'X yields the quotient T' (V ~) "'" E; if 71'X

>

< 0.

°

and a summand V~ of type (0, 1)

FinallY,71" is an equation for E~ by the same arguments as above.

~ E6 . {E

°

F

= OFF F 0,

E'

°

= OFF F F,

E"

°° F

= OFF F} is a generating set.

First, consider the contraction to the diagram

On :a 2

---+

c2 i Z

+-b 2 .

If 71X > 0, then in a decomposition of R(X), there must be a summand of dimension type 1 101 0, or 1 0, or 1, or 1 1 1. But all the first three summands will determine a sub1 object of X of positive defect. Hence, there is V 0 of type 1 1 1 which determines the subobjects 011 l'(V0) "'" Ep . If 71X < 0, then there must be a summand of type 1 0, or 1 0, or 1 1 0, or O I l 1 1, or 1 2 l, But again, with the exception of a summand VI of type 1 2 1, all other would lead to a quotient of X of negative defect. And VI determines the quotient T"(V 1) "'" E 1 .

°° °

° °°

°

°

31

REPRESENTATIONS OF GRAPHS AND ALGEBRAS

Second, consider the contraction to the diagram A 3 : az -+ z ~ b l . If 71'X > 0, then a direct summand V~ of R(X) of dimension type 0, 1, 1) must occur, which detennines the subobject T'(V~) ~ E~. And a summand V~ of type (0, 1,0) determines the quotient T"(V~) ~ E'l'

Finally the proof that 71" is an equation for E~ follows the same lines. F F 0, I)F E 7 . {E = 0 F F F F F 0, E' = 000 F F F 0, E" = 0 F x 0 F x 0 F x F F x F o x FOx F} is a generating set. First, consider the contraction to ~

1 0 If 71X > 0, then there must be a direct summand of R(X) of dimension type 0 0 0, or 1 0 0, 0 1 1 or 0 0 1, or 1 1 1. Using a by now standard argument, a summand V 0 of type 1 1 1 must 010 occur and T'(V o ) ~ Eo' If 71X < 0, then a summand of type 0 1 0, or 0 1 0, or 0 1 1, or 1

1 2 1 must occur. All but the last one lead to a quotient of X of negative defect. And the 1

summand VI of type 1 2 1 determines T"(V I ) ~ E I . Second, consider the contraction to

> 0, a summand o

If 71'X

1 V~ of type 1 1 must occur and T'(V~) ~ E~. If 71' X

< 0, V~

of

type 1 0 must occur as a summand and T" (V~) ~ E~ . Finally, consider the contraction to

c .j,

Os : az

-+

z ~ b3 ~ bl

.

o

If 71"X > 0, then there must be a direct summand of R(X) of dimension type 1 0 0 0, or 00101 1 1 1 o 0 1 0, or 0 0 0 1, or 0 0 0 0, or 1 1 1 1, or 1 1 1 0, or 0 1 1 1, or 1 1 1 1, or 1 2 2 1. Again, by routine elimination of all types with the exception of the last one, we deduce ex1

istence of V~ of dimension type 1 2 2 1 which determines T'(V~) ~ E~. Similarly, if o 0 0 0 71"X < 0, then all but the last one of the dimension types 0 1 0 0, 1 1 0 0, 0 1 1 0, 0 0 1 1, 1 1 1 o 1 0 0 and 1 2 1 0 can be eliminated. .nd the direct summand V; of dimension 1 2 1 0 determines the quotient T"(V;) ~ E~. ~ F 0,1)F E s . {E = 0 0 0 0 F F F 0, E I = 00 F x 0 F x 0 F x 0 F x F F x FOx F,

0, E"

=0

ox

F x 0 x 0 F x 0 x 0 F x F x 0 F x F x 0

0 x F} is a generating set.

1,0)F+(0, 1, 1)F F x F x FOx F x F

32

VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL

Consider the contraction to c

04 :a s

~

--+

and proceed as in the case of 1/ in E or E 6

z

+-

bZ '

7.

Second, consider the contraction to c

Os :a 3

~

--+

z

+-

E

b z +- b 1 ,

and proceed as in the case of 1/" in 7 . Finally, consider the contraction to c

E6 :a z If 1/"X

> 0, there are

--+ a4 --+

~

z

+-

b z +- b 1 .

18 possible dimension types of direct summands of R(X), for which the 1.

Jl"value is positive. However, all, with the exception of the type 1 2 3 2 1 determine a subobject of negative defect. And, the direct summand V~ of the mentioned type yields the subobject T'(V~) ::::; E~. If 1/"X < 0, there are 10 possible dimension types of direct summands of R(X) having 1/-value negative. Again, by simple elimination, we conclude that a 1

summand V~ of dimension type 1 2 3 2 1 must occur and T"(V~) ::::; E'~. 41' {E = 0 0 F F F. E' = 0 G G F F} is a generating set. First, consider the con-

F

.

(l,Z)

tractIOn to 8 3 : a3 ---+ z +- b. If 1/X > 0, there must be a summand V0 of type (2, 1, 1) in the decomposition of R(X) and T' (Vo ) ::::; Eo' If 1/X < 0, there is a summand VI of type (2,2, 1) and T"(V 1 ) ::::; E 1 . Second, consider the contraction to az V~ of type (1, 1, 1): T'(V~) ::::; E~. If 1/'X

~ z +- b.

< 0, then

If 1/'X > 0 there is a summand a summand V~ of type (1, 1,0) oc-

curs and T"(V~) ::::; E~. F4Z' {E = 0 F F G G, E' = 0 F F F O} is a generating set. First, consider the contraction to 8 3 : a z --+ z~ b 1 • If 1/X > 0, then there is a direct summand V0 of R(X) of dimension type (1, 1, 1) and T'(Vo ) ::::; Eo. If 1/X < 0, then the existence of a summand VI of type (0, 1, 1) yields the quotient T"(V 1 ) ::::; E 1 . Second, consider the contraction to az --+ z ~ b z . If 1/'X > 0, we establish easily the existence of a direct summand V~ of R(X) of type (1, 1, 2) : T'(V~) ::::; E~. And, if 1/'X < 0, there must be a summand V~ of type (1, 2, 2): T"(V~) ::::; E'l' GZ1 ' {E = 0 F F} is a generating set. Consider the contraction to the Dynkin dia(1,3)

.

gram Gz :az ---+ z. If 1/X > 0, then there must be a dlfect summand V0 of R(X) of dimension type (3, 1) and T' (Vo) ::::; Eo' If 1/X < 0, then there is a summand VI of type (3,2) and T"(V 1 ) ::::; E 1 . Gz z. {E = 0 F G + fG with f E F\ G} is a generating set. Consider the contraction

to G z :z ~ b. If 1/X > 0, then there is a direct summand Vo of R(X) of dimension type (1,2) determining T'(Vo ) ::::; Eo. And, 1/X < 0 implies the existence of a summand VI of type (1, 1). Thus there is a quotient T"(V 1 ) ::::; E 1 of X.

33

REPRESENTATIONS OF GRAPHS AND ALGEBRAS

5.

HOMOGENEOUS REPRESENTATIONS

In this section we assume again that ('JJ/ , Q) is a realization of an extended Dynkin diagram (r, d). We want to show that the study of the homogeneous representations can be reduced to the study of the homogeneous representations of a realization of a diagram of type All or A 1 2 . The realization of a diagram of type Allor Ai 2 will simply be called a bimodule, and (d 12 ,d 21 ) i is such a diagram with the oriendenoted by FMG; more precisely, if (r, d) = i tation n defined by i ~ 2' and the modulation IJR = (Fl' F 2• 1M2' 2M1)' we write simply F = F 1 , G = F 2 and FMG = (1M2) . Then FMG determines completely the F1 F2 realization ('JJ1, Q) and, in this way, we will consider FMG as the realization of (r, d). Thus, a representation (UF , VG , tp) of FMG consists of two vector spaces UF , VG , and a G-linear mapping tp: UF ® FMG ---+ V G' Note that, for every FMG' R(FMG)

=

H(FMG)'

5.1. Let (IJJ/, Q) be a realization of an extended Dynkin diagram, and let R('JJ/, Q) = H(IJJ/, Q) x R(l) x ... x R(h). Then there exists a bimodule FMG of type THEOREM

Allor A 12 , a full exact embedding T: H(FMG) ---+ R(IJJ?, Q) and h simple objects R(2), . . . ,R(h) in H(FMG) such that

R(l),

(i) the image of objects of H(FMG) under T have continuous dimension types; (ii) for all t, T (R(t») E R(t); (iii) the full subcategory of H(FMG) of all objects without subobject of the form R R(t), I :0;;;; t :0;;;; h, is equivalent to H('JJl, Q) under T.

As a consequence, we get also some information about the category H(FMG)' Namely, H(FMG) is the product of h + I categories; h of these are uniserial categories with a unique simple object R(t) and the remaining one is described in (iii); the objects have no composition factors of the form R(t). The proof of the theorem will consist in a case by case inspection. In the tables of Chapter 6, there are listed a biMlodule FMG' a functor T: L(FMG) ---+ L(IJR, Q), and the representations R(t) of FMG' As we will show, these data satisfy the following conditions: (0) T is a full and exact embedding (or, at least, the restriction of T to the category Le(FMG) of all representations (UF , V G' tp) with a surjective mapping tp is a full and exact embedding). (i) If X is a representation in Le(FMC)' then T(X) has continuous dimension type if and only if X has continuous dimension type. (ii)' T(R(t») contains a simple object of R(t) as a subobject, and End R(t) is a division ring.

(iii)' Every homogeneous representation of (9)/, Q) is an image under T. We claim that these conditions imply the assertions of the theorem. For, since T: Le(FMG)---+ L(IJJ/, Q) is a full embedding, any representation X in Le.(FMC) is indecomposable if and only if T(X) is indecomposable. Now, by (ii)', T(R(t») is indecomposable, and has no subobject in R('JJl, Q) of continuous dimension type. Therefore R(t) has to be simple, because T is exact and satisfies (i). Since T(R(t») is indecomposable and contains a simple subobject of R(t), it belongs to R(t). Let C be the full subcategory of all

34

VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL

objects of H(pMd without subobjects of the form R(t), I ~ t ~ h. If Y =1= 0 belongs to C, then Hom(R(t), Y) = 0 (since R(t) is simple) for all t; thus, also Hom(T(R(t), T(Y)) = 0, and therefore T(Y) is homogeneous (using the remark at the end of Chapter 3). This shows that the restriction of T to C gives a full and exact embedding of C into H(Wl, n). But every indecomposable homogeneous representation of ('JJl, n) is of the form T(Y) with an indecomposable representation Y and of continuous dimension type. Also, no R(t) can be embedded into Y; for, otherwise, T(R(t)) c... T(Y~and T(Y) would belong to R(t). This implies that the functor T: C --+ H(IJR, n) is dense and therefore T is an equivalence. Now, we are going to consider the individual extended Dynkin diagrams. In all cases, condition (i) is satisfied trivially and, in most cases, it is also very easy to see that the condition (iii)' is satisfied: one uses the properties of a simple regular representation X with T/(X) = 0 which are listed in the last column of the tables. These properties are satisfied for every simple homogeneous object, and therefore for every homogeneous representation at all. As a r~ult, we are mainly concerned with conditions (0) and (ii)'. An' It is obvious that T(R) contains

o --+ 0 ... --+ 0 --+ N F "'" 0, 0-----" "'" 0 --+ 0 ••• --+ 0 --+ 0 ? whereas T(R') contains

o -----"

0--+0--+ '" 0--+0

"'"

O.

"'" 0 --+ 0 --+ '" 0 --+ F F-----"

. mult Smce F Cl ® Cl F C2 --+ F C2 comes, in fact, from an F-linear mapping F Cl ® ClFF ~ FF' its kernel K is an F-subspace of F Gl ® GlFF' and dim K F = 1. Thus T(R) contains 0--+ 0 --+ ••• 0 --+ K F --+ 0 ~B

n"

as a subobject.

en'

is K F2

Obviously, the kernel of 1 ---> 1 --->

4

-

0 or b = 0 and a < 0, C xC

for b

=0

(~ ~)

: C xC,

and a ;> 0,

C

: C, a

and for the point "", again

C

o

: C.

The endomorphism ring of a representation corresponding to (a, b) with b

> 0 is C,

57

REPRESENTATIONS OF GRAPHS AND ALGEBRAS

to (a, 0) with a

< 0 is

H, to (a, 0) with a

> 0 is

R and to the points (0, 0) and

00,

it is C.

From the above paper of C. M. Ringel, it follows easily that, given a K-realization

('JJ/, Sl) of a connected valued graph (f. dl. L('JJ/, Sl) is of tame representation type if and only if (r, d) is an extended Dynkin diagram. Namely, for all valued graphs with an indefinite quadratic form and for all K-realizations

('JJ/, Sl), the category L(IJJ/, Sl) is of wild repre-

sentation type in the sense that there is a full exact embedding of the category of all modules over a free associative algebra with two generators over a commutative field.

VlASTIMI l

DlAS (CARLETON, OTTAWA) AND

CLAUS MICHAEL RINGEL (BONN)