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
> -- ..... _-.<; '._._1_._.. '
._._._1_._._ .., ._._1_._._._._ .. (1,2) .--.--.--.--., , · _
G21
. _
. (2,1). _
(1,3)
• --. --.;
G22 . _ . (3,1).
or
.',
3
4
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
then the category L('JJl , il) has two kinds of indecomposable representations: those of discrete dimension types and those of continuous dimension types. The mapping dim: L('JJ?, il) ~ Or induces a blj'ection between the isomorphism classes of indecomposable representations of ('JJ/, il) of discrete dimension types and the positive roots of cr, d). The continuous dimension types are the positive integral multiples of the least stable positive integral vector of Or. Moreover, the indecomposable representations of continuous dimension types can be derived from the indecomposable representation of continuous dimension type of a suitable realization of the graph Allor A l2 . In fact, in the case of an extended Dynkin diagram, we can give a more detailed description of the category L('JJ?, il) as follows: The full subcategory of all direct sums of indecomposable representations of
c~ntinu
~ur dimension type is closed under images, kernels and cokernels only for diagrams All and
A l2 . However, the full subcategory of all images of maps between direct sums of indecomposable representations of continuous dimension type is an abelian exact subcategory of
L('JJ/, il) which will be called the category R('JJ?, il) of regular representations of em, il). The largest direct factor of R('JJ?, il) containing only representations .of continuous dimension type will be called the category H = category U satisfying R('JJ?, il)
=Hx
H('JJ?, il) of homogeneous representations. The
U can be described completely: It is an abelian cate-
gory with a finite number of simple objects, and every indecomposable representation belonging to U is serial (i. e. has a unique composition series in U). In fact, U is the direct product of at most three indecomposable categories, and each of these has global dimension one. Furthermore, given ('JJ/, il), there exists a realization ('JJl I, n') of either Allor A l2 which determines all indecomposable representations of continuous dimension type of
L('JJ?, il) in the following sense: There exists a full exact embedding T: L('JJ/', il') ~ L('JJ/, il) and (at most three) representations X in L('JJ/', il') such that the full subcategory of L('JJI', il') of all regular representations without subobjects of the form X is equivalent to H('JJ/, il). The indecomposable representations of ('JJl, il) which are non-regular can be described in the following way: There are two endofunctors C + and C- on the category L('JJl, il) called the Coxeter functors, having the property that the list of all representations C- m P and C+ m Q where m ~ 0, P is an indecomposable projective representation and Q is indecomposable injective representation, is just a complete list of all non-regular indecomposable representations. The dimension types of these representations correspond to c-m(dim P) and cm(dim Q), where c is the corresponding Coxeter transformation. Also, there is a numerical invariant, called the defect, characterizing the behavious of indecomposable representations X in L('JJ/, il): the defect of X is negative, or zero, or positive if and only if X is of the form C-mp, or X is regular, or X is of the form C+mQ, respectively. Let us observe that a change in orientation of a valued graph results in a change of the indecomposable projective and injective representations, as well as the Coxeter functors. Thus, also the dimension types of the regular representations depend on the orientation. However, the categories R('JJ/, il) and R('JJ1, il') with an arbitrary modulation '))/ and orientations il
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
5
and n', are equivalent except in the case of the diagram An; in fact, the equivalences are given by functors similar to the Coxeter functors. For the diagram An' the category R( ,n) is determined by the number of arrows pointing in one direction in the orientation n. Thus, in summary, we may consider in the tables of Chapter 6 a fixed orientation. The results on the representations of valued graphs can be translated to the theory of representations of finite-dimensional associative algebras over a field or, more generally, to certain classes of artinian rings. Indeed, given an artinian ring R, we obtain a valued graph r R = (r, d)R of R as follows: Assuming, without loss of generality, that R is basic, we have R/Rad R = TI 1 <;; i <;; n F j and Rad R/(Rad R)2 = TI 1 <;; j.; <;; n jN; with uniquely determined division rings F j and F(Frbimodules jN;; let {l, 2, ... , n} be the set of vertices of the graph
rR
*
(d'd")
and let, for each jN; 0, i -'- j with d' = dimCN;)F; and d" = dimFjCN;) be a Now, if for each pair {i, jN; = or ;Nj = 0, this graph is a valued "valued" edge {i, graph and 'J)~ = {F j , jN].} ., is a modulation of this graph provided that there exist
n.
°
n,
1~
I,} ~
n
numbers I j with dj;ij = d;Jj and that the dualization conditions HomF /jN;, F j ) "'" HomF;CN;, F;) hold. In particular, these are satisfied in the case that R is a finitely generated algebra over a central field K; then, also the numbers I j can be interpreted as the in-
*
dices [F j : K]. Define the orientation n of r R by i ~ j if jN; 0. In this way, one can assign to many artinian rings a modulation of a valued graph and an orientation. Conversely, given a modulation m of a valued graph and an orientation n, the category L(m, n) is equivalent to the category of all right modules over the tensor ring R = R('1J1 ,n) defined as follows: R = EB t ;;> Mt), where S = MO) = TIl <;; j<;; n F j , Ml ) = TIl <;; j,;<;;n jN;, and Mt) = M t - l ) ® S M l ) for t;;;;' 2, with the component-wise addition and the multiplication induced by taking tensor products. If n is, in addition, an admissible orientation, then R( '1)/,n) is always an artinian hereditary ring. From here, one can deduce the fact that a hereditary finite-dimensional K-algebra R if of finite representation type if and only if (r, d)R is a disjoint union of Dynkin diagrams. In fact, through the above translation, one can obtain in this case a complete description of the category of R-modules. In a similar
°
way, a complete description can be also given for the category of all modules over a finite-
°
dimensional K-algebra R with (Rad R)2 = using the so-called separated diagram of R. For a more detailed account on these questions, we refer to [9] and [4]. The definition of a valued graph excludes graphs with loops and multiple edges. Also, we consider only orientations which are admissible. In particular, the graphs
..,
-
and An (n;;;;' 1) with the cyclic orientation
° •
~
.. ------+ • ------+ ••• ------+ •
1 2 n do not appear explicitly in the paper. Some ~f these are of importance in applications; indeed, the classification of representations of A
° leads to Jordan normal forms of square
6
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
matrices and the classification of representations of· =4 . leads to the Kronecker's classification of matrix pairs. However, it is easy to see that the representations of An (n;;;;' 0) with respect to the orientation •
o
~
• ----+ • ----+ ••• ----+ •
1
n
2
can be identified with the representations X of IS.. n + I with respect to the orientation
• ----+ • ----+ •
o
such that the mapping XII of the graph
+-
~
••• ----+ • +----
•
1 2 n n+l X n + I is the identity. Also, the category of all representations
(d l ' d'l)
a
with respect to the realization F 1
is precisely the category of all representations of the valued graph
(~ dt, ~ cit) ffi F M
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
7
transformations, the Coxeter functors are defined and studied in the second chapter. The results of these chapters furnish, in particular, the arguments for the proof of part (a) and the description of all indecomposable representations of non-zero defect in part (b) of the Theorem. The third chapter deals with the theory of indecomposable representations of zero defect which is then applied to determine all simple representations of zero defect in the following fourth chapter and to advance the proof of part (b) of the Theorem. Moreover, the results of Chapters 3 and 4, provide a further refinement of the part (b) which is stated in Theorem 3.5. The description of the homogeneous representations in terms of representations of the graphs
A11
and
Al 2
is given in Chapter 5. And, finally, the results on extended
Dynkin diagrams are tabulated in Chapter· 6. 1. VALUED GRAPHS: COXETER TRANSFORMATIONS, DEFECT AND LISTING OF ROOTS Let (f, d) be a valued graph and Q r the corresponding rational vector space. Define a symmetric bilinear form B = B r on a r as follows: For x, yEar, 1 B(x, y) = fixiYi - 2" ~ dij!jxiYj'
1; 1
I,
J
Thus, the corresponding quadratic form Q = Q r can be expressed in the form
Q(x)
= B(x, x) = Lfix; - L
dij!jxixj'
{i, j}
i
Note that Band Q are defined up to a positive rational multiple. One can easily verify the following LEMMA 1.1. For every kEf and every x Ear, the linear transformation Sk of a r
defined in Introduction has the form SkX Consequently, s~
=X
-
2
B(x, k) ( ) k. B k, k
= 1 and B (SkX, SkY) = B(x, y) for all x, y afar.
The following proposition can be easily derived from Theorems 1 and 4 ofN. Bourbaki
[3] , VI, 4. PROPOSITION 1.2. (a) (f, d) is a Dynkin diagram if and only if its quadratic form is posi-
tive definite. (b) (f, d) is an extended Dynkin diagram if and only if its quadratic form is positive semidefinite. For the benefit of the reader, we give a brief proof of Proposition 1.2. First, if (f, d) is a Dynkin, or an extended Dynkin diagram, then a direct calculation shows that Qr is positive definite or semidefinite, respectively. (The quadratic forms Qr of extended Dynkin diagrams are listed in Tables, Chapter 6.) Thus, let Q r be a positive quadratic form of a connected graph (f, d). Clearly, every edge {j, k} of (f, d) satisfies d jk x d kj ~ 4; for, taking x E a r with x j = Yz d kj , x k = 1 and Xi = 0 otherwise, Q(x) = Yzfk (4 - d jk x d kj ). By a similar argument, every pair of edges
Nl -
{j, k} and {k, l} with d jk x d kj = 3 satisfies d kl x d lk = 1: Q(x) = d kl x d lk ) for x E af' with x j = d kj x d lk , x k = 2dlk , Xl = 1 and Xi = 0 otherwise. Also, every circuit
8
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
= io are its vertices, then, = 1 (0 ~ r ~ m) and xi = 0 otherwise, Q(x) = L~=O I Ii . (1- di i I. r r r+ I r' Therefore, all d i i = 1 and subsequently also all d i i = 1. r+I r r r+ I
contained in (r, d) is of type Am' m ~ n; for, if i o ' iI' ... , i m for x E Or with Xi
Now, in order to complete the proof, it is sufficient to show that the quadratic form Qr', of every (full) subgraph (r', d') of (r, d) is necessarily positive definite. For, as a consequence, no extended Dynkin diagram can appear as a proper subgraph of (r, d) and a straightforward combinatorial argument establishes both (a) and (b) of Proposition 1.2. Thus, assume that (r', d') be a minimal subgraph with the property that Qr' is not positive definite. Let
U,
k} be an edge of
(r, d)
such that j E
r'
and k f/=.
Qr'(x') = O. Then, in view of minimality of (r', d'), the vector x" whose components are
x;
I I, i
E
r',
r'.
Let 0 =1= x' E Or' satisfy
x; =1= 0 for all i E r'.
In fact, since
always satisfies Qr'(x") ~ Qr'(x'), we de-
duce that all components of x' are positive or that all are negative. Assume elat all i E and Xi
r', and consider the vector = 0 otherwise. Then
x E Or defined by Xi = x; for all i E
r',
x; > 0 for
x k = Y2djk xj
(' 1 2'2 1 ~ f' d ' _ ( ) -Qr,X)-J/kdjkXj Qrx -2~JkjkdikXjXi' I
where the summation runs through all edges {i, k} with i E
r'.
Thus Qr(x)
< 0 in contra-
diction to our assumption and the proof is completed. Following N. Bourbaki [3], the elements of the Weyl group W which are of the form
c
= Sk n ••• Sk2skl'
where k 1 , k 2 , ... , k n are all vertices of r (in a certain order) will be called Coxeter transformations. Obviously if c is a Coxeter transformation, so is c- 1 . Note that given c = sk
n
we may prescribe a unique orientation Dc to (r, d) such that the ordering k I' k n is admissible. On the other hand, to every admissible orientation D of (r, d),
... Sk 2 SkI'
k2 ,
... ,
there exists (in an obvious way) a Coxeter transformation c such that Dc
= D,
and c is again
uniquely determined. For, if k l , k 2• ... , k n and k'I' k~ • ...• k~ are two admissible orderings of r with respect to D, then Sk n ••• sk2skl = Sk~ ••• skisk'l' since Si and Sj commute for any pair of vertices i, j E r which are not neighbours. LEMMA 1.3.
Let (r, d) be a valued graph and z E Or. Then the following statements
are equivalent: (i) cz
= z for a Coxeter transformation c; = z for all w E
(ii) sk z = z for all k E r (i. e. wz (iii) B(z, y) = 0 for all y E Or.
W);
If, moreover, the corresponding quadratic form Q is positive, then the above statements are equivalent to (iv) Q(z) = O. Proof Let c the equality
= sk n
•••
Sk2skl' The fact that (i) implies (ii) follows inductively from
Zk = (sk t
n
••• sk
The latter yields immediately that sk z 1
t+l
sk z)k t
t
for all 1 ~ t
< n.
= sk 2 Z = ... = sk n Z = z.
Also, using Lemma 1.1,
9
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
we see that Skz
=Z
is equivalent to Br(z, k)
basis of Or, the statements skz
=Z
= O.
Consequently, since the vectors k form a
for all k E rand B(z, y)
=0
for all y E Or are equiva-
lent. And, the equivalence of (iii) and (iv) is obvious. DEFINITION.
Given a valued graph (r, d), define its radical subspace by
= N r = {x E Or Iwx = x for all wE W}. Thus, if (r, d) is a Dynkin diagram, then N = {O}. And, if (r, d) N
is an extended Dyn-
kin diagram, then N is a one-dimensional subspace of Or generated by the "canonic" vector
= n r ; these vectors are listed in the tables of Chapter 6. Note that all components of the vectors n are positive and that, in each case, at least one component of n equals 1. Moren
over, it has been shown in [1] that the existence of a positive vector in N implies that (r, d) is an extended Dynkin diagram. Notice that the set R
= {x E 0 r I x = wk
for some w E Wand k E r} of all roots
satisfies the defining properties (but the finiteness condition) for a reduced root system of [3] : R generates 0 r; for y E R, the linear transformation
B(x, y) x 1--+ x - B(y, y) y
maps R into R; for x, y E R, 2B(x, y)/B(y, y) is an integer; and, if x E R, then 2x ~ R. In particular, if x E Or is a root and x
= wk, then -
x
= w(- k) = (wsk)k
is also a
root. Furthermore, as an immediate consequence of the following more general result, one of the roots x or - x is always positive (i. e. LEMMA
xi ;;;;,
0 for all i E
n.
1.4. Let (r, d) be a valued graph such that the corresponding quadratic form
Q is positive. If x is a positive root and k E Proof. Let x
= wj
r,
then either Skx is positive or x
for a certain j E r. Since 0 :s;;; B(x ± k, x ± k)
± 2B(x, k), we get
IB(x, k) I :s;;;
k(B(k,
k)
+ BU, j))
=
= k.
= B(x, x) + B(k, k)
k(fk + fj).
Thus, we have for the integer 2B(x, k) the following inequality
+ fj):S;;;
- (fk First, if 2B(x, k) = f k
+ fj,
2B(x, k) :S;;;fk
+ fj.
then B(x - k, x - k) = 0, and thus x - kEN. Thus, if
= k. Otherwise, x = k + hn with a suitable integer h. SkX = hn - k > 0, because all components of n are;;;;' 1.
(r, d) is a Dynkin diagram, then x Therefore, in case that h =1= 0,
Second, if 2B(x, k) :s;;; 0, then
sk x
= X -
B(x, k) 2 B(k, k) k
has all components greater or e.qual to those of x, and is therefore positive .. Finally, let 1 :s;;; 2B(x, k)
< f k + fj. 2B(x, k)k
we get
Then, calculating the kth component of = (x -
skx)B(k, k),
10
VLASTIMIL DLAB AND CALUS MICHAEL RINGEL
hence, x k
> 0, and
therefore x k
Now, 2B(x, k)IB(k, k) sequently,
;;;,
1.
< 1 + fjlfk , and Sk X =
therefore, if fj <,.fk' 2B(x, k)IB(k, k) ~ 1. Con-
B(x, k) 2 B(k, k) k
X -
is positive. Thus, we have so far proved our Proposition in the case that fj shall reduce the problem to this case as follows: Consider the valued graph (f', d') defined by f' = IIf;. Further, define the delation ~: Or choose
f;
Yi
= (~x)i = fix i
=f
and d;j
~
f k . 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
st X = Y = (Yi , jl/J;) as
let Yk be the kernel in the diagram 5
0K. k )j 0-+ Yk~ jl/J i Also, if a = (0:;): X {3i
= O:i
= j
-+
for i :f= k and
(k
EB x® .Mk~Xk' jEr J J
for i:f= k and jl/J k
= jK k : Y k
® kMj
X' is a morphism in L(Wl , D.), then
(3k :
ffi
Yk
-+ Y~
(O:j ® 1):
jEr
-+
lj.
st a = (3 = ({3;) is defined by
as the restriction of
ffi jEr
X j ® jMk
-+
ffi jEr
X; ® jMk ·
st X E
L(9Jl, skD.) with changed (!) orientation. Similarly, if k is a source of (9R, D.) and X = (Xi' j
Note that Y
= (Yi , jl/J i ) as
follows:
5 Here, and elsewhere, put
k'Pj
=0
for all
jE r
which are not neighbours of
k.
=
16
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
= Xi
Yi
for all i
=1= k;
let Y k be the cokernel in the diagram C-:{;k)i
Xk~
(k 1T )i Xi 0 i M k - Y k ~O;
EB iEr
j 1/1 i
= j'Pi
for j
=1=
k
and In this way, we get S; X E L(''1JI, sk,Q), again with a changed orientation. And, if a : X ~ X' is a morphism in L(:Dl, ,Q), then S; a = {3 = (13 i ), where f3 i = (Xi for i =1= k and 13k is the map induced by ffijO::j 0 1. It can be easily seen that is left adjoint to st; however, this fact will not be used in the paper.
S;
r,
Now, for each k E
define the representation F k E L(9JI, ,Q) by F k
= (Xi'
j'P i)'
where Xi
=0
for i
= Fk
k, X k
=1=
and all j'Pi
= O.
Observe that the representations F k are just the simple objects of L(9JI, ,Q). PROPOSITION
r
(i) Let k E
2.1. Let (9JI,,Q) be a realization of(r, d). Let X E L(9JI, ,Q). be a sink. Then there is a canonical monomorphism
J1.
:S;stx~ X.
In fact, J1.(S; st X) has a complement in X which is a direct sum of copies of F k' Thus, if X is indecomposable, then either
st X = 0,
X"'" Fk and or
J1. is an
isomorphism in which case
End (st X) "'" End X and (ii) Let k E
r
dim (st X)
= sk(dim X).
be a source. Then there is a canonical epimorphism €:X~sts;x
Again,
€
has a section
€':
st S; X ~ X and
X is a direct sum of €'(st
s; X) and of copies
of F k' Thus, if X is indecomposable, then either
X or
€
= F k and S; X = 0,
is an isomorphism in which case
End (S;X) "'" End X and dim (S;X)
= sk(dimX).
Proof (i) In the diagram
o ~ (st X)k
~
EB jEr
Xj 0 \
.Mk ~ (S; st X)k J(k'P.). •••• JJ,
.' J1.k Xk
J<'
~ 0,
REPRESENTATIONS
OF
17
GRAPHS AND ALGEBRAS
Ilk is obviously always a monomorphism. Now, if (k'fJj)j is not surjective, then X is a direct sum of the representation Y = (Yi , jl/J), where Yi = Xi for i -=1= k, Y k = Im[(k'fJj)j] and all jl/J i = j'fJi' and of copies of F k . Thus, if X is indecomposable, then X"'" F k (and then S; X = 0) or (k'fJj)j is surjective. In the latter case, Ilk is an isomorphism, and we have S;; S; X "'" X. Consequently, also S; S;; S; X "'" S; X, and therefore the composition of any two maps in S+ SS+ End X -':... End (S; X) ~ End (S;; S; X) ~ End (S; S;; S; X) is a bijection. Hence, End(S;X)k "'" End X. Finally, the exact sequence
0-+ (S;X)k -+
E9 iEr
Xi ® iMk -+ X k -+ 0
yields the formula dim[(S;X)k]p
k
= iEr L =-
dim(Xi®iMk)p -dim(Xk)p k k
dim (Xk)Pk
+
L
iEr
dimCMk)p • dim(X)p.; k
I
thus, since dim'CMk)Pk = d ik and (S;X)i = Xi for i -=1= k, we get dimeS; X) = sk(dim X). (ii) The second part of Proposition 2.1 can be proved in a similar way as (i) considering the diagram
Xk
,,'" j(,, ,), 0-+ (S; S;; X)k -+
E9
Xi ® iMk -+ (SI: X)k -+ O.
iEr
Let us illustrate the use of Proposition 2.1 on the following example. EXAMPLE. Let G be a subfield of F with dimG F = dim F G = d such that there exists a nontrivial bimodule mapping
f
:
GFG -+ GGG' Then
1m = (F 1 = G, F2 = F,
1M2 =
GFp, 2M1 = pFG) is a modulation of (r, d) = i (1,d) 2' Let Q be the orientation i -+ 2' Now, f defines a bimodule embedding w: pFp -+ pFG ® GFp (such that (1 ® €)(w ® 1) is the identity of F G = Fp ® pFG = F G ® GG G)' If V G is a G-subspace of a vector Fspace Vp, then the mapping V G c.... Vp determines a representation of (9J1 , Q). We claim that there exists a unique indecomposable representation X = (VG , Vp , 'fJ) of(9JL Q) of dimension type (d, d - 1) given by the canonical mapping
and that this mapping is an inclusion. Moreover, the endomorphism ring of X is F, which operates on pFG ® GGG and (pFG ® GFp)/(w(pFp) from the left canonically. For, consider the obvious representation Y = (FG, F p , F G ..l- F p ® pFG ) of (9J1, S2 Q) and apply the functor S~. Thus, we consider the mapping w = T: Fp -+
F G ® GFp, and form the cokernel F G ® GFp -+ (FG ® GFp)/w(F). This shows that X = SlY' Since we know that S;Y is indecomposable, the mapping V G -+ Vp has to be
18
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
an inclusion. Since End Y = F with the canonical operation from the left, the same is true for X. Also the representation X is uniquely determined because of the fact that X = SlY' Here, we make use of the fact that the only indecomposable represenX= tation of dimension type dim X is Y. Later, in Proposition 2.4, we shall determine the indecomposable objects of L(IJJl, Q) which are projective or injective. The first step in this direction is the following rather obvious
S;Si
Si
2.2. Let em, Q) be a realization of (r, d). (i) If k E r is a sink, then F k E L(IJJl, Q) is projectilJe. (ii) If k E r is a source, then F k E L('JR, Q) is injective. LEMMA
2.3. Let X and X' be indecomposable representations in L('JR, Q). is a sink and X' "* 0, then induces an isomorphism Ext1(X, X') ~
PROPOSITION
st
st
(i) If k E r Ext 1 (St X, X'). (ii) If k E r is a source and S;; X"* 0, then S;; induces an isomorphism Ext1(X, X') ~ Ext1(S;;X, S;;X').
st
st
Proof Let k E r be a sink. If X = 0, then X ~ F k is projective and therefore O. Ext1(X, X') = O. Thus, we may assume that We want to show that carries exact sequences 0 ---+ X' ~ E ~ X ---+ 0 into exact sequences. Obviously, we need only to verify that the kth component of
st
stX"*
S+ #.L
S+ e
o ~S+X' ~S+E ~S+X ---+ 0 k k k is exact. To this end, consider the following commutative diagram
-----+lO
!
a
1
a
o
By assumption, the middle and the lower rows are exact. The first and the third columns X "* 0 "* S+ X'; therefore, also the are exact, because X and X' are indecomposable and second column is exact (the extension of two epimorphisms is an epimorphism). Finally, the upper row is exact because the formation of kernels is left exact, and a dimension argument e)k is surjective. implies that In this way, defines a map Ext1(X, X') ---+ Ext1(Stx, and obviously S;; defines the inverse map. This proves (i). The second part of Proposition 2.3 follows in a similar fashion.
st
(st
st
stX'),
19
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
Now, let k 1 , k 2 ,
... ,
k n be an admissible ordering of f' with respect to the orienta-
tion n. Then
are functors from L('JR, n) to L('1Jl, n) (the same orientation!) and are called the Coxeter functors of L('JR, n). Note that both C+ and C- depend only on the orientation nand not on a particular admissible ordering of f'. Obviously, they depend on a particular choice of the bimodule mappings jMi "" HomFiCMj , F i ) "" HomF/iMj , F j );
however, the choice of these mappings is irrelevant for the theory. Also, for I ~ t ~ n, we introduce the representations Pk and Q k t
Pk
t
= Sk-1 Sk-2
... Skt- 1
Fk
t
'
where F k E L(9Jl, t
from
t
sk sk ••• sk t t+ 1 n
L(~)l,
n) by
n),
and
PROPOSITION
2.4. Let C+ and C- be the Coxeter functors for L(9Jl, n) and c the
corresponding Coxeter transformation. Let X be an indecomposable representation from L(iJR, n). Then the four statements, both in (i) and (ii), are respectively equivalent. (i)
(1) (2) (3) (4)
X is projective; X"" Pk for some kEf'; C+X = 0; c(dim X) ~ O.
(ii)
(1) (2) (3) (4)
X is injective; X "" Q k for some kEf'; C-X = 0; c-1(dim X) ~ O.
Proof Let k 1 , k 2 , Proposition 2.1, we get C+p
kt
= S+k
n
... ,
k n be an admissible ordering of f' with respect to n. Using
... S+ S+ S- S- ... Sk2
kl
kl
k2
kt-l
F
kt
"" S+ ... S+ F kn
st
kt
kt
= O.
st st
Conversely, if C+ X = 0 and if t is the smallest index such that t ... 1 X = 0, then 2 S+k ... Sk+ Sk+ X"" F k ,and therefore X "" Sk- Sk- ... SkF = Pk · The relations t- 1 2 1 t 1 2 t- 1 k t t between the corresponding dimension types result immediately in the equivalence of (3) and (4). Now, if we show that all Pk • kEf', are projective, then these n pair-wise nonisomorphic indecomposable representations must be just all indecomposable projective objects of
L(9Jl, n). Indeed, L(m, n) is equivalent to the category of all right R-modules, where R = R(''J)/ .n) is the tensor algebra constructed from the corresponding realization ('JR, n) (see [9] or [4]), and since there are no circuits with cyclic orientation, R is semiprimary. Therefore, the indecomposable projective representations are precisely the projective covers of the simple representations. Thus, let X' be an indecomposable representation of (9Jl, n). First, assume that
20
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
"*
Sk+ ••• Sk+ S; X' O. Then, by Proposition 2.3, t-I 2 1 Extl(P X') = Extl(S- S- ••• SF X') ~ Ext l (Fk S+ ••• S+ S+ X') kt' kl k2 kt-l k t ' t' kt - l k2 k l ' and this is zero because Fk E L(IJ)~, Sk sk .•. sk Sl) is projective according to Lemma t t t+ 1 n 2.2 (k t is a sink with respect to sk sk ••• sk Sl). But, if Sk+ .•. S; S; X' = 0, t t+ 1 n t- 1 2 1 then X' = Pk with s < t, and making use of Proposition 2.3 again, we get s
Extl(P k ,X') t
= Extl(Pk t 'Pks )
~ Extl(S; .•• S; s
t- 2
Skt-I F k t , F k s ).
Thus, assume (without loss of generality) that s = 1 and consider an extension 0
---+ F
kl E ---+ Pkt ---+ 0 with E = (E i , jtflJ We claim that the mapping EBjET E j ® jMkl ---+ E k 1 is not surjective; this follows from the fact that it is an extension of the corresponding map in F k ,which is zero, by the corresponding map in Pk ' which is an iso1 t morphism. This shows that E has a direct summand of the form Fkl , and, in this way, we get a splitting of the given sequence, as required. The second part of Proposition 2.4 can be proved by dual arguments. ---+
2.5. Let (~, Sl) be a realization of(r, d). Let C+, C- be the Coxeter functors and c the Coxeter transformation with respect to Sl. Let X E L(lJ)l, Sl) be indecomposable. Then (i) either X ~ Pk for some k E rand c+ X = 0 or X ~ cC+X, End(C+X) ~ End X and dim(C+X) = c(dim X); PROPOSITION
(ii) either
X
~
Q k for some k E rand C-X
= 0,
or
Proof Both statements follow immediately from Proposition 2.1 and 2.4. The preceding Proposition 2.5 together with Proposition 1.9 imply further PROPOSITION 2.6. Let (lJ)l, Sl) be a realization of a valued graph (r, d). (a) If (r, d) is a Dynkin diagram, then the mapping dim: L(IJ)~, Sl) ---+ Or provides a one-to-one correspondence lJetween all positive roots of (r, d) and all indecomposable representations in L(~, Sl). (b) If (r, d) is an extended Dynkin diagram, then the mapping dim provides a oneto-one correspondence between all positive roots of (r, d) of nonzero defect and all indecomposable representations in L(IJ)~, Sl) of nonzero defect.
In the final part of this chapter, we want to study in greater detail the End X- End Xbimodule structure of Ext l (X, X') for certain representations X and X'.
2.7. Let (9)/, Sl) be a realization of (r, d) and C+, C- the corresponding Coxeter functors. (i) If k E r is a sink, then Extl(C-F k, Fk ) ~ Fk(Fk)Fk. (ii) If k E r is a source, then Extl(F k, C+F k ) ~ Fk(Fk)Fk. PROPOSITION
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
21
Proof (i) Let k be a sink. We shall construct an exact sequence O~Fk ~T~C-Fk ~O
with a projective representation T which does not possess any direct summand of the form F k . Then, obviously Hom(T, F k ) = 0 and Ext1(T, F k ) = 0, and therefore the exact sequence Hom(T, F k ) ~ Hom(F k , F k ) ~ Ext1(C-F k , F k ) ~ Ext1(T, F k ) yields the required isomorphism Ext1(C-F k , F k ) "'" Hom(F k , F k )
= F/Fk)Fk'
Now, in order to construct T and the above exact sequence, consider an admissible ordering k = k 1, k 2 , . . . , k n of r with respect to n. Let W kt be the representation of (~l, sk sk ••• Sk n) such that t
t+ 1
n
(Wk)k
= (k
ttl
M k )Fk and (W k )i t
t
t
= 0 for
i =1= k t ·
Put Vkt=S;lS~ "'S;t_lWkt'
l';;;;;'t';;;;;'n.
Since W k is the direct sum of d k k copies of F k ' it follows readily that V k is the direct t I t t t sum of the same number of copies of Pkt ; therefore, V kt is projective. Also, Vk1 = O. Observe that, for all s =1= t,
where the summation runs over all neighbours k r of k s such that k r ~ k s in the orientation n (and therefore, a fortiori, r > s). On the other hand, the components of C-F k satisfy, fors=l=l,
Finally, we consider the exact sequence
o ~ Fk where T = For s
EB t
V k and t
> 1,
€ =
(E k
s
)
~ T ~ C-F k ~ 0,
is defined by induction on s = n, n - 1, ... ,las follows:
[
Ek s = EBE k r ®
] l( k
M
r ks
ffi 1(kl M k
)
),
s
which means that Ek identifies Tk with (C-F k)k . And, for s = 1, we have the exact ses s s quence
O~Fk ~EB(C-Fk)k 1
k
r
r
®kMk
r
1
~(C-Fk)k ~O; 1
here, EB k (C-Fk)k ® k M k = T k and we put Ek ="fr. Thus, we have defined e whose r r r 1 1 1 kernel is just F k , as required. This completes the proof of (i). The assertion (ii) can be established by a similar argument.
2.8. Let (9)~, n) be a realization of (r, d) and C+, C- the corresponding Coxeter functors. (i) For X = C-rP k with C-X =1= 0, PROPOSITION
22
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
(ii)
Proof. Both statements are straightforward consequences of Propositions 2.7 and 2.3.
3.
REPRESENTATION OF DEFECT ZERO: GENERAL THEORY
Let (Wl, D) be a (fIxed) realization of an extended Dynkin diagram (r, d). The defect ac(dim X) of a representation X E L(W~, D) will be denoted simply by ax. The general theory of representations of zero defect follows closely the theory of quadruples by I. M. Gelfand and V. A. Ponomarev. First, we have the following equivalence. The following properties are equivalent for X E L(W~, D).
LEMMA 3.1.
(i) X is a direct sum of indecomposable representations of zero defect in L(W~, D); (ii) ax = 0 and ax' ~ 0 for every subobject x' of X in (iii) ax = 0 and ax" ~
L(Wl, D); 0 for every quotient X" of X in L('JR, D).
Proof (cL [10]). Assume (i) and let X' be a subobjec-t of X such that ax' noting the order of the induced Coxeter transformation
> 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
(4) Ext 1 (X, Er ) = 0 for all simple regular representations
1.
r
and X
r
and X
'* '*
Er + l ' Er _
l,
O
Proof (1) For arbitrary natural r, let Er
= E9 Xt
be a direct decomposition into
24
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
indecomposable representations. Since the orbit of dim Er is finite, all orbits of dim X t are finite and thus all aX t = O. Hence, all Er are regular representations. Furthermore, since 17(C+ 1E) = 17E > 0, E can be embedded into C+1E by (ii) of Definition; hence C+1E = E. Moreover, all E r , 0 ~ r < " are simple in R(1JJl , S"l); it is sufficient to show this for E because E ~ C+O-r)E r . But 17 is additive on extensions and therefore, in view of 17E > 0, there is a composition factor X of E in R(Wl, S"l) with 17X
> 0:
by (ii), E can be embedded
into X and thus E = X is simple. Also, E is nonhomogeneous, because dim E rF N. (2) Now, let X be a simple regular representation; then by (ii) or (iii) of Definition 17X = 0 unless X ~ E or X ~ E I . In fact 17EI = - 17E because LO
'*
'*
o -----+ X -----+ V
-----+ E -----+ O.
Since 17 V = 17X + 17E > 0, the regular representation V contains a representation Y isomorphic to E. In view of 17X = 0 and 17E > 0, X and Yare nonisomorphic. Consequently, X n Y = 0, and E ~ Y c..... V - - E is the identity of E, i. e. the sequence splits. According to Proposition 2.3, Extl(E r , X) = Extl(E, C-rX) and the statement (2) follows.
(4) To prove the last statement, consider the extension 0-----+ E I -----+ V -----+ C-(r-I)X -----+ 0,
<
*
*
E and C-(r- I)X ~ E I . Then 0 and E I is a quotient of V. Hence the sequence splits and, again by Proposi-
with a simple regular representation X such that C-tr-1)X T/V = 71EI tion 2.3,
I ~r~"
ExtI(X, Er) = ExtI(C-(r- I)X, E I ) =0,
0
for all X Er _ I and X ~ E r · The following lemma provides the final argument in the proof of Theorem 3.5.
3.4. Let (1JJl, S"l) be a realization of an extended Dynkin diagram (r, d) and C+ be the corresponding Coxeter functor. Let i be a source with respect to S"l. Let E be a representation of (WI, S"l) such that C+E =1= 0 and E i = O. Then there exists k E r such that LEMMA
End E = F k , ExtI(E, E) = 0 and Ext1(E, C+E) = Fk(Fk)Fk'
Proof. Denote by L;C~I, S"l) the full subcategory of L(~l, S"l) of all representations X of (1JJl, S"l) such that Xi = O. Since i is a source, there is an admissible ordering k l' k 2 • . . . , k n of r such that k n = i. Thus Y=S+ ···S+S+EEL.(IJ)I S"l) kn-I k2 kl I' and C+ E is an extension of Y by a direct sum Z of copies of Fi
o -----+ Y -----+ C +E -----+ Z -----+ O. From here, we get the exact sequence
:
25
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
Hom(E, Z) --+ Ext1(E, Y) --+ Ext1(E, C+E) --+ Ext1(E, Z), in which both the first and last terms vanish: the first one triviaIly and the last one because of injectivity of Z. Thus Now, there is a canonic isomorphism
T: Li(~m, D)
--+
L(9Ji' , D ' ),
where (\]R', D') is the restriction of ('JJl, D) to the graph (r' , d'), where r'
= r\{i}
and d'
is the induced valuation. Observe that (r', d') is a disjoint union of Dynkin diagrams. It turns out that T(Y) = C' + T(E), where C' +
= s+kn-l
••. S+ S+ is the Coxeter functor on L(9Ji', D'). Now, by Propositions k2
kl
1.9 and 2.6, T(E)
= C'+YQk
for some k E
r',
and by Proposition 2.8 (ii) Ext1(T(E), C'+T(E)) "'"
Fk
(Fk )
.
Fk
Also, by Propositions 2.5 and 2.3, End T(E) = F k and Ext1(T(E), T(E)) = Q. Lemma 3.4 foIlows. 6 Now, given E which possesses an equation Put
n
Ker
1],
c*Y1]
observe that
c*Y1]
is an equation of E y •
= KTj'
O«y
By Lemma 3.3 (2), always KTj
2 N.
DEFINITION. A finite set {e I I ~ t ~ h} of representations of (9Jl, D) is said to be a generating set if (i) the orbits of dim E(t) under the action of c are nontrivial and finite; (ii) any two dim E(t) and dim E(t') for t = t' belong to distinct orbits of c; (iii) each E(t) possesses an equation 1](t); (iv) there exists a source i with respect to D such that each E(t) has the property that
E~t) I
= Q' '
(v) n1«t«h KT)(t) =N. Now, Lemmas 3.3 and 3.4 in combination with Proposition 2.3 yield the foIl owing results which refines the second part of Theorem in the Introduction.
3.5. Let ('In, D) be a realization of an extended Dynkin diagram and C+ the corresponding Coxeter functor. Let E = {E(l) I I ~ t ~ h} be a generating set of regular representations of (9Jl, D). Then the category R(9Jl, D) is the product of H(\]R, D) and h THEOREM
6 Here, T(E) is a representation of a disjoint union of Dynkin diagrams. However, since it is indecomposable, it is, in fact, a representation of one connected component and so we may apply the pre.Yious results.
26
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
subcategories
R(t)
corresponding to the orbits
t ~ h. The indecomposable objects in
R(t)
O(t)
= {E(t), C+E(t),
... , c+1nE(t)}, I ~
are serial with composition factors from
O(t)
or-
dered in a (going down) sequence corresponding to the action of c+ : each of them is fully determined by its (simple) socle and its length which both can be arbitrary.
The last statement follows from a well-known characterization of serial abelian categories which (in the case of a module category) is due to T. Nakayama (see e. g. [9]). Since L (9Jl, Q) has global dimension ~ I, the same is true for each R(t). Since each simple object has a nontrivial extension by a suitable simple object, there are indecomposable objects of arbitrary length with a prescribed socle. As an immediate consequence, we get the following statement which will be used in Chapter 5: If X is an indecomposable representation of continuous dimension type belonging to some R(t), then End X is a division ring if and only if X has no proper subobject of continuous dimension type. This happens if and only if every simple representation of R(t) appears precisely once as a composition factor of X. Also, if Y is another indecomposable representation of continuous dimension type in R(t), then
Hom(X, Y)
*- o.
The existence of a generating set E of Theorem 3.5 is proved, for each of the extended Dynkin diagrams, in the next chapter.
4.
SIMPLE REGULAR NONHOMOGENEOUS REPRESENTATIONS
In this chapter, we shall exhibit a generating set of regular representations for each of the extended Dynkin diagrams with a suitable orientation. The results will then be incorporated in the tables of Chapter 6; in particular, all simple regular nonhomogeneous representations will be listed there. As a consequence of these results, we can formulate the following theorem. THEOREM 4.1. Let (r, d) be an extended Dynkin diagram and Q an admissible orientation. Let n + I be the number of its vertices and h the number of elements in a generating set of regular representations. Then 0 ~ h ~ 3 and the number I of all simple regular nonhomogeneous representations is given by the formula
l=n+h-1.
-
The number h is independent of Q except in the case of the diagram An (n;;" 2) when h = I or 2.
The formula in the theorem can be reinterpreted as follows: If Otis an orbit of a simple regular nonhomogeneous representation under the action of C+ , denote by It the length of 0t; then
L(lt - I)
=
n - 1.
t
We shall use letters a, b, Z, . . . , possibly with some indices, to denote the vertices of the diagrams; the vector spaces of a given representation attached to these vertices will be denoted by the corresponding capital letters A, B, Z. ... (with the respective indices). For convenience, a~m), . . . will denote also dim A~m), . . . . For further notation, such as using the letters Hand G to denote subfields of F, we refer to the tables.
27
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
In order to show that a given set is a generating set, we have to verify conditions (i)(v) of the definition. However, the conditions (i), (ii) and (iv), (v) can be checked in a routine manner, and thus we shall be concerned only with the condition (iii): In each case, we shall prove that the linear form .,.,(m) listed in the tables is an equation for E~m). Throughout the proofs, X stands always for a regular representation, X' for a (not necessarily regular) sub object and X" for the quotient of X by X'. We shall use frequently the fact that, ax' ~ 0 and axil;;;;. o.
, ~.
Before we proceed to establish the existence of a generating set for each individual extended Dynkin diagram, we are going to give an outline of the proof that .,.,(t) is an equation for E(f) which will be systematically followed in each case. It is trivial to verify that .,.,(t)E(t) > O. Hence, we need only to show that the other two conditions concerning regular representations X with .,.,(t)X =1= 0 are satisfied. This will be achieved by the method of contraction to a compatible realization of a Dynkin diagram in the following sense. We shall outline the method of contraction in general, for arbitrary valued graphs. DEFINITION. Let (9J~, .S1) be a realization of a valued graph (r, d). A realization (m', Q') of a valued graph (r', d') is said to be a contraction of ('D~, Q) if r' <; rand (i) for every two k, IE r', for which there is an oriented sequence of edges k = io - i1 ••• i q = I in (r, d) (i:- the orientation .S1), there is an edge k - I in (r', d'); (ii) for every oriented (with respect to Q') edge k --- I in (r', d'), there is a unique sequence of edges L kl = (k = i o - ••• - ip - ip + 1 - ••• - i q = I), and it is oriented in Q by i p --- ip + 1 for all 0 ~ p ~ q - 1; (iii) L kl n Lk,l' =¢for {k. I} n {k'. t} = ¢; (iv) F~ = F k for every k E r'; and (v) k~ = 01il 0···0 ipMip+I 0···0 iq_1MI' where Up' i p + l } runs through Lkl,for every edge k -I of(r', d'). Now, for each k E r', define
+k = {i E r Ii = i o --- i 1
iq
--- ••• -
=k
with ip
ff- r' for
all 0 ~ p < q}
and
t k = {i E
r Ik
= i o --- i l -
i q = i with i p
.•• -
ff- r' for all 0 <
p ~ q}
Thus, k belongs both to tk and t k' Let X = (Xj , j'P) be an arbitrary representation of ('1R, Q) and (9J~', Q') a contraction of ('De, Q). Define the representation U = R(X) of ('Dc', Q') as follows: U = (Uk' It/Jk)' where Uk
= Xk
for all k E
r'
and It/J k : Uk 0 k~ -
with
,t,
l'Yk
ip+I'P'i p
=. '''~
'qYlq_l.
= ip+I'Pjp
• ••.
,,,~
'p+I'/"'p
UI for all k - I (in Q')
•••. '''~ . '''~
'2'/"'2'1'/"'0
and
0 I 0 ... 0 l:Xip 0 ipMip+I 0 ip+IMip+2
@"·0.'q-l M.'q
-x.'p+l
0.'p+l M.'p+2 0· .. 0 iq_1 Miq forallO~p
28
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
On the other hand, if V is a representation of (9Jl', [2') which is a subobject of U = R(X), we can define canonically two subobjects of X: T(Y) and T'(V). But first, given V = (Vk , /l/Ik)' introduce the following notation: If i E r belongs to ~k' and L k = (i = i 1 ip
$ r ' for all 0
-:p'
~
iq iq - l
p
~
---+ i 1 ---+ ... ---+ i q
= k)
is a path satisfying
q, define ViLk as the inverse image of Vk under the mapping
••• i -:P'i - l •••.1110 -:p~ : X. 10 p p
---+
X. 0· M. Iq
Iq
Iq -1
0 ... 0 . M, 11
10'
where
,
-:p', =.Ip+ 1 -:p,I 0 1 0 ... 0 I : X.I 0,IpM.I
Ip + 1 I p
---+
x.Ip+ 1
p _l
p
p
0.
Ip + 1
M. 0. M. Ip
Ip
and put V~k I
Ip - l
0 ... 0 . M. 11
0 ... 0 . M. 11
= n v.L k
10
10
for all
O~p~q-l,
C - X.l '
I
Lk
where the indices run through all possible paths L k described above. In a similar fashion, ifi E r belongs to t k , and L k = (k = io -- i 1 ---+ ••• ---+ iq = i) is a path satisfying ip $ r ' for all 0 ~ p ~ q, define V. k as the image of Vk under the mapIL ping i"iii defined above. And again, put q 0 Vi
1= L L
k
V.I Lk C; Xi'
where the indices run through all possible paths L k • Now, we define the representation T(V) of (9J~, D) as follows: T(Y)
= (Vi, ;'Pi)'
where
(and thus, if no such k exists, Vi
= X;)
Vi = n V;k' k where the indices run through all k such that i E
~k
and ;'Pi are induced by X. Similarly, we define the representation T' (V) of (9J~, D) by T' (Y)
VI~
=
= (V;, ;'P;), where
L vit, k
where the indices run through all k such that i E t k (and thus, if no such k exists, Vi = 0) and the mappings j'P;' are again induced by X. It is easy to verify that both T(U) and T'(D) are subobjects of X. The following consequence will be repeatedly used. 4.2. Let (9J1', D') be a contraction of (9J1, D). Let X be a representation of (9J~, D) and U = R(X) the corresponding representation of (9J~ I, [2'). If V is an indecomposable direct summand of U, then V determines a subobject T'(y) of X and a quotient T"(y) = X/T(W) of X, where W is a complement of V in U. Now, our proof will consist of considering, for each extended Dynkin diagram and for each fI(f) , a contraction (911', [2') of (911, [2) defined by the nonzero components of fI(f), and in decomposing R(X) into indecomposable representations of (911', [2'). Since (9,n', D') is a realization of a Dynkin diagram, the direct summands are determined by their dimension LEMMA
29
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
type, and these are the positive roots of the diagram. From the condition 11(t)X > 0, we deduce that one particular summand V0 must occur in the decomposition and show that the subobject T' (V 0) of X is isomorphic to E(t) = E~t). Similarly, using the condition 11(t)X < 0, we single out another particular direct summand VI of R(X), and show that the quotient T"(V 1 ) of X is isomorphic to E~t) = C+E~t). These conclusions are obtained in each case as a result of a very simple elimination process of those direct summands V which would lead to subobjects of X of positive defect or to quotients of X of negative defect. In the case of diagrams E, F and G we shall make use of an additional information that the orientation is chosen in such a way that all mappings jVi i: X j ---+ X j 0 Ifi are monomorphisms; for, otherwise we would get a sub object of X of a positive defect. Although we shall deal in the proofs with the dimension types only, the actual description of the simple regular nonhomogeneous representation given in the tables follows immediately from the form of the corresponding representation V0 or VI of the contracted Dynkin diagram. and 2. Obviously, there are no regular nonhomogeneous representations. An' n > 1. {Eo = Ec ,E~ = F d } is a generating set (of course, the set contains p q only Eo if q = O}. First, 11 is an equation for E. Consider the contraction to the Dynkin diagram ~ : cp ---+ b (more precisely to the realization F ---+ F of the Dynkin diagram cp - b), and decompose R(X) into the direct sum of indecomposable representations. The dimension types of the direct summands are (0, 1), (1, 0) or (1, 1). Now, if 11X > 0, then there must be a direct summand Vo of dimension type (1, 0); for, 11 is additive and 11V < for the summands V of type (0,1) and 11V = for those of type (1,1). And, evidently X' = T'(V o) ~ Eo. If 11X < 0, there must be a direct summand VI of dimension type (0, 1). And, we get X" = T"(V 1 ) ~ E 1 . The same arguments show that 11' is an equation for E'. Bn. {E = F z } is a generating set. Consider the contraction to the diagram 8 2 :
All
Al
°
°
~ b.
n-1
The dimension types of the direct summands of R(X) are (0, 1), (1, 0), (1, 1) and (1, 2). If 11X > 0, a summand V0 of type (1, 0) must occur and T'(V 0) ~ Eo. If 11X < 0, R(X) has a direct summand of type (0, 1) or (1, 2). But, there is no direct summand V of type (0, 1), because it would determine a quotient T"(V) of X of negative defect -1. Hence, there is a summand VI of type (1, 2) and the quotient T"(V 1 ) is isomorphic Zn
_ 1
to E 1 ;....,
en.
{E
~ b.
= F z n-1 }
is a generating set. Consider the contraction to the diagram 8 2
:
If 11X > 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
----*
(Fl/G)c ® c(F2 )F2
~ (F2)F2
= Gc
0--+ 0 --+ '" 0 --+ G c --+ 0 as a subobject. "'---' . mult Ben' The kernel of the mappmg GH ® HGC ~ Gc is a nonzero G-subspace K c of GH ® H Gc; therefore o --+ 0 --+ ••• 0 --+ Kc --+ 0 is contained in T(R).
BOn· The kernel of the mapping FF ® FFc ® c FF ~ F F is a nonzero F-subspace KF of FF ® FFc ® CFF' and thus KF EB 0 <; r..,; therefore, 0--+ 0
--+ •••
0 --+ KF EB 0 --+ 0
is contained in T(R). Then
On the other hand, let
r.., n F F
----*
(F/G)c ® CFF ~ FF'
35
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
o
~ ... ~ FF
: : FF $ 0 FF
$ 0
~ (FIG)o
$ 0
is contained in T(R').
COn'
Since the kernel K of F 0 ® of0
lows that K $ 0
~
~ F0
is an F-subspace of F o ® OFF' it fol-
r.p is an F-subspace; thus,
0 .......... 0/
O~O·"O~KEBO~O
is contained in T(R). For the projection mapping F 0 ® 0 F 0 - - (FIG)o ® 0 F 0 "'" F 0' we see that
f.p n (Fo EB 0) ® oFF
= (Go
EB 0) ® OFF and hence T(R') contains
O~ _ _ (Go EB 0) ® OFF ~ .•. ~ (Go EB 0) ® oFF ~ (FIG) ® GPo· Go ® OFF
On.
Obviously, T(R) contains
0.......... 0--'" where rid
= {(f, f) If E
.--0
O~O"· O~f'd I
~O'
F} is the graph of the identity mapping. If'l'l
= 0, then r 1 =
U EB 0; thus
0.......... --",FEB
F
O~FEB O~
is contained in T(R'). Similarly, E~
E6 .
.. •
~FEB
0
--",0 .......... F
<; T(R").
First, we show that T: L(FMF) ~ L('JR, n) is a full embedding. Let
C1
'1
C2
'1 T(UF , VF ,('I'1,'I'2)=A 14 A 2 4
Z ~L'2~Bl'
n B 2 = 0 U 0; thus, all three direct components of Z are determined. Furthermore, (AI + C1 ) n B 2 identifies 0 U 0 with 0 0 U, (CI + B I ) n A 2 defines the graph of '1'2' whereas C2 n A 2 is the graph of '1'1 . Then A 2
Note that T(R) contains a copy of Eo determined by the central vector space 0 U 0,
T(R') contains a copy of E~, with the central vector space {(O, u, u) lu E U}, and T(R") contai~s E~ with the central vector space 0 0 U
E 7' Again, we show that T is a full embedding. Obviously, the direct components of Z in T( UF' VF' ('1'1 ' '1'2)) are determined, and (A I tween 0 U 00,00 U 0 and 000 U
+ C) n B 3
determines the diagonals be-
Also '1'1 is determined by (C
+0
U 00)
n
V 0 U 0,
V 0 U o. T(R ) contains a copy of Eo, with the central vector space generated by the elements
and '1'2 is determined by (C
+0
0 0 U)
n
(0, U, U, 0), U E U T(R') contains a copy of E~, with the central vector space generated by the elements (0, 0, U, u), u E U Finally, T(R") contains E~, namely
36
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
{(O, U. 0, - U) Iu E U}
0000
c...
E8.
0 U 00
c...
'1
0 U 00 C-.O U 0 U......:JO U 0 U ~ 000 U..--:J 000 U
To show that T is a full embedding, one notes that the components V 00000,
o0
U 0 0 0, and 0 0 0 0 U 0 of the central vector space Z in T(UF' Vpo ('PI' 'P2)) are determined using only the subspaces Ai and Bj . Also, (C + At) n A 4 n B 2 = 000 U 00, and then (C + 000 U 00) n A 2 = 0 U 0000, and (C + V 00 U 00) n B 2 = 00000 U This shows that all components are determined. Obviously, C provides the identifications of the different copies of U. as well as the definition of the graphs of 'PI and 'P2. Now, T(R) contains a copy of Eo, since (0, 0,1,1,1,0) belongs to As n B 2 n C. Similarly, T(R') contains a copy of E~, since (0, 0, 1, 0, 1, 1) = (0, 0, 1, 0, 0, 0) + (0,0,0,0, 1, 1) belongs both to C and to (A 3 n B 2 ) + B t . Finally, T(R") contains a copy of E~ whose central vector space is 0 U 0 U 0 U, and whose vector space C is generated by the two elements (0, 0, 0, 1, 0, -1) and (0, 1, 0, 1, 0, 0). In the remaining four cases, we shall use the concept of an interior and that of a closure of a G-subspace in a vector F-space (where G is a subfie1d of F) defined in the introduction to the tables in Chapter 6 (for details, see [4]). F41' The functor T is a full embedding: If T(UG • VG , 'P) = (AI ~ A2 ~ A 3 ~ Z +- B), then UG = AI' VG = A 2 n ~3' and B is just the graph of 'P. We claim that T(R) contains a copy of Eo' The kernel of the mapping F G 0 G F F (F/G)G 0 GFF is the F-subspace K F = GG 0 GFF. Thus 0 x K is an F-subspace of
-
F G 0 G FF ~ F G 0 G FF' and T(R) contains O~O~O xK~O
xK+-O xK.
On the other hand, let 'P be the identity mapping 'P: GG 0 GFF ~ GG 0 GFF· Then rIP n GG x GG = (1, l)G and therefore T(R') contains a copy of E~, namely
o ~ (1,
I)G
~
(1, l)G
~
rIP
+-
rIP·
Perhaps we should point out that the category of all images under T is given by all representations which satisfy the conditions ~2 = 0, .42 = Z. ~3 E9 B = z. Al E9 :13 = W. The conditions A 1 E9 B = Z and A 2 E9 B = Z are not satisfied for all images under T. but rather for the homogeneous ones. 42. First, we show that the functor T is a full embedding. Since UF 0 ~G 0 G GG = 0,
F
we see that ~2 = 0 X VF · Also UF 0 FNF =8 1 n (AI + ~2)' and UF = UF 0 FNF 0 ~. Here we use the fact that dimFN = dim N F = 1, so that N* exists with FNF 0 FN'f. = FFF. Finally, 'P is determined by A 2 . Second, T(R) contains Eo, because (10 1) x 0 belongs to A 2 n B 1 in this case. On the other hand, T(R') contains a copy of E'l' because the kernel of the multiplication map-
~ FF is just N F' so that Al n 8 1 =1= O. The functor T: L(G (F/G)G) ~ L(IJR, .Q) defines an equivalence of categories
ping F G 0 GFF
<321 .
between the category LdG(F/G)G) of all representations 'P: UG 0 G(F/G)G ~ V G with a surjective 'P, and the category C of all representations (A 1 c.... A 2 c... Z) of ('))1, .Q) which
37
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
have the property that (A 1 c.......... Z), considered as a representation of •
~
• , is a direct sum
of copies of (GG c..-. F p ). It is obvious that H(G(F/G)G) S; Le(G(F/G)G)' Also, using the fact that a homogeneous representation X does not allow a nonzero homomorphism from E 1 into X, we get H(9.U, Q) S; C. Thus (A I c.... Z) has to be a direct sum of copies of (0 c-Fp) and (G G c.......... Fp). However, if there is a copy of (0 c....... Fp), then dim(A 1 )G < dim Zp, and, using the defect argument, dim(A 2 )G > 2dimZp , which would imply the existence of a subobject of the form Eo' Now, consider T(R), where R = (Fp , (FG ® GFG)/(FG ® GGG + w(Fp 1T). Since (FG ® G G) n w(F) = 0 according to the example following Proposition 2.1, dim T(R) = (3,6,9). Obviously, T(R) contains a copy of Eo, namely
»,
0--)0 w(Fp ) --)0 w(Fp ), and the quotient is E 1 . But T(R) is not the direct sum of Eo and E 1 ; for, in T(R), the closure Al of Al = FG ® GGG is Z = FG ® GFp. Thus EndR "'" End T(R) is a division ring. G22 · Let pMp = (pFG ® GFp)/w(F) and FNG = (pFG ® GGG + wF)/w(F). According to the example following Proposition 2.1, the endomorphism ring of the pair p
(Mp
N G ), considered as a representation of F •
G, is F. Now, the direct sums of copies of (Mp 2 N G) form a category C which is equivalent to the category of all vector F-spaces, under the functor which maps an F-space Up to (Up ® pMp 2 Up ® pNG ). [In order to show that it is an equivalence, one easily checks that it is a dense full embedding.] Thus, there is a reversed functor which associates to a direct sum (Zp 2 B G ) of copies of (Mp 2 N G ) a vector space Up such that +--
Up ® p(Mp 2Ne) "'" (Zp 2Be).
We claim that T provides an equivalence between the category Le(pMp ) and the category C of all representations (A pC- Z p .-::; Be) such that (Z .-::; B) is a direct sum of indecomposable representations of dimension type (2, 3). Of course, T maps L e(pMp ) into C. But, obviously, T: Le(pMp ) --)0 C is also dense. Moreover, T is a full embedding, since (A p --)0 Zp - Be) determines Ue and the kernel of cp, and therefore also
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
38
6. TABLES The following tables summarize the results of Chapters 4 and 5 on the indecomposable regular representations. Each of the tables provide the following information on the respective extended Dynkin diagram: I. The type of the valued graph including the notation for the vertices. 2. The quadratic form of the graph. 3. The vector n stable under the action of Weyl group, and the tier number g.
ac with respect to an orientation which is indicated here. Wl = (Fi, iMj)' As a rule, if iMj is one-dimensional on both sides,
4. The defect vector S. A modulation
we assume Fi = FJ. and F.CMJ·)F' = F.(F)F" This is not, in general, possible for the diagram ~ I J I I An' because it is not a tree; here, we assume that Fi = F for all i E r and that iMj = FFF for all but one edge. Also, if iMj is one-dimensional as a left Fi-module, we may assume that Fj is a subfield of F i and iMj = F/F)Fj; and, similarly for the case when iMj is a onedimensional right Frmodule. In the tables we omit to display these bimodules explicitly. 6. The simple nonhomogeneous regular representations. The list is divided into the orbits of C+. Besides the representations, we incllde their dimension type, their image under
C+ , the equation 17
= 170
for E
= Eo
and the derived equations 17r for other representations
Er . The last column in the tables points out the consequences of the equality 17~t)X = 0 for a simple regular representation X. In particular, a simple regular representation is homogeneous if and only if all conditions given in the last column are satisfied simultaneously. [The consequences follow immediately from the proofs of the tables. However, they can be easily derived from the tables alone: that is to say, the assumption that a particular consequence, for a regular X, is not satisfied implies that, for the respective E~t), either E~t) c..... X or X -
E~t).] Here, the capital letters A, B, Z, ... (possibly with some indices) denote
the vector spaces of the representation X in the corresponding vertices a, b,
Z, . . .
of r. In
case that G is a subfield of F and the mapping A c ---+ ZF ® FFc is a monomorphism, we shall assume that A c is a G-subspace of the F-space Z F' In such a situation, we shall denote by ~ the maximal F-subspace {a E A laF <; A} contained in A (the "F-interior" of A) and by A the F-subspace ~aEA aF (of ZF) generated by A (the "F-closure" of A). Observe also that the source i needed in the proofs of Chapter 4 is always a or aI ' 7. A bimodule FMc whose representations determine the homogeneous representations of the corresponding extended Dynkin diagram. 8. A functor T: L(FMC) ---+ L(Wl, .Q) which defines the correspondence. 9. The simple regular representations R of FMc such that T(R) is not simple. Always, E&t) <; T(R(t») with the exception of the diagram 1=42' where T(R') contains (E'l)' Finally, let us make the following convention: If FMc = M I EB M2 , we shall decompose
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
a(1,4)b; Q(x)
= (x a -
2x b )2
= (a
- 2b)2;
= 2 - l;g = 2; ac = I ~- 2;
n
F with [F: G] = 4;
'JJb G -
All regular representations are homogeneous.
a (2, 2) b; Q(x) = (x a -X b )2 = (a - b)2;
n=l-l;g=l;
ac = 2 ~- 2; P1Mp2 IJR= F 1
F 2 with dimp1 M = dimMp2 = 2;
All regular representations are homogeneous.
39
40
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
An a
/' C 1 - C 2 -
... -
Cp
"-
"- d -d - ... -d /' 1
2
b, where p ?i' q and p
+q
=
n - 1;
q
Q(x) = L(X; - X;)2 where the summation runs through all edges i I
1-1-"'-1
n = I / ' ' - l' g '1-1- ... -1 . . . . '
I
= 1;
i;
0-+0-+"'-+0
la = n + 1 / < - ' " ....,..0-+-0-).0 ••• -+0/1'
:C
(n
+
1)' '
F-F-'''-F
Wl
= F :
where FN = FF and the right action of F on N is given by a field automorphism of F;
'- F, F - F - ••• - F
N
E (f) r 00 .. , 0 F
Eo
for
= 0 00
E 1 =F
p?i'1
1
F 0 .•. 00
E
=0
E~
=0
E' -F 1-
0
0
0
0
0
F
1
0
0
0
0
00 ... F 0
00 ... 00 00'" OF FF"'FF
00 ... 00 00 ... 00
E~
=0F
E'
=0
q
0 ..• 00
00 .. , 00 00 "'FO
00 '" 0 0
1 1 ."
11
10 ... 00
0
.. , 00
00"'00
00 ... 0 1
00 ... 00
00 '"
10
00 ... 00 00···00 00 '" 0 1 1 1 ." 1 1 00···00 00 '" 0 0 1 0 '" 0 0
r
(f)
00 ... 01
1
U
T(UFI VF , (
= UF
o
U
1
F
---+
1
E2
1
0
E3
0
0
Eo
0
0
E'1
o 00·"
1
E'2 1
-1 0 ... 00
0
E'3
0
0 0'" 0 0 0 1 -1 •.. 00
D 1 -+ D 2
0
E'0
0
00 ... 0 0 0 00 ... I -1
D q _ 1 -+ D q
00 .•. 00
-1 0 •. , 00 00 ... 00
1 ---+
0 0'" 0 0
00 .. , 0 0
S; T(R') for q ?i' 1.
00
00··' 01
0
C 1 -+ C2
0
Cp _
-1
00· .. 00
U
101 F~
.vF ;
= (FF' F F' (1.0)) satisfies E~
A -+ C 1
1 -1 .•. 00
U -+ ... ---+ U ~2 F 1 F 1 1 F ---+
R = (FF' NF • (0,1)) satisfies Eo S; T(R) for p?i' 1;
R'
0
00 .•. 1 -1
1 U -+. " F
Cp -+ B
0
M=FFF$FNF; /'"
-1
E1
00 ... 00 00 ... 1 0
1/
0
00 ... 00
. F FF"'FF
= 0 00
P
for q ?i' 1
0 0
.. , 00
00 '" 0 0
E2
C+Er(t)
dimE/f)
isomorphisms for simple X with 1/r(f) X = 0
0
1
-+ Cp
D q -+B A -+ D 1
41
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
(1,2) a--z i Q(x)
n
=
= (a 1-
zl)2
1-
Dc = 2 -
-Z2 -
+
1-
0-
0
..
(2,1) -zn_l - - b ;
L
(Zt - Zt+ 1)2
l
..
0-
'Dl=G1 - F - F -
0
1-
-
000
000
1; g
0-
-
+ (Zn_1
-
b)2;
= 2;
- 2;
=
- F - G 2 with [F:Gd
[F:G 2 ] =2;
isomorphisms for simple X with llyX = 0
Eo
=
000
E1 E2
=F F F =0F 0
En _ 1
= 000
000
0
..
000
000
0 F 0 000
000
0 1 0 E 1 000
F F F 2 1 1 .. 1 1 2 E2 0
000 0 1 0
F 00 000
0 1 -1
000
1 -1 0 .. 0 0 0
000
000 E3 0 1 -1
000
1 00 Eo 000
0
000
000
000
1 -1 0
Zn_l -Hom(pFc2 ,Bc2 ) A0
C1
Fp -Zl
Zl
---*
Z2
42
VLASTIMIL DLAB AND CALUS MICHAEL RINGEL
(2,1) a - - z 1 -z2 -
0c
=
L
= (2a-z 1 )2 +
Q(x) n
(l,2) •.• -zn-l - - b
1-
2-
2-
(Zt-Zt+l)2 +(zn_l -2b)2;
1';;t';;n-2
... -
= 2 -+ 0 -+ 0 -+ ...
9Jl = F 1 - G - G -
2-+
••• -
1; g
0
G-
Er
1;
-2; F 2 with [F 1 .. G]
dim Er
=
[F2 .. G]
C+Er
l1r
= 2 .• isomorphisms for simple X with l1rX = 0
000"'010
E1
000 ... 0 1 -2
Zn_l -+
E1
= 000 .•• 0 GO = F 1 G G ... G G F 2
111"'111
E2
2 -1 0 ... 000
Ae
E2
= 0 GO'"
000
010"'000
E3
o 1 -1
Zl -+ Z2
••• GOO
000"'100
Eo
000 ... 1 -1 0
Eo
En _
M
-+
=
1
=000
= PI (F1)e
... 000
Be
-+ Zl
Zn_2
-+Zn_l
® e{F2)P2;
T(UP1 ' UP2 ' l,O) = (UP1
-L UP1
® Pl(F1)e
R = ((F 1)Pl ' (F2 )P2' (F1)e ® e{F2)P2 Eo <; T(R);
-L ... -L UP1
® Pl(F1)e ..:£,. VP2 ;
(Ft!G)e ® e{F2)P2
R::
(F2 )P2) satisfies
43
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
(l, 2) a - - z 1 -z2 Q(x) = (a - ZI)2
(l, 2) ... -zn_l --b;
+
L
(Zt - Zt+ 1)2
+ (Zn_l
- 2b)2;
l
de
=
~m
=H-
2-
1 ---+ 0
2---+
G-
0
... -
2-
---+ ••• ---+
G-
... -
Er
0
1; g ---+ -
G-
=
2;
2;
F with H C; G C; F and [F: G] = [G: H] = 2;
dim Er
C+Er
isomorphisms for simple X with TlrX = 0
Tl r
Eo = 000 '" 0 G 0
000"'010
E1
000 ... 01-2
Zn_l ---+ B G
E1 = G G G .. · G G F
211"'111
E2
1 -1 0 ... 000
AH®~G~ZI
E2 = 0 G 0 ... 000
010'''000
E3
01-["'000
ZI ---+ Z2
000"'100
Eo
000 ... 1 -1
En-l =OOO"'GOO
a
Zn_2 ---+ Zn_l
44
a1
a2
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
----Z /"'" 1
L:
=
Q(X)
Z2 -
t=I,2
(2, 1)
... -
Zn-2 - - b''
L:
+2
(2a t - a t )2
(Zt - Zt+l)2
+ 2(zn_2
- b)2;
l<;;t<;;n-3
1 ____ n =
1-
2-
2-
... -
2-
2' g = l' "
ac = 1 --....0~0~···~0~-1·, 1~
9J~
= F .......... F - F - ••• - F FE
,(t)
dimE
a a
E2
a Fa'" a a a a
a 10"'000 a
E3
a 1 -1 a
a
a 00"'100 a
Eo
a 00 a
=
E2
=
F F
F F," F F F
= a OO"'FOO 0
Eo
=
E'
= a FF'"
F
F F'" F FG
F'
FFG
M= pFe 0 eFp; Vp T(Up, Vp • l,?) = Up
--....
./"
a 1 1
a
11'''111
E'1
1 1 ••. 1 1 1
E'0
1
1
x a
a ... a a a
-1
1
a 00 2 0
•.•
aaa
••• 1 -1 0
Al EB A 2 ~ ZI
ZI
~
Z2
Zn-3 ~ Zn_2
a -1
Al ::::: A 2
00"'00-1
Al :::::A 2
2
1
isomorphisms for simple X with 1I,u l =
11, (t)
11'''112
E1
1
C+E,(t) E1
= a a ... a F a
,
,(t)
= 2;
a 00"'010 a
Eo
E n-2
G with [F: G]
... 0
1
Up EB Vp ~ Up EB Vp ~ ... ~ Up EB Vp ~ (Ue EB Ve)/r""
r", = {(u,
l,?(u 0 1 0 1)) lu E U} is a G-subspace of Ue EB Ve ; mult R = (Fp , F p , F p 0 pFe 0 eFp --)- F p ) satisfies Eo ~ T(R); where
R'= (FF' F p . F p 0 pFe 0 eFp ::::: Fe 0 eFp ~ (F/G)e 0 eFp ::::: F p ) satisfies
E'o C - T(R')·,
45
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
3 = c
1"" 0--+0--+···--+0--+-2' 1---" ' G
m = --.. . . G -
G-
G./'
E E
=
Ez
=
En-Z E
, o
, I
G G
0 0
0 0 1
G G··· G G F
1 0
GO··· 000
0
= 0 OO· ..
0 0
GOO
0
0
0
= F FF···FFF F
=0
F with [F: G] = 2;
dim Er (t)
0
o
G-
(t)
= 0 OO···OGO
EI
E
r
.•• -
2 2
FF···FFF
0
C+E
r
00···010
EI
11·"111
Ez
10···000
E3
00·"100
Eo
22·"221
E'I
22···221
E'0
1'I (t)
(t)
t
0
00 ... 01-2
Zn_z --+ B G
-1 0"· 0 0 0
Al
1 -1 ... 000
ZI --+
00 ... 1 -1 0
Zn_3
00 ... 00-1
Al
0·0 ... 00-1
AIR::A z
0 1 1 0 0 0 0 0 1 1
isomorphisms for simple X with 1'I r (t)X = 0
0
E9 A z --+
ZI
Zz
--+
Zn_Z
R::A z
= G FG ; representations of GF G are considered both as mappings UG ® GFG --+ VG and as mappings UG ® GFp --+ VG ® GFp;
M
T(UG• VG• 1,0) where
=
VG ® GFp---,. ---"
UG ® GFp
--+
r", = graph of UG ® GFp
= (FG' F G' F G ® GF G R' = (FG' F G , F G ® GFG R
(UG E9 VG) ® GFp --+ ••• --+ (UG E9 VG) ® GFp
muIt ---)0
--+
(UG E9 VG) ® GFp/r""
VG ® GFp; ,
F G) satisfies Eo <; T(R);
---- (F/G)G ® GFG
R::
F G) satisfies E~ <; T(R');
46
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
L
Q(x)==
t"'\'2
n==
1 -........ 1 ...-
3 == c
1
L
(20 t -Z I )2+ 2
(Zt- Zt+I)2+
l';t';;;n-4
~ 1 g==I' ........ l' ,
F-........
-1
. ---"-1'
I?
'))/ = F -- F -
F -
'" -
...-F
F ........ F'
C+E,
dim Er(t)
o 0 E=OO"'OF 0 00 1
E == 2
F F
a a
FF"'FF
Fa· .. oo
o
1
F
1
1
a
F F'" F F
a
a
a
a
a a 0'" o a 1
0
o
" F a E = FF"'FF I 0 F
1
a ...
0
00
0
a a
1a
1 1 ... 1 1
a 1
0
1
1
1 1
1
o
1
1 1 ... 1 1
a 1 1 ...
0
11
1 1
EI E2
1
F
" 0 F E = FF"'FF o F a
T(Up , Vp , ('P1' .;'2)) ==
1 1 ... 1 1
a
, a 0 E = FF"'FF OFF F
a a 00'" a 1 o 0
F
En-3 == 0 OO"'Fa 0
E' =
(zn_3- 2b t)2;
2-2-"'-2~'
---"O~O~"'~O'--*
E ==
L t=I,2
a 1
E3
Tlr(l) 0
E'1 E'o E"1 E"o
-1
00 ... 0 1
-1
0 1
-1 0 .,. 00
1 0
a a 1 1
a o 1
0 0
1 -1 ...
0 0
Eo
isomorphisms for simple X with Tl,
a 0 ...
aa0 a 1 -1
0
a a· .. a 0 a0
... 00
Al ffiA 2
Z1
~Z1
~Z2
Zn-4 ~ Zn_3
-1
0 0 -1
a a .. · 00
1
o 00'"
0
Zn-3 ~BI ffiB 2
0 -1 -1
00
a
Vp ---..
.--*
Up
---.. (Up El1 Vp )/f2 '
.--or Up El1 Vp ~ Up El1 Vp ~ ... ~ Up El1 Vp
where f; is the graph of 'P; (i = 1. 2); R = (Fp , F p , (1,1)) satisfies Eo
~
T(R);
R' = (Fp, F p, (0, 1)) satisifes E~ ~ T(R');
R" == (FF' F p , (l, 0)) satisfies E~ ~ T(R");
(Upffi Vp)/f l
47
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
Cz
I al-az-z-bz-bl Q(x) = ( 6a l - 3a + (6b l z
f
-
3b )z z
+ (6c I
-
3c z )z
+ 3 [(3a z
1
1
I
t
- 2z)z
+ (3b z
- 2z)z
+ (3c z
- 2z)2];
1
2
I n = I - 2 - 3 - 2 - 1 ; K=I;
ac
t == 1 -----> 1----->-3 <--1 <--1;
F I F
'lJ/
= F -
I
F -
F -
F -
F;
E (t)
dim Er(t)
r
o F
F
F
F
o
(I,l)F (I,I)F E I = F x 0 F x 0 F x FOx FOx F
E~ ==
E'I --
0
F
F
0
F F F
F
F F
0
F F
F
F
F
o
1 1 001 1 0
o
0 1 1 1 1 00
F
0 1 001 1
o
0 0 11110
0 E~ ==
F
o
0
E"0-E"I -E"2-
0
F
0 0 F
0
F
F
F
F F F
M == F(F x F)F;
F
o
1 1 112 11 0 0 011 1
0 0 0
0 1 o1 1 1 0
o
1 01 100
C+E (t) Tl (r) r r
consequences for simple X with Tl r (t)X = 0
EI
0 1 o 1 -2 1 0
Z/A z ffiZ/B 2 ffiZ/C2 == Z
Eo
1 0 10-101
A I ffi B 1 ffi CI == Z
E'1
0 0 o 1 -1 0 1
AZffiBI=Z
E'Z
1 0 00-1 1 0
B z ffi CI = Z
E'0
0 1 1 0 -1 00
C z ffi A I == Z
E"I
0 1 00-1 00
B I ffi C z == Z
E"2
0 0 1 0-110
Al ffi B z == Z
E"0
1 0 01 -1 00
C I ffiA z =Z
CI t
Cz R == (FF' FF' (0,1)); R' == (FF' Fp. (1, 0»; R" == (FF' FF' (1,1»;
t T( UF' VF' ('1'1' 'l'z)) == V 0 0 -----> V U 0 -----> V U U <-- 0 U U <-- 0 0 U,
where C I == {('I'2(1I), 11, 11) Iu E U} and Cz == C I
+ {('I'I (u),
U,
0) Iu E U};
48
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
c
I
Ia. - a 2 - a 3 - z - b 3 - b 2 - bl·'
Q(x) = 6[(2a. - a2?
+ (2b. - b 2? + 2[(3a 2 - 2a 3 )2 + (3b 2 - 2b 3 )2] + (4a 3 - 3z)2 + (4b 3 - 3z)2 + 6(2c - z)2;
2
I
n = I - 2 - 3 - 4 - 3 - 2 - 1 ; g=l; 2 .j.
3c = 1 ---> 1 ---> 1 --->
4
-
<--
1 <-- 1 <-- 1;
F
I
9R=F-F-F-F-F-F-F E,(t)
consequences for simple X with 1'/,(t)X=O
dim E,(t)
C+Ey)
1 0011100
EI
(I,OF 1 E. =FxO FxO FxO FxF OxF OxF OxF 1112111
E2
(A 3 nB 3 )EBC= 1 001-2100 Z!A 3 EBZ!B3 EBZ!C=Z 1 100-1001 Al EBB I EBC=Z
Eo =
0
0
F
F F
F
0
0
1'/,(t)
0
F
F
0 F
F
F
0
0 0111110
E3
0 010-1010
E'0 -- 0
0
0
F F
F
F
0
1 0001110
E'I
1 000-1010 B 2 EB C = Z
E' --
F
F
F
F F
F
0
0
1 1 1 1 1 100
E'2
0 100-1100 A. EBB 3 =Z
E~ =
0
F
F
F F
0
0
0
1 0111000
E'3
1 010-1000
A 2 EB C = Z
F
F F
1 00 1 1 1 1 1
E'0
0 001 -1 001
(A 3 EB B. = Z
(I,l)F 1 FxO FxO FxF FxF OxF OxF 01 122 1 1
E"
o 1 0 -2 1 0 1
E"0
1 1 0 1 -20 1 0
E2• =
.
E'3 --
0
E"0--
0
0
F
F
(I,I)F E~ = FxO FxO FxF FxF OxF OxF
M = F(F x F)F;
F
0
1 1122110
•
1
A 2 EB B 2 = Z
(A 2 n B 3 ) EB (B 3 nC)EBB. =Z (A 3 nB 2 )EB (A 3 n C) EB A. = Z
C f).
T(Up. VF , (10.,102)) = vooo c.. VUOO c.. VUUO c.. VUUU ~ OUUU .-.J OOUU .-.J OOOU, where C is generated by the elements (10. (u), u, U, 0) and (102(u), 0, u, u) with u E U; R = (Fp. FF' (0, 1)); R' = (FF' FF' (1, 0)); R" = (FF' FF' (1,1));
0
0
0
FxO
0
==
E~ ==
E~ ==
E~ ==
FxO
FxO
0
FxO
FxO
FxO
0
F
6-
FxO
FxF
FxF
FxF
FxO
F
F
FxO
F
F
FxO
F
-
4a s f
R" == (FF' FF' (I, 1»;
R' == (FF' FF' (I, 0»;
R == (FF' FF' (0, 1»;
M == F(F x F)F;
I
3
F 0
F
0 F
F F 0 F
OxF
OxF
FxF
OxF
(I,I)F FxF
(I,I)F FxF (I,I)F FxF (I,I)F FxF
F
F F
OxF
0
OxF
F
0
0
OxF
0
<-
E"0
E"I
2 01010-3 1 1 I I 0 I 0 I -3 I I
I o 0 1 0 0 -2 I I I 1 00 1 0 -2 1 0 1 o 1 0 0 1 -2 0 1
0 o 0 0 I 0 -I 0 I
1 00 1 00 -I 00
0 01 000 -I I 0
1 1 0000 -I 0 1
1 o 0 0 0 1 -2 1 0
Tl, (t)
C
U, U,
0), ('P2(u), 0, u, 0,
U,
OOOOUU, u) and (0, u, 0, U, 0,0) with U E U;
...;:J
AI EDCED(B I nA S)ED(B2 nA 3 )==Z
A 2 ED B I ED (B 2 n C) ED (A 4 n C) == Z
A 2 ED (As n B I ) ED (As n C) == Z
A I ED (A 4 n B2) ED (A 4 n C) == Z
B I ED (A 3 n B2) ED (B2 n C) == Z
A 4 ED B I == Z
A 3 EDC==Z
A 2 ED B 2 == Z
AI EDB I
C/.l
-EDC==Z
""
.I>\0
C/.l
;.-
I:l:I ;tI
~
Cl
t""
";.-
;.Z
C/.l
::I:
""
;.-
;tI
Cl
>rj
Z
0
C/.l
(3
Z ..., ;....,
~
~
;tI
Z/A s ED Z/B 2 ED Z/C == Z
~
;tI
(As n B 2 ) ED C ==
T/, (t)X == 0
consequences for simple X with
~==F-F-F-F-F-F-F-~
I
F
fl c... VUOOOO c... VUUOOO c... VUUUOO c... VUUUUO c... VUUUUU..:J OOUUUU
2 01122321 I 11223321
E'0
E'2
E'I
Eo
0 000 I I I I 1 I 00111221 I I 1 1 222 1 0 1 o 1 1 1 22 1 I
E4
E3
E2
EI
I 001 1 I 100
0 01111110
1 1 I II 1 2 1 1
1 00001110
C+E,(t)
2;
where C is generated by the elements (
T(UF' VF , ('PI'
2
dim E,(t)
<-
+ 30(2b l - b 2 )2 + 2(6a s - 5z)2 + 1O(3b - 2z)2 + 30(2c _ z)2; 2
Es
0c == 1 ---+ 1 ---+ 1 ---+ 1 ---+ 1 ---+ -6
3a 4 f + 3(5a 4
g == I;
F
2;
F
FxO
0
E,(I)
4-
-
(I, I, O)F + (0, I, I)F E"o -FxOxO FxOxO FxFxO FxFxO FxFxF OxFxF OxOxF 0 (I, I, I)F E'; == FxO xO FxOxO FxFxO FxFxO FxFxF FxFxF OxFxF OxOxF
£4
0
0
E 3 ==
F
F
0
E2 ==
0
5-
FxO
4-
FxO
FxO
0
E 1--
3-
Eo ==
2-
0
n == 1 -
I
Q(x) == 30(2a l - a2)2 + 10(3a 2 - 2a 3 )2 + 5(4a 3 3
a l - a 2 - a 3 - a 4 -as - z - b 2 -bl;
I
c
50
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
al -
(1,2)
a2 -
a3 - - z -
= 3(2a l
b;
a2 )2 + (3a 2 - 2a 3 )2 n = 1 - 2 - 3 - 2 - 1 ; g=2;
Q(x)
3c
= 1 ----+
1
-
----+
1 ----+ -4
+-
+ 2(2a 3
-
3z)2
+ 6(2b - z)2;
2;
~l =G-G-G-F-Fwith [F:G] =2;
E
Eo
= 0 = GxG
0
r
(t)
F
F
F
GxG GxG FxF (1, l)F
consequences for simple X with l1/t)X=O
dim E(t) r
C+E
002 1 1
EI
00 1 -2 1 13 (fJ B
2 2 221
E2
100-1 1 Al (fJ B
r
(t)
l1 (t) r
=Z =Z
EI E2 --
0
F
F
F
0
022 1 0
Eo
o 1 0 -lOA 2 = 0, A 2 = Z
E'0--
0
G
G
F
F
o1 1 1 1
E'1
o 1 0 -2 2
E'1 --
G
G
F
F
0
112 10
E'0
1 0 1 -2 0 A I (fJ:i3 = Z
~
A 2 (fJB=Z
M= GFG; T(UG, V G•
= (FG' F G• F G 0 GFG - (FIG)G 0 GFG ~ F G ); ~-(G G W G' G, GG 0 GFF ~ GG 0 GFF);
R
-
51
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
(2, 1)
a 1 -a 2 - z - - b 2 - b 1 ; Q(x)
= 6(20 1
-
+ 3(2b 1
a 2 )2
b 2 )2
-
+ 2(3a 2 -
2z)2
+ (3b 2 -
4z)2 ;
n = 1 - 2 - 3 - 4 - 2 ; g=l; Dc = 1
~
1 ~ -3
<-
1 <- 1;
IJJ/=F-F-F-G-Gwith [F:G] =2;
E
r
E0--
0
(t)
F
F
G
G
dim Er(t)
C+E (t)
11/ t )
consequences for simple X with 11}t)X = 0
o 1 111
E1
02 -2 0 1
A 2 E9 B 1 = Z B 1 E9!!.2 = Z
r
E 1 --
0
0
F
F
G
00 1 2 1
E2 =
0
F
F
G
0
11 1 10
E2 00-2 1 1 Eo 20-2 1 0
E'0--
F
F
F
F
0
o 1 1 20
E'1
o 1 -2
10
A 2 E9!!.2 = Z
112 2 2
E'0
1 0 -1 0 1
Al E9B1 =Z
E~ = (1,f)F (1, f)F FxF GxG GxG
Al E9B 2 =Z
with fE F\G M= FFc 0 CFF; T(UF . VF ,
with FKF = ker(FFc 0 CFF ~ FFF); R = (FF' Fpr FF 0 FFc 0 CFF "'" Fe 0 CFF - - (F/G)c 0 CFF "'" FF); , mult R = (FF' FF' FF 0 FFc 0 CFF ~ FF);
52
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
(1,3)
at -a z - - z ; Q(x)
= (2a t
-
az )2
+ 3(a 2 - 2z)2 ;
n=1-2-1; g=3;
ac = 1 9Jl
Eo
=G-
=
1-
- 3;
G-
F with [F: G]
0
F
E t =GxG+ GxG+
(e. f)G
= 3; dim Er
C+E r
Tl r
consequences for simple X with TlrX = 0
F
031
Et
o 1 -2
At = Z, ~2 = 0
FxF
332
Eo
1 0-1
Ai ® eFp-Z is an isomorphism
(e, f)G
such that {I, e, f} is a basis of Fe
M
= e(F!G)e;
representations of e(F!G)e are considered as the mappings
<{!: Ue ® eFe - Ve with
Fp ® pFe - Fe with respect to some Proposition 2.1);
€:
eFe -
eGe (see the example after
REPRESENTATIONS OF GRAPHS AND ALGEBRAS
53
z (3, 1) b; Q(x)
= 3(2a -z)2 + (2b
n=1-2-3;
ac
=
- 3z)2;
g=l;
1--+-2 +--1;
IJR = F - F -
G with [F: G] = 3; consequences for simple X with C+ Er
Eo
F G + fG withfE F\G
=0
E1 = F
M
F
G
l1 r
l1rX = 0
o12
E1
0 -3 2 B EB Bf = Z as abelian groups
III
Eo
3 -3 1 A EB B
=Z
where W = T: FF --+ F G ® GFF is the mapping canonically attached to the identity FF ® FFG --+ F G with respect to some € : G F G --+ GGG (see the example after Proposition 2.1);
= (FFG ® GFF) / w(FFF)'
T(Up. VF, tp) = Kertp 4 UF ® FMF ....=> UF ® FNG' where FNG = (FFG ® GGG + w(FFF))/w(FFF); R
= (FF' MF/xF,
FF ® FMF ~ MF -
MF/xF) with 0
=1=
x EN;
54
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL REFERENCES
1. Berman, S., Moody, R. and Wonenberger, Maria: Certain matrices with null roots and finite Cartan matrices. Indiana Univ. Math. J. 21, 1091-1099 (1971/72). 2. Bernstein,1. N., Gelfand, 1. M. and Ponomarev, V. A.: Coxeter functors and Gabriel's theorem. Uspechi Mat. Nauk 28, 19-33 (1973); translated in Russian Math. Surveys 28, 17 -32 (1973). 3. Bourbaki, N.: Groupe et algebres de Lie, Ch. 4, 5 et 6. Paris: Hermann 1968. 4. Dlab, V. and Ringel, C. M.: On algebras of finite representation type. J. Algebra 33,306-394 (1975). 5. , Representations des graphes values. C. R. Acad. Sc. Paris 278, 537 -540 (1974). 6. , Representations of graphs and algebras, Carleton Mathematical Lecture Notes No.8, 1974. 7. Donovan, P. and Freislich, M. R.: The representation theory of finite graphs and associated algebras. Carleton Mathematical Lecture Notes No.5, 1973. 8. Gabriel, P.: Unzerlegbare Darstel!ungen 1. Man. Math. 6, 71-103 (1972). 9. - - - , Indecomposable representations II. Symposia Math. 1st. Naz. Alta Mat., Va!. XI, 81-104 (1973). 10. Gelfand, 1. M. and Ponomarev, V. A.: Problems of linear algebra and classification of quadruples of subspaces in a finite-dimensional vector space. Coll. Math. Soc. Bolyai 5, Tihany (Hungary), 163-237 (1970). 11. Kleiner, M. M.: Partially ordered sets of finite type. Zap. Nauen. Sem. Leningrad. Otde!' Mat. Inst. Steklov. (LOM!) 28, 32-41 (1972). 12. Moody, R. V.: Euclidean Lie algebras. Can. J. Math. 21,1432-1454 (1969). 13. MUller, W.: Unzerlegbare Moduln iiber artinschen Ringen. Math. Z. 137, 197226 (1974). 14. Nazarova, L. A.: Representation of quadruples. Izv. Akad. Nauk SSSR, ser. Mat. 31, 1361-1377 (1967). 15. ,Representation of quivers of infinite type. Izv. Akad. Nauk SSSR, ser. mat. 37,752-791 (1973). 16. Nazarova, L. A. and Roiter, A. V.: Representations of partially ordered sets. Zap. Nauen. Sem. Leningrad. Otde!' Mat. Inst. Steklov. (LOM!) 28, 5-31 (1972).
V. Dlab Department of Mathematics Carleton University Ottawa Kl S 5B6, Canada
C. M. Ringel Mathematisches Institut Universitat Bonn Federal Republic of Germany
55
REPRESENTATIONS OF GRAPHS AND ALGEBRAS ADDENDUM
We should like to use this opportunity and point out some of the developments in the theory of representations of graphs which took place after this paper was prepared for publication. In order to complete the description of the category UIJ)?, [2) for a realization (9Jl, [2) of an extended Dynkin diagram, one would need the classification of the homogeneous indecomposable modules. In general situation, this seems to be very difficult. However, the subcategory H(IJ)?, [2) can be described in the case of a K-realization (or "K-species"), that is in ~ill,
the case of a realization
[F; : K]
<
00,
[2) such that all F; contain a common central field K with
which operates on each ;Mj centrally. For these, H(9J(, [2) is always a direct
sum of uniserial subcategories of global dimension one with a single simple object (c. M. Ringel, Representations of K-species and bimodules, to appear in J. Algebra.) Thus in order to complete the classification of all indecomposable representations of K-realizations of extended remai~
Dynkin diagrams it only
to
desc~be
all simple homogeneous representations of all
K -realizations of the diagrams All and Al 2 .
This can be done when the field K is the field R of the real numbers. For, obviously, there are only six different R-realizations of this kind (C and H denote the fields of complex numbers and quaternions, respectively):
RHH
HH R
1) R----+H and H ~R FFF
2) F
EB
FFF
) F with F = R, Cor H; and
C EB C-
3) C c c
c
«. c, where
the right action of C on cCe is given by conjugation.
This follows from the fact that a bimodule FMc can be considered as a left F
Cl9R
HOp is a simple algebra and C 0
Cl9K
GOp_
C is the product of two copies of C. R In 1), the complete classification is given in V. Dlab and C. M. Ringel, Real subspaces of a vector space over the quaterpions, to appear: The simple homogeneous representations
module; now, H
are of the form U R ~ H R , where U is a two-dimensional R-subspace of H containing the sub field R, and thus the set of such representations is the two-dimensional real projective space. The endomorphism ring of every such representation is the field C. In 2), H('.m, [2) is the product of the module category of the corresponding polynomial ring F [x] and a uniserial category with a single simple object whose dimension type is (l, 1). Here, if F = R, the set of all simple homogeneous representations is a compact real
2-hemisphere
K: the endomorphism ring of a representation corresponding to a point on the
boundary is the field R, and that of a point in the interior is C. The correspondence can be described as follows: Consider the hemisphere
K as the one-point compactification of the
closed upper real plane. For the points (a, b) with b R x R
> 0,
; R x R;
(- ~ ~ )
the corresponding representation is
56
VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
for (a, 0),
R
a
R·,
and for the point "",
0 R
~R.
If F = C, the set of simple homogeneous representations is the real 2-sphere and the endomorphism ring of every such representation is C. We consider the real 2-sphere as a one-dimensional projective C-space and then a correspondence is given, for c E C, by 1
C
c
:C
and for "", by
0 C If F
= H, then
: C.
the set of all simple homogeneous representations is again the compact
real hemisphere K and the endomorphism rings of such representations are either H or C, depending whether the corresponding point lies on the boundary or in the interior of K. Here, the correspondence between the points of K and the representations is given as follows: For (a, b) in the real upper half-plane (b ;> 0),
H and for the point "",
H
1 ===: H, a + bi
o
H.
Finally, in 3), H('JJl, Q) is the product of the category of all modules over the twisted polynomial ring C(x, -] with respect to the conjugation, and a uniserial category with a single simple object, again of dimension type (1, 1) (V. Dlab and C. M. Ringel, Normal forms of real matrices with respect to complex similarity, to appear in linear Algebra and App!. Again, the set of all simple homogeneous representations is K; the correspondence is given as follows: for c = (a, b) with b > 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)
