Proceedings of the International Congress of Mathematicians August 16-24, 1983, Warszawa

CLAUS MICHAEL BINGEL

Indecomposable Representations of Finite-Dimensional Algebras

Let 1c "be a field and A a finite-dimensional 7c-algebra (associative, with 1). AVe consider representations of A as rings of endomorpliisms of finite-dimensional 7c-spaces, and thus JL-modules, and we ask for a classification of such representations. More generally, we may consider the following problem: given an abelian category # and simple ( = irreducible) objects F(l),..., B(n) in #, what are the objects in # of finite length with all composition factors of the form jßf(l),..., 22/(w). Problems of this kind arise naturally in many branches of mathematics, in particular, classifica­tion problems for linear representations of other algebraic structures (groups, Lie algebras, etc.) may be reinterpreted in this way. We will always assume that we know the simple -A-modules B(i), 1 < i < w, and also their first extension groups Ext1(5/(i), F(j)), and thus the modules of length 2, and our aim is to study the indecomposable modules of greater length. Note that any (finite-dimensional) A-module can be written as a direct sum of indecomposable modules, and the Krull-Schmidt theorem asserts that these indecomposable direct summands, as well as their multiplicities, are uniquely determined. Besides the semisimple algebras (with all indecomposable modules being simple), there are other algebras with only finitely many (isomorphism classes of) indecomposable modules (they are said to be representation finite). However, there will usually be large families of (pairwise non-isomorphic) indecomposable modules, indexed over suitable algebraic varieties. As Drozd [16] has shown, for representation infinite algebras, there is a strict distinction between the tame and the wild representation type, the tame algebras being character­ized by the property that there are at most one-parameter families of indecomposable modules. Por a wild algebra, it seems difficult to obtain a complete classification of all indecomposable modules, since it would involve the (unsolved) problem of classifying pairs of square matrices

[425]

426 . Section 2: C. M. Ringel

with respect to joint similarity. A typical tame problem is the classification of all matrix pencils, which was solved by Kronecker in 1890. This result was needed for classifying pairs of symmetric bilinear forms, but it is also of interest for solving differential equations. Eote that matrix pencils

[ Tc Jc2

For simplicity, we will usually assume that Jc is algebraically closed. Also, let A be connected (without central idempotents ^ 0,1). Given a finite-dimensional algebra A, we denote by K0(A) the Grothendieck group of all finite length J.-modules modulo all exact sequences. The equivalence class in K0(J.) of an ^.-module M will be denoted by dimM and called its dimension vector. 'KQ(A) has a canonical basis given by the elements dimF(i), 1 < £ < M , and, in this way, we may identify K0(A)

n

with Zn. Evaluating dimM = J* midimB(i), we see that the integer

mi = (dimM)t is non-negative, it is just the Jordan-Holder multiplicity of F(i) in M (the multiplicity of E(i) in a composition series of M ; the invariance of mi is usually called the Jordan-Holder theorem). If all mi

are positive, M is called sincere, and A is called sincere provided there exists an indecomposable sincere A -module. The Cartan matrix CA has as the (i, j)th entry the number (dimP(j))i, where P(i) is a projective cover of F(i). If GA is invertible (for example, if gl.dim. A < oo), consider the bilinear form <x,y)> =xG2TyT on K0(A). The quadratic form %(x) = (x,xy is called the Fuler characteristic of A. BTote that if X, Y are J.-modules with proj .dim.X< oo or inj.dim. Y < oo, then

<dîmX,dimr> == J £ ( - l ) M i m E x * ^ X , Y).

I t turns out that for some classes of algebras A the indecomposable A-modules are controlled by %. The first result of this kind was Gabriel's theorem on hereditary representation finite algebras [17, 7]. In the same way, the hereditary tame algebras were first considered by Gelfand-Ponomarev, Donovan-Ereislich and JSTazarova [18, 15, 23, 14], and those which are wild by Kac [22]. We are going to outline that at least the representation finite and the tame cases can be well unterstood by consider­ing the corresponding Auslander-Eeiten quivers, and at the same time we want to consider some classes of tame algebras of global dimension 2.

1. The Auslander-Reiten quiver F (A) of A

The present representation theory is based on some fundamental concepts due to Auslander and Eeiten. These concepts are incorporated in the

Indecomposable Representations of Finite-Dimensional Algebras 427

notion of the Auslander-Beiten quiver T(A) of A. We are interested in the set of isomorphism classes [Jf] of indecomposable -4-modules M, and we consider it as the set of vertices of a translation quiver F(A), with an arrow [Jf]->[J7"] provided there exists an irreducible map M->N, and using the Auslander-Beiten translation r.

1.1. (Auslander-Beiten [3]). For any indecomposable JL-module Z, there exists a map g : Y->Z (unique up to an isomorphism) with the following three properties:

(i) g is not split epi, (ii) if gf: Y'->Z is not split epi, then there is an r\\ Y'->Y satisfying

9' « m (iii) if f : Y->Y satisfies g = lg, then £ is an automorphism. Such a map g: Y->Z will be called a sinlc map for Z (Auslander-Beiten

used the term "minimal right almost split map"). Dually, for any indecom­posable J.-module X, there exists (again unique up to an isomorphism) a source map f: X~> Y'.

A niap h: M->N is said to be irreducible provided Ji is not a split map and has only trivial factorizations (i.e., 7i = h'K" implies that li' is split mono, or %" is split epi). Let M, N be indecomposable. Let rad( M, N) be the set of non-invertible maps M->N9 and

rad2(Jf, N) - £ r a d ( l f , X)rad(X, N)

where X runs through all indecomposable modules. Then li: M->N is irreducible if and only if h erad(Jf, N)\mà2(M, N). We call Irr( M, N) = rad( M, JV)/rad2( M, N) the bimodule of irreducible mapB. If Y~>N is a sink map, or M-^Yf a source map, then dim7cIrr(Jlf, N) measures the multiplicity of M in a direct decomposition of Y into indecom­posable modules, and also the multiplicity of N in such a decomposition of Y'.

There is a strong interrelation between sink maps and source maps:

1.2. (Auslander-Beiten [3]). Let Z be an indecomposable -4-module, and g: Y->Z a sink map. Then, either Z is projective, Y is its radical, and g the inclusion map, or else g is epi, rZ: = Ker(#) is indecomposable, and the inclusion map %Z->Y is a source map.

We see that for Z indecomposable and not projective, we obtain (uni­quely) an exact sequence

0^rzUY^yZ^0

with xZ also indecomposable, / a source map, g a sink map. These sequences are now called Auslander-Beiten sequences (Auslander-Beiten used the

428 Section 2: C. M. Ringel

term "almost split sequences".) There is a direct recipe for constructing

xZ for a given Z. Let Px-^P0->^->-0 be a minimal projective presentation of Z, and let v = DHom( —, AA), then xZ = Ker(^p). In case the Cartan matrix G of J. is invertible, the dimension vector of xZ is given as follows :

dimxZ = (d im^)0~(dimKer^)^ + d im^,

with 0 = —C~~TG. Thus $ measures the change of dimension vectors under the Auslander-Beiten translation [2].

Since .T(J.) is locally finite, any component of r(A) is either finite or countable, and finite only in case A is representation finite:

1.3. (Auslander [1]). Assume that r(A) has a component containing only modules of bounded length. Then A is representation finite and r(A) is connected.

This result of Auslander strengthens a theorem of Bojter which had established the first Brauer-Thrall conjecture. On the other hand, it also yields a method for showing that a given finite list of indecomposable .A-modules is complete.

We will consider the possible shapes of components in the next sections. A component ^ of T(A) will be Said to have a trivial modulation provided dimÄIrr(X, Y) < 1 for all indecomposable modules X, Y in ^.

There are several ways of considering T(A) as a two-dimensional simplicial complex, with O-simplices the vertices of r(A). Following Gabriel and Biedtmann, we take as 1-simplices both the pairs ([Jf], [JV]) with [M]-> IN] in r( A) and the pairs ( \xZ~\, \Z~\) where Z is indecomposable and not projective, and as 2-simplices the triples ([xZ], [Y], \Z~\), again for Z indecomposable and not projective, and [3T]->[#] in r(A). In this way, we may speak of the underlying topological space of r(A). As an example, the Auslander-Beiten quiver of the algebra of all upper triangular n xw-matrices over ft is just a large triangle, it will be denoted by @(n).

2. Directing modules and preprojective components

A cycle in the category J.-mod of A-modules is a finite sequence M0, Mx,... . . . , Mn = MQ of indecomposable modules satisfying rad(ilf£_1, M^ ^ 0 for 1 < i < n. Of course, any cyclic path in T(A) gives a cycle in A-mod; however, there may exist additional cycles in A-mod, both containing modules composed of a single component and containing modules of different components. An indecomposable module M not contained in

Indecomposable Representations of Finite-Dimensional Algebras 429

Sb cycle is called a directing module. An element x of K0(A) will be called a root if %(x) = 1 and a null root if %(x) = 0.

2.1. If M is a directing module, then dim M is a positive root, and M is uniquely determined by dimilf.

2.2. The existence of a sincere directing A-module implies that gl. dim. A < 2 .

2.3. A component ^ of r(A) containing no cyclic path and such that any module in # is of the form x~%B for some indecomposable projective module P and some t > 0 is called a preprojective component. The modules belonging to a preprojective component are all directing modules, and they can be constructed inductively from the indecomposable summands of the radicals of the indecomposable projective modules as iterated cokernels. Given a preprojective component #, its orbit quiver is a quiver with labelled arrows: its vertices are given by the x-orbits in (ê, or, equivalently, by the indecomposable projective modules in #, and the number of arrows from P(i) to B(j) with label t is nijt, where radP(i) = 0 ( r~ 'P( j )p ' .

In case ^ is a preprojective component not containing indecomposable injective modules, then the full subcategory of all modules belonging to # is uniquely determined by ft and the orbit quiver. (There is a general notion of preprojective modules due to Auslander and Smal0 [4], the modules in a preprojective component being always preprojective.)

2.4. If A is a (finite-dimensional) hereditary algebra, then A is the path algebra of some finite quiver A (A). In this case, A has a unique prepro­jective component, and its orbit quiver is just A (A), with all labels being 0.

2.5. A representation finite (connected) algebra A is called directed pro­vided there is no cycle in J.-mod, or, equivalently, the indecomposable JL-modules form a preprojective component which is finite. A quadratic form g on a free abelian group with a distinguished basis is said to be integral provided q takes only integer values, and q takes the value 1 on the elements of the distinguished basis; and q is said to be weaJcly positive provided q takes positive values on positive elements. If A is a directed algebra, then the Euler characteristic %A is obviously an integral quadratic form on K0(A). In addition, if A is sincere, then %A is weakly positive, and dim furnishes a bijection between the indecomposable A-modules and the positive roots of %A [20]. (Actually, as Bongartz [9] has shown,

430 Section 2: C. M. Ringel

in dealing with directed but not necessarily sincere algebras, one may replace the quadratic form % by some truncated form %' which is always weakly positive, integral and with dim defining a bijection between the indecomposable modules and the positive roots.) Now, an interesting theorem of Ovsienko [24] asserts that, for a positive root x = (x19..., xn) of a weakly positive integral quadratic form on Zn

9 all the coordinates-satisfy Xi < 6, thus we see that the Jordan-Holder multiplicities of an indecomposable module over a directed algebra are bounded from above by 6. (A different proof of this bound was given by Bongartz in [8].) It will be possible to classify the sincere directed algebras. In [8], Bongartz determined all such algebras with more than 336 simple modules: there are 24 different series of them, and its is interesting to note that for these algebras the Jordan-Hölder-multiplicities of the indecomposable modules are bounded even by 3. There are only finitely many additional sincere directed algebras, and the corresponding list should be furnished with the help of a computer. Finally, let us note that the possibilities for the orbit quivers of sincere directed algebras are rather restricted: the orbit quivers form a tree (Bautista-Larrion-Salmeron [6]) with at most four end-points (Bongartz [8]).

2.6. An algebra A is said to be minimal representation infinite provided A is not representation finite, but for any non-zero ideal! the algebra A\I is rep­resentation-finite. The minimal representation-infinite algebras having a preprojective component have been classified by Happel and Vossieck [21]. The underlying graph of the orbit quiver of the preprojective component is a Euclidean diagram (An9 Dn, E69 E79 or JB8). In case An, the algebra is hereditary, in case Dn, there are 4 different series of them, and the numbers of the isomorphism classes in the cases JB6. E19 JE8 are 56, 437, 3809, res­pectively. /

The minimal representation infinite algebras with a preprojective component will be called concealed algebras.

3. Representation finite algebras

3.1. Given J., representation finite, one may construct the universal cove­ring r(A) of r(A), say with the covering group G (A) and the covering map n: F(A)->r(A). Bongartz and Gabriel [10] have shown that F(A) itself is an Auslander-Beiten quiver, although usually not of a finite^

[ndecoinposable Representations of Finite-Dimensional Algebras 431

äimensional algebra, but only of a locally bounded algebra Ä9 called the universal cover of A. Given an indecomposable A-module M9 we choose some indecomposable Jt-module M with n([M]) = [Jf] and consider the support algebra Ä(M) = Äj(e = e2 eÄ\eM == 0>. Then Ä(M) is a directed algebra with M being a sincere J[(i&)-module, and many pro­perties of the JL-module M can be dealt with considering Jf as an Ä(M)-module. Note that the silvie J!( Jf)-modules are partitioned into subsets which correspond to the simple JL-modules so that the Jordan-Holder multi­plicities of a simple JL-module in Jf is given by the sum of the Jordan-Eölder multiplicities of the simple Jl(Jf)-modules in the corresponding subset. Since Bretscher and Gabriel [12] have shown that Ä and G(A) 3an be constructed directly from A as soon as its quiver and a set of relations are known, one may use Ä efficiently in order to determine r(A). Namely, we construct P(A) inductively, using one-point extensions [note that this construction leads rather quickly to a periodic behaviour with respect to G(A) since (dirnJET̂ . < 6 for indecomposable M and any i), and we obtain r(A) from r(A) by factoring out the operation of G(A) on r(A). We remark that Bongartz and Gabriel [10] have shown that G (A) is always a free (non-abelian) group.

5.2. There is a general result concerning the Auslander-Beiten quiver of a representation finite algebra A. According to Bautista and Brenner [5], my element x of T(A) has at most four direct predecessors and at most LOUT direct successors, and the number four occurs only if one of the four •nodules is projective-injective. This may be rephrased as a statement concerning subquivers of T(A) of form D4, and is a special case of a more renerai statement dealing with subquivers of r(A) which are Euclidean liagrams: namely, one can determine a universal bound for their "repli­cation length".

J.3. As a special case one obtains the following result of Biedtmann [25] : an indecomposable JL-module Jf is called periodic provided r*Jf w Jf cor some t > 1. The translation subquiver of T(A) consisting of all periodic aiodules will be denoted by rp(A). Then, for a representation finite A, jhe components of r(A) are of the form ZAjG, A being a Dynkin diagram [An, Dn, E6J ET, or Ea) and G being a group of automorphism of ZA. Por a representation infinite algebra A, it was shown in [19] that there s only one further possibility, namely A = A^. We recall the construction Df Z0 for an arbitrary quiver 0: we start with disjoint copies of 0 indexed by Z, say with elements (z9 i) where zeZ, and i is a vertex of (9. Then

432 Section 2: C. M. Eingel

we add arrows (x,i)->(y, i+1) for any arrow y->x and any i, and define riyy x(z,i) = (#, i— 1). If the underlying graph & of 0 is a tree, Z0 only depends on 0 .

4. Separating tubular families

A component of F(A) with trivial modulation is called a fajfee provided it contains a cyclic path and its underlying topological space is of the form. jS1 x R£ (where 8l is the 1-sphere, and R£ the set of non-negative real numbers). The regular tubes of rank r (where r > 1) are the components of the form Z J.00/<T' ,>. We want to outline a procedure for obtaining families of tubes F(X) with X e PxJc. With any family of positive integers % (X e I), we associate its type: form the disjoint union of diagrams Anx (A e I ) , choose in any A^ one end-point and identify these end-points in order to form a star. The type of the sequence n19 ...9nr will be denoted by Anv...,n • Given a family of tubes, its type will be the type of its rank function. A family of tubes &(X)9 A e I, will be said to separate 0* from 3,, provided the remaining indecomposable A -modules fall into two classes &, SL such that Hom(.S,^) ==Hom(^, &*(X)) = Hom(«r(A), @) = 0 for all 1, and, moreover, such that, for any X any map from a module in 0> to a module in St factors through a direct sum of modules in &~(X).

We consider an algebra A which is a one-point extension of an algebra-

A0, say A = J.0[JS]: = I ft ° , where B is an J.0-module. The JL-modules.

can be written as triples (X0, Xm ,y),Xoi being a &-space and y : B ® X^XQ, being an J.0-linear map. The indecomposable J.0-module WQ is said to be a wing module provided it belongs to a component with trivial modulation and such that any arrow #->[W0] in r(A0) is contained in a full convex translation subquiver 0Q (a "wing") of r(A0) isomorphic to some 0(nx)r

[TF0] being the projective-injective vertex of 0X. The type of W0 is the type of the function nx. A wing module WQ is said to be separating provided the indecomposable J.0-modules not belonging to the wings fall into two* classes ar, 9 with Hom(^ , X) = Hom(^, W0) = Hom(W0, X) = 0, and such that any map from a module in X to a module in <$f factors through a direct sum of copies of W0. Finally, W0 is said to be dominated by B,. provided dimJB = (dimWr

0)(27-^-1) and, for any 0 ^ Q: JB->TF0, t he J.0[i2]-module W'0(g): = (WQ, Jc, Q) satisfies proj. dim.TF0(e) < 1.

4 .1 . Let W0 be a sincere separating wing J.0-module of type A dominated by B. Let A = AQIB], let w = dimW0+dim(0, Jc, o) in K0(JL), and let

Indécomposable Representations of Finite-Dimensional Algebras 433

iw — <w, —>. Let 0>w, ^w, Mw be the sets of indecomposable ^.-modules satisfying ^(dimjf ) < 0, = 0, > 0, respectively. Then ^w is a P^-family of regular tubes of type A separating &w from lw. Also, Tw is controlled by the restriction of %A to Ker^ (for Jf in &*w9 dimJf is a root or a null root; conversely, given a positive root a in Ker^, there is a unique inde­composable Jf with dimJf = a, and given a positive null root a in Ker vw, there is a one-parameter family of such modules).

4.2. If A is neither a Dynkin diagram, nor a Euclidean diagram, then both @w and &w are wild. Note that one may use (4.1) to construct tubular families of arbitrary type A.

4.3. Assume A0 is a directed algebra and Jf a sincere maximal indecompos­able module. Then Jf is dominated by some projective module B. Thus, if the T-orbit of Jf is the only possible branching point in the orbit quiver, then we can apply (4.1) to W0 = Jf.

In particular, if A0 is hereditary and representation finite, then A0

is directed and has a unique maximal module Jf, and we may apply (4.1) to WQ = Jf. Let Jf be dominated by B. Then A = J.0[i2] is hereditary, too. The orbit type of A0, and therefore the tubular type of A, is a Dynkin diagram A. In this case @>w is a single component, namely the preprojective component of A, and its orbit diagram is the corresponding extended diagram A. Similarly, £lw is the preinjective component of A, also with the orbit diagram A. In this way, we obtain a full classification of the representations of a tame hereditary algebra [15, 23, 14]; actually, one may consider in the same way also the general case of an arbitrary base field as considered in the joint work of Dlab and the present author [14], A similar structure theory holds for all concealed algebras. The indecom­posable modules 8 belonging to the mouth of a tube (thus those with Hom(r#, 8) = 0) are said to be simple regular.

4.4. Let A0 be a concealed algebra with a tubular family ^0(X) of rank r0(X), and 8 a simple regular -40-module in some f0(e). Then A = A0[8] is called a simple tubular extension of A0. The extension type of A is, by definition, the type of the function rA: PJc-^N with rA(X) = r0(X), for X ̂ Q, and rA(g) = r0(g) +1. Inductively, one may define arbitrary tubular extensions, and their extension types (see [13, 27]). Note that all those algebras have global dimension < 2. The dual construction is that of a tubular coextension.

Let A be a tubular extension of a concealed algebra A09 and let A

434 Section 2: C. M. Eingel

be the extension type of A. If A is neither Dynkin nor Euclidean, then A is wild. If A is Dynkin, then A has a preprojective component &, namely the preprojective component of AQ. I t has also a preinjective component SL containing all indecomposable injective modules with the orbit diagram A, and the remaining indecomposable modules form a PjMamily of (not necessarily regular) tubes separating & from SL. Also, .A-mod is controlled by %. Thus, it remains to consider the case of a Euclidean A, so that à = Ani nr, with (%,. . . , nr) = (2, 2, 2, 2), (3, 3, 3), (4, 4, 2) or (6, 3, 2).

4.5. Let A be a tubular extension of a concealed algebra A0 of the Eucli­dean extension type A. Then A is also a tubular coextension of a concealed algebra A^. Let w0 be the minimal positive null root of A0, and w^ that of AQQ9 and define ^ = <,awQ

JrßwQQ9 —>. We have the following com­ponents of r(A):

(1) a preprojective component ^ 0 ( = the preprojective component of A0)9

(2) a separating tubular P1Ä-family &~Q (obtained from the tubular family of A0 by ray insertions),

(3) for any y e Q+9 a separating regular tubular P^-family fY of type

A consisting of all indecomposable modules X with iatß(dimX) = 0 , where y = ßfa,

(2)* a separating tubular Pxifc-family S*^ (obtained from the tubular family of A^ by co-ray insertions),

(1)* a preinjective component SL^ ( = the preinjective component of

Also, the category A-mod is controlled by %.

4.6. One may use this in order to obtain a corresponding result for subspace categories of vector space categories, in particular for the non-domestic partially ordered sets of finite growth. In fact, for these posets, the one-parameter families of indecomposable representations have been determined before by Nazarova and Zavadskij [30], and Zavadskij has independently constructed the remaining representations. The algebras considered here were studied by Brenner-Butler [11] and in [26]. The proof of the results in 4. are outlined in [27], they will appear in [28].

5 . Addendum

We have tried to outline the use of the Auslander-Beiten quiver F (A) for classifying indecomposable A-modules, and for getting some insight into the module category A-mod. Along these lines we have reported on

Indecomposable Eepresentations of Finite-Dimensional Algebras 435

some of the advances in representation theory in recent years. However, we should stress that this paper covers only a small portion of the present theory. For earlier results, we refer to the report of Bojter at the Helsinki Congress 1978 [29]. In the discussion of the material presented here, we have avoided the more technical notions (such as tilting functors) even if we are sure of their usefulness for a better understanding. Also, we have not discussed at all some of the major recent developments which would have justified a separate report. We only want to mention Biedtmann's classification of the representation finite self injective ( = quasi-Frobenius) algebras, and the work of Bojter, and Bautista, Gabriel, Salmeron on the existence of a multiplicative basis for representation finite and minimal representation infinite algebras. In fact, it follows from the above work that for a basic representation finite algebra A, the Auslander-Beiten quiver r(A) uniquely determines A, provided the characteristic of Jc is different from 2.

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32 — Proceedings...

436 Section 2: C. M. Eingel

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