Group actions on algebras and module categories

J. A. de la Peña

 Introduction and notations

Let be a field and a finite dimensional (associative with ) -algebra. By we denote the category of finite dimensional left -modules. In many important situations we may suppose that is presented as a quiver with relations (e.g. if is algebraically closed, then is Morita equivalent to ). We recall that if is presented by , then is a finite quiver and is an admissible ideal of the path algebra , that is, for some , where is the ideal of generated by the arrows of , see [6].

It is convenient to consider as a -linear category with objects (= vertices of ) and morphisms given by linear maps , where is the trivial path at (for ). In this categorical approach we do not need to assume that is finite (therefore the -algebra may not have unity). Ocasionally we write if we do not need to explicit the quiver .

The purpose of these notes is to present an introduction to the study of actions of groups on algebras and their module categories and to consider associated constructions that have proved useful in the Representation Theory of Algebras.

A symmetry of the quiver is a permutation of the set of vertices inducing an automorphism of . We denote by the group of all symmetries of . Those symmetries inducing a morphism such that form the group . In natural way, any induces an automorphism of the module category and on the Auslander-Reiten quiver of (since the action commutes with the Auslander-Reiten translation of ).

In section 1, we present some basic facts about the actions of groups on (orbits, stabilizers, Burnside's lemma) and show that for a representation-finite standard algebra , we have , where is formed by the symmetris of commuting with the translation . We recall that is standard if is representation-finite and for a choice of representatives of the isoclasses of indecomposables, the induced full subcategory of (denoted by ) is equivalent to which is the quotient of the path algebra by the ideal generated by the meshes of the almost split sequences . We recall that for , any representation-finite algebra is standard.

In section 2 we consider relations between the structure of and the Coxeter polynomial of (recall that, is the Coxeter polynomial associated to the Coxeter matrix which is -invertible in case ).

In section 3 we present the main constructions associated to groups acting on algebras:

if acts freely on (that is, and for a vertex , implies ), then the Galois covering is a -invariant functor of -categories.

if is a -graded -category, the smash product is a -category accepting the free action of . Moreover, .

Galois coverings were introduced by Bongartz and Gabriel [6, 7, 2] for the study of representation type of algebras. Smash products is a well-known construction in ring theory (see [1]), but only recently was observed by Cibils and Marcos [4] that it yields the inverse operation to Galois coverings. Indeed, if acts freely on , then is a -graded category such that .

In section 4, following [2], we introduce functors relating the module categories of and when acts freely on . The main results in these notes (section 5) relate the representation types of and (which was the original purpose of the introduction of Galois coverings). Indeed, given a Galois covering with a finite dimensional -algebra, then is representation-finite if and only if is locally representation-finite (that is, for each , there are only finitely many indecomposable -modules , up to isomorphism, with ). The proof of this result was partially given in [7] and completed in [10], and provides an efficient tool to deal with representation-finite algebras.

The representation-infinite situation is more involved. We recall that is said to be tame if for every there are finitely many -bimodules wich are free finitely generated as right -modules and such that any indecomposable -module with dimension is of the form for some and some . We say that a tame algebra is domestic (resp. of polynomial growth) if can be chosen (resp. for some ) for all . It is not hard to show that is tame if is tame for a group acting freely on . The converse was shown to be false in [8]; nevertheless there are many interesting, general situations where it holds true.

In section 5 we give examples of results [5, 12] showing that for a Galois covering , the category is tame if and only if is tame, provided certain restrictions on the group or on the category are satisfied.

We denote by the Grothendieck group of , freely generated by representatives of the simple -modules, where is the number of vertices of . We denote (resp. ) the projective cover (resp. injective envelope) of . With the categorical approach an -module is a functor , and a morphism is a natural transformation. In case , the Coxeter matrix is defined by .

These notes follow closely the lectures given at the Workshop on Representation Theory in Mar del Plata, Argentina in March 2006. The intention of the lectures was to present an elementary introduction to the topic which would serve as a source of motivation and information on the techniques used. While we cannot provide complete proofs of every result, we tried to sketch some representative arguments. We thank the organizers of the Workshop for his hospitality.

1. The group of automorphisms of an algebra

1.1. Let be a finite dimensional -algebra:

denotes the group of automorphisms of . By we denote the group of symmetries of and by the group of symmetries of fixing (that is, induces such that ).

Lemma.

is a subgroup of .

Each symmetry gives rise to a matrix sending to . This representation is called the canonical representation.

1.2. Let be a subgroup. For a given , denotes the orbit of and subgroup of , denotes the stabilizer of . Clearly is a subgroup of .

Lemma.

The mapping , is a bijection from the orbit to the set of left cosets .

where the sum runs over representatives of the orbits.

1.3. Let be the Grothendieck group of . Consider

the -space of -invariant vectors. Then the number of orbits of in equals .

Let be a set of representatives of the irreducible -representations of (here is the number of conjugacy classes of ). Let be the trivial representation.

Consider the character corresponding to (that is, , ). The characters form an orthonormal basis of the class group , with the scalar product .

Lemma.

[Burnside's lemma]. where , is the character of the canonical representation .

Proof. Observe that is the number of fixed points of .

where . □

1.4. Let be a subgroup of then acts on as follows:

such that for , we have . Similarly, for , we define .

Clearly, this action preserves indecomposable modules and induces an action of of the Auslander-Reiten quiver , satisfying:

1. the action preserves projective, injective and simple modules;
2. the action preserves Auslander-Reiten sequences (in particular, );
3. is a subgroup of , the group of automorphisms of the quiver (commuting with the -structure).

For (c), let be an element inducing a trivial action on ; we shall prove . Indeed, implies for every vertex .

Let , be all arrows between and , then establishes a permutation of the . Let be the dimensional -module with . Since is indecomposable, or . Therefore .

Proposition.

Let be an algebra satisfying:

(a) is representation finite, (b) is standard

then .

Proof. We already know that is a subgroup of . Let and consider the induced automorphism of the mesh category . Clearly, restricts to an automorphism of (considering the full embedding , ). By definition there is an automorphism inducing , and therefore inducing . □

2. The canonical representation and the Coxeter matrix

2.1. Let and assume that . For example, this happens if is triangular, that is, has no oriented cycles.

Recall that , is the Coxeter matrix if , which is -invertible. In case (i.e. ), then for any non-projective indecomposable .

The characteristic polynomial is called the Coxeter polynomial.

Examples:

1. Dynkin type, then .
2. extended Dynkin type, then and is a root of multiplicity .
3. 

   is irreducible (over ).

Proposition [11].

1. is an automorphism of the representation .
2. If is not trivial, then is not irreducible.

Proof. (a): It suffices to observe that for any .

(b): Let be a set of representatives of the irreducible -representations of . Let be the trivial representation. Up to conjugation (with )

, then .

By Schur's lemma , where is an automorphism. Hence . Therefore if is irreducible then for .

If , then . Moreover, the characters

form an orthonormal basis of with scalar product

Then

for , .

Finally, , with implies the existence of with . Therefore is not irreducible. □

2.2. Example:

Consider and the canonical representation. It is not hard to calculate the character table of :

where corresponds to irreducible -representation of (with the trivial representation).

 

Then

Since , which is also a -decomposition. Moreover, is an irreducible factorization.

3. Constructions of algebras asociated to groups of automorphisms (coverings and smash products)

3.1. Let be a -category given as . Let be a subgroup of . We say that acts freely on if for some implies .

Lemma.

Let be a group acting freely on . Then there exists a -category and a functor satisfying:

1. (-invariant): for every
2. (universal -invariant): for any functor which is -invariant, there exists a unique functor such that
3. is formed by the -orbits of vertices in and for any and

   

Proof. Let be defined as in (c) with composition maps

Define , in the unique possible way (since implies ). □

3.2. If acts freely on , the functor as in (2.1) is called a Galois covering defined by and is a Galois quotient of .

We say that a -category is -graded if for each pair of objects there is a vector space decomposition such that the composition induces linear maps

Lemma.

is a -graded -category.

3.3. For a -graded -category we define the smash product as the -category with objects and for any pair , the morphisms are

with composition given by:

Theorem

[4]. The category accepts a free action of such that .

Moreover, if acts freely on , then .

Proof. Clearly there is a -invariant functor inducing a functor which is the identity on objects. Check that is an isomorphism.

Assume acts freely on and let be the induced Galois covering.

Since acts freely, there is a bijection which commutes with the -action. Moreover, if , , then

is compatible with the composition. □

In other words, smash products and Galois quotients are inverse operations in the class of -categories.

3.4. Galois correspondence. Let be a -category. There is a correspondence

( Galois covering defined by ) group grading , such that for any

two Galois coverings defined by and defined by , there exist a commutative diagram

if and only if there exists a normal subgroup , such that .

A Galois triple of is a Galois covering defined by the (free) action of a group . There is a universal object in this set of Galois triples. Indeed, let and consider the set of all walks in starting and ending at a fixed vertex . Let be the equivalence relation induced by the following elementary relations:

1. and for any arrow ;
2. if with , such that for any we have , then for ;
3. if , then , whenever the products are defined.

Then has a group structure such that is -graded. The group is called the fundamental group of .

Proposition

[9]. Let and be defined by . The triple is a universal Galois covering, that is, for any Galois covering defined by the action of a group , there exists a covering defined by such that .

4. Actions induced on module categories

4.1. Let be a Galois covering of -categories defined by the action of . We shall denote by

• the category of left -modules;
• those with for every ;
• those with .

There are naturally defined functors:

called the pull-up functor;

and such that for , then , sends to , called the push-down functor. We observe that is a left adjoint to . Similarly, there is a right adjoint to . For modules , the modules and coincide.

Recall that acts on and is -stable if for every . The category of -stable -modules is denoted by .

4.2. Proposition [7, 2]. Let be a Galois covering defined by the action of . Then the following happens:

1. The categories and are equivalent.
2. For any and , we have . Moreover,
   as -modules.
3. Let be a subgroup of and . Then has a natural structure as -module. If is a finite group and , then decomposes as a direct sum of at least factors.

Proof. (a): clear.

(b): Observe that canonically.

Hence and correspondingly in morphisms.

(c): Choose a set of representatives in of the right cosets . Then and correspondingly in morphisms.

Hence we get a group homomorphism such that for each idempotent of , is idempotent and there is a factorization of .

In case is a finite group with , by Mashké theorem, the group algebra is semisimple (with idempotents). The result follows. □

4.3. Assume acts freely on and is the corresponding Galois covering.

Let , the stabilizer is the subgroup of formed by those such that . That is, if .

Proposition [7].

1. If and is torsion free, then .
2. If and , then is indecomposable and for any module with , then for some .

Proof. (a): Let for some , then establishes a permutation of (a finite set). Then for some , on . Since acts freely on , then . Since is torsion free, then and .

(b): Assume , then . Assume is a direct summand of , then and . Therefore is indecomposable.

If , then is indecomposable and for some . □

5. Coverings and the representation type of an algebra

5.1. Examples:

(a) Let

Let be the infinite -category with quiver

and generated by all relations of the form . Then we get a Galois covering defined by the action of on .

The module

is indecomposable.

It is stable under the action of , .

The universal Galois covering of is given by

where the free group in two generators acts on . The normal subgroup of acts on inducing the covering .

Observe that for , we have and therefore .

(b) [13] Let be a standard -algebra of finite representation type. Then the Auslander-Reiten quiver is finite and equipped with the mesh relations: , for each almost split sequence .

There is an universal Galois cover of translation quivers defined by the action of a free group . Moreover, is the Auslander-Reiten quiver of a -category such that .

We illustrate the situation in the following example:

A full subcategory of is convex in if for a quiver which is path closed in (i.e. with implies ).

Theorem.

Let be a representation-finite -algebra. Then

1. [13] There exists a universal Galois covering defined by the action of a free group . Moreover for a -category . If is standard, then is a universal Galois covering.
2. [3] For every finite convex subcategory of , we have and is representation-directed (i.e. is representation-finite and is a preprojective component).

5.2. We say that the group acts freely on indecomposable classes of -modules if for implies . Observe that for the algebra with quiver

the indecomposable representation has non-trivial stabilizer.

Theorem.

Let be a Galois covering defined by a group

1. [2] If acts freely on indecomposable classes of -modules, then induces an injection ( and preserves Auslander-Reiten sequences.
2. [2, 10] If is finite (hence a -algebra), then is representation-finite if and only if (i)  acts freely on and (ii)  is locally representation-finite (i.e. for each , is finite). In that case, .

Proof. (a): Let , then we know and . Moreover implies for some . For an almost split sequence with we get an exact sequence

with , indecomposable. For a non-invertible epi in , we get a map

There is some such that . From the adjunction we get

Then and is almost split sequence in .

(b): Assume is representation-finite and are representatives of . Let and suppose is such that . We shall prove that . Let be a covering defined by a free group as in (5.1), that is, we get commutative diagrams:

Then induces an automorphism on such that and for some with . We get an automorphism of acting freely (since acts freely on ).

Let be the full subcategory of formed by . It is easy to see that is convex in and therefore is a preprojective component. Since , then we get with . Since commutes with , there is a projective -module with , which is a contradiction unless .

We check that is locally representation-finite: let and let be all indecomposable -modules which are direct summands of for some and . This set is finite [indeed, , then and , for all . Since , the set is finite].

Finally, let . Then and

hence for some . Therefore . □

5.3. Let be a Galois covering with group . In case is a representation-finite -algebra, we have seen that covers all indecomposable modules. The situation is different for representation-infinite algebras. We shall briefly discuss the new occuring phenomena. By we denote the full subcategory of formed by all objects isomorphic to for some (these modules are called -modules of the first kind). The remaining indecomposable modules (called of the second kind) form the category .

If is such that is not finite but is finite, then is called a weakly -periodic module.

Lemma [5].

Let be a Galois covering such that acts freely on . For the following conditions hold:

1. if and only if , where all .
2. if and only if , where each is weakly -periodic.

Proof. (1): If for some , then . Assume now that is a direct summand of . Let and be such that . We get morphisms and which yield a pair of maps and . Since for , then the endomorphism of is invertible.

Since induces an equivalence of categories , then is a direct summand of . Thus .

(2): Let . By the proof of (1), where . Then we have to show that an indecomposable direct summand of is weakly -periodic. Indeed, let be a set of representatives of the cosets of with respect to . Then is a direct summand of . Since is contained in a finite number of -orbits, then is finite. The converse follows from (1). □

5.4. A category is called locally support finite if for each , the full subcategory of , consisting of the vertices of all with and , is finite.

Proposition.

Let be a Galois covering. Then

1. If acts freely on , then induces a bijection .
2. If is locally support finite, then . In particular if acts freely on , then induces a bijection between and . Moreover, in this case is tame (resp. domestic, poynomial growth) if and only if so is . For the Auslander-Reiten quivers we have .

Proof. (a): Follows from (4.2) since there are no weakly -periodic indecomposable -modules.

(b): Let and consider with . Let be the (finite) full subcategory of consisting in the objects such that or for some . Let be an indecomposable direct summand of the restriction such that . Hence and it is easy to check that is a direct summand of in .

Thus . The last claim is easy to prove. □

Example:

The category given by the quiver with relations

with , is locally support finite (why?). Moreover the group generated by the action acts freely on and on . Hence the Galois covering yields a bijection . The algebra is given by the quiver with relations

Since is tame (resp. polynomial growth for ), so is .

5.5. Given a natural number , a group is said to be -residually finite if for each finite subset there is a normal subgroup of finite index such that and . For example, free groups are -residually finite.

Theorem [12].

Let be a Galois covering given by the action of a -residually finite group , where . Assume that is locally support finite. Then is tame if and only if is tame.

Proof. Without loss of generality, we assume that is finite. We denote also by the induced covering functor. We divide the proof in several steps.

(1) Let be the push-down functor. Consider a sequence of finite full subcategories of such that and if or , then . The restriciton functor has a left adjoint such that . Therefore, the functor

is right exact, .

(2) Let , we shall prove that is a direct summand of . Indeed, since is locally support-finite, by (5.3), the pull-up decomposes as , where . Thus . We proceed in two steps.

(2.1) There exists a normal subgroup of with finite index not divisible by , such that the Galois covering induces a bijection .

Since is finite, the set of vertices of is a disjoint union of a finite number of -orbits , . With the notation introduced above, we consider the full subcategory , of . Let be the set of all elements such that , for some . Since acts freely on , then is finite and there is a normal subgroup of such that is finite of order not divisible by and . Hence , . By (5.1), the induced Galois covering yields an injection . We show that this map is also surjective. Let and a vertex of such that . The restriction is a bijection and there is an indecomposable -module with and such that is an indecomposable direct summand of the restriction of to . We conclude that and , showing that .

(2.2) For the proof of (2), consider a normal subgroup of as in (2.1) and a factorization of Galois coverings

Then and . For an indecomposable , we get

Consider the indecomposable decomposition with for . Then and the result follows from (4.2). In particular, is a direct summand of for some .

(3) There is some and a function such that

has the property that for every with , there exists some with and such that is a direct summand of .

Indeed, the set of vertices is the disjoint union , then there is some such that contains all the categories for , . By (2), is a direct summand of for some . For some , and satisfies that is a direct summand of . Clearly, (where denotes the number of objects of ) satisfies the desired property.

(4) Assume that is tame, we show that is tame. Indeed, let and be as in (3). Since is right exact (1), there is a -bimodule such that . Let .

Since is tame, there is a family of -bimodules which are finitely generated free as -right modules and such that any indecomposable -module with is isomorphic to for some and a simple -module. By (3), for every with , there exists some and a simple -module such that is a direct summand of . It is easy to see that this (aparently) weaker property is equivalent to the tameness of .

(5) Assume now that is tame. It is enough to show that each is tame. Consider the right exact functors

Let . Thus and

where is the finite set of such that . As in (3), (4) we get that is tame. □

5.6. We consider several examples:

(a) Let be a Galois covering induced from a Galois covering of quivers with relations and defined by the action of a group . Assume that is locally support-finite.

• If is a finite group such that does not divide the order of , then (5.5) applies and is tame if and only if so is .
• If is a free group, then both (5.2) and (5.7) apply.

(b) Consider example (5.4). For , the category is tame and locally support finite. Hence the algebra is tame.

(c) [8] Consider the Galois covering

given in (2.2.c) and assume that . We know that is tame. We show that is a wild algebra.

Set , , , . Then is isomorphic to the algebra given by the quiver with relations.

The universal covering constructed as in (2.3) admits a full convex subcategory as follows

Since is wild (observe that the Tits form takes value in the indicated vector), then is wild. Then is also wild.

(d) The algebra of example (c) provides another example of a wild algebra whose Tits form is weakly non-negative. Indeed,

5.7. We consider briefly the situation of coverings where is not necessarily locally support-finite.

Let be a Galois covering. A line in is a full convex subcategory isomorphic to the path category of a linear quiver (of type , or ). A line is -periodic if its stabilizer is non-trivial (then is of type ).

Let be a -periodic line in , we construct an indecomposable weakly -periodic -module by setting for and for and for any arrow in . Then is isomorhic to . Let be a set of representatives of the -orbits in , for any object in . Then is a -bimodule such that for each , is a free -module of rank . We consider the functor

Let be the set of all lines in and be the set of representatives of the -orbits in .

Theorem [5].

Let be a Galois covering such that acts freely on . Assume that for any weakly -periodic -module , is a line. Then

1. Every module in is of the form for some and some indecomposable -module .
2. , where is the translation quiver of finite dimensional indecomposable -modules, consisting of a -family of stable tubes of rank one.
3. is tame if and only if so is .

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 J. A. de la Peña
Instituto de Matemáticas, UNAM.
Circuito Exterior.
Ciudad Universitaria.
México 04510, D. F. México
jap@matem.unam.mx

Recibido: 6 de octubre de 2006
Aceptado: 3 de junio de 2007