Servicios Personalizados
Revista
Articulo
Indicadores
-
Citado por SciELO
Links relacionados
-
Similares en SciELO
Compartir
Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.48 n.2 Bahía Blanca jul./dic. 2007
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
.
1.3. Let be the Grothendieck group of
. Consider
![ℚ](/img/revistas/ruma/v48n2/2a02189x.png)
![G](/img/revistas/ruma/v48n2/2a02190x.png)
![t0(G)](/img/revistas/ruma/v48n2/2a02191x.png)
![G](/img/revistas/ruma/v48n2/2a02192x.png)
![Q0](/img/revistas/ruma/v48n2/2a02193x.png)
![dim ℚInvG (Q )](/img/revistas/ruma/v48n2/2a02194x.png)
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
.
![Qg = {j ∈ Q0 : gj = j}](/img/revistas/ruma/v48n2/2a02214x.png)
1.4. Let be a subgroup of
then
acts on
as follows:
![α i -→ j](/img/revistas/ruma/v48n2/2a02220x.png)
![X(gα ) Xg (i) = X (gi) -→ X (gj)](/img/revistas/ruma/v48n2/2a02221x.png)
![f ∈ HomA (X, Y )](/img/revistas/ruma/v48n2/2a02222x.png)
![f g ∈ HomA (Xg, Y g)](/img/revistas/ruma/v48n2/2a02223x.png)
Clearly, this action preserves indecomposable modules and induces an action of of the Auslander-Reiten quiver
, satisfying:
- the action preserves projective, injective and simple modules;
- the action preserves Auslander-Reiten sequences (in particular,
);
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:
-
Dynkin type, then
.
extended Dynkin type, then
and
is a root of multiplicity
.
is irreducible (over
).
- Proposition [11].
-
is an automorphism of the representation
.
- 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
)
![m L ⊕ r(α) γ = R α α=1](/img/revistas/ruma/v48n2/2a02301x.png)
![R : G → GL (dim R ) α ℚ α](/img/revistas/ruma/v48n2/2a02302x.png)
![n = m∑ r(α)dim R α=1 α](/img/revistas/ruma/v48n2/2a02303x.png)
By Schur's lemma , where
is an automorphism. Hence
. Therefore if
is irreducible then
for
.
If , then
. Moreover, the characters
![X (G )](/img/revistas/ruma/v48n2/2a02313x.png)
![α = 1](/img/revistas/ruma/v48n2/2a02316x.png)
![∑ r(1) = |G1| χ γ(g) = t0(G ) g∈G](/img/revistas/ruma/v48n2/2a02317x.png)
Finally, , with implies the existence of
with
. Therefore
is not irreducible. □
Consider and
the canonical representation. It is not hard to calculate the character table of
:
![χi](/img/revistas/ruma/v48n2/2a02327x.png)
![Si](/img/revistas/ruma/v48n2/2a02328x.png)
![ℝ](/img/revistas/ruma/v48n2/2a02329x.png)
![G](/img/revistas/ruma/v48n2/2a02330x.png)
![S1](/img/revistas/ruma/v48n2/2a02331x.png)
Then
![dim γ = 6](/img/revistas/ruma/v48n2/2a02333x.png)
![γT = S1 ⊕ S1 ⊕ S2](/img/revistas/ruma/v48n2/2a02334x.png)
![ℚ](/img/revistas/ruma/v48n2/2a02335x.png)
![p (t) = (1 - 3t + t2)(1 + t)4 A](/img/revistas/ruma/v48n2/2a02336x.png)
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:
- (
-invariant):
for every
- (universal
-invariant): for any functor
which is
-invariant, there exists a unique functor
such that
is formed by the
-orbits of vertices in
and for any
and
- (
Proof. Let be defined as in (c) with composition maps
![F : A → B](/img/revistas/ruma/v48n2/2a02368x.png)
![f ∈ A (i,j) ⊂ ⊕ A (i,gj) = B (a, b) g](/img/revistas/ruma/v48n2/2a02369x.png)
![gj = j](/img/revistas/ruma/v48n2/2a02370x.png)
![g = 1](/img/revistas/ruma/v48n2/2a02371x.png)
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
![g h gh B (a,b) ⊗ B (b,c) → B (a,c)](/img/revistas/ruma/v48n2/2a02383x.png)
- 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
- 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
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
![H ⊴ G ′](/img/revistas/ruma/v48n2/2a02430x.png)
![G ′∕H = G](/img/revistas/ruma/v48n2/2a02431x.png)
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:
and
for any arrow
;
- if
with
, such that for any
we have
, then
for
;
- 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:
![⊕ f = (fg) ∈ B (a,b) = A (i,gj) g∈G](/img/revistas/ruma/v48n2/2a02481x.png)
![F λX (f): FλX (a) → F λX (b)](/img/revistas/ruma/v48n2/2a02482x.png)
![(ag )](/img/revistas/ruma/v48n2/2a02483x.png)
![( ) ∑ h X (fh-g1)(ah) h g](/img/revistas/ruma/v48n2/2a02484x.png)
![Fλ](/img/revistas/ruma/v48n2/2a02485x.png)
![F.](/img/revistas/ruma/v48n2/2a02486x.png)
![F ρ: MODA → MODB](/img/revistas/ruma/v48n2/2a02487x.png)
![F.](/img/revistas/ruma/v48n2/2a02488x.png)
![X ∈ modA](/img/revistas/ruma/v48n2/2a02489x.png)
![FλX](/img/revistas/ruma/v48n2/2a02490x.png)
![F ρX](/img/revistas/ruma/v48n2/2a02491x.png)
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:
- The categories
and
are equivalent.
- For any
and
, we have
. Moreover,
as
-modules.
- 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].
-
- If
and
is torsion free, then
.
- If
and
, then
is indecomposable and for any module
with
, then
for some
.
- If
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
(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
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
- [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.
- [3] For every finite convex subcategory
of
, we have
and
is representation-directed (i.e.
is representation-finite and
is a preprojective component).
- [13] There exists a universal Galois covering
5.2. We say that the group acts freely on indecomposable classes of
-modules if
for
implies
. Observe that for the algebra
with quiver
![X](/img/revistas/ruma/v48n2/2a02647x.png)
- Theorem.
- Let
be a Galois covering defined by a group
- [2] If
acts freely on indecomposable classes of
-modules, then
induces an injection (
and preserves Auslander-Reiten sequences.
- [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,
.
- [2] If
Proof. (a): Let , then we know
and
. Moreover
implies
for some
. For an almost split sequence
with
we get an exact sequence
![FλX](/img/revistas/ruma/v48n2/2a02672x.png)
![FλτAX](/img/revistas/ruma/v48n2/2a02673x.png)
![f: Y → F λX](/img/revistas/ruma/v48n2/2a02674x.png)
![modB](/img/revistas/ruma/v48n2/2a02675x.png)
![g ∈ HomA (F.Y, E )](/img/revistas/ruma/v48n2/2a02677x.png)
![βg = π1F.(f)](/img/revistas/ruma/v48n2/2a02678x.png)
![(F.,F ρ)](/img/revistas/ruma/v48n2/2a02679x.png)
![βg′ = f](/img/revistas/ruma/v48n2/2a02681x.png)
![F η λ](/img/revistas/ruma/v48n2/2a02682x.png)
![mod B](/img/revistas/ruma/v48n2/2a02683x.png)
(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:
![B˜ `— - → ind = k(˜Γ ) ˜Γ - -˜g → ˜Γ || ||B˜ B ||B B|| | |k(π) π| |π F |↓ |↓ |↓ |↓ g B `— - → indB = k(Γ B) Γ B - - → Γ B](/img/revistas/ruma/v48n2/2a02693x.png)
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
![Y ≃ F λXα](/img/revistas/ruma/v48n2/2a02735x.png)
![α](/img/revistas/ruma/v48n2/2a02736x.png)
![Γ B = Γ A ∕G](/img/revistas/ruma/v48n2/2a02737x.png)
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:
if and only if
, where all
.
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
- If
acts freely on
, then
induces a bijection
.
- 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
.
- If
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
, 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
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
![n ∈ ℕ](/img/revistas/ruma/v48n2/2a02883x.png)
(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
![F λ = Fλ′F¯λ](/img/revistas/ruma/v48n2/2a02938x.png)
![F.= F¯.F. ′](/img/revistas/ruma/v48n2/2a02939x.png)
![Y ∈ indA](/img/revistas/ruma/v48n2/2a02940x.png)
![′ ⊕t F.Y = Yi i=1](/img/revistas/ruma/v48n2/2a02942x.png)
![¯ ¯ F λF.Yi ≃ Yi](/img/revistas/ruma/v48n2/2a02943x.png)
![i = 1,...,t](/img/revistas/ruma/v48n2/2a02944x.png)
![F F.Y ≃ F ′F.′Y λ λ](/img/revistas/ruma/v48n2/2a02945x.png)
![Y](/img/revistas/ruma/v48n2/2a02946x.png)
![F λXj](/img/revistas/ruma/v48n2/2a02947x.png)
![j ∈ L](/img/revistas/ruma/v48n2/2a02948x.png)
(3) There is some and a function
such that
![Y ∈ indA](/img/revistas/ruma/v48n2/2a02952x.png)
![dimkY = d](/img/revistas/ruma/v48n2/2a02953x.png)
![X ∈ indΓ n](/img/revistas/ruma/v48n2/2a02954x.png)
![dimkX ≤ f(d)](/img/revistas/ruma/v48n2/2a02955x.png)
![Y](/img/revistas/ruma/v48n2/2a02956x.png)
![F.X](/img/revistas/ruma/v48n2/2a02957x.png)
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
![Y ∈ ind Γ n](/img/revistas/ruma/v48n2/2a021007x.png)
![X = F λɛnλY ∈ modA](/img/revistas/ruma/v48n2/2a021008x.png)
![S](/img/revistas/ruma/v48n2/2a021010x.png)
![g ∈ G](/img/revistas/ruma/v48n2/2a021011x.png)
![supp (ɛnY )g ∩ Γ n ⁄= ∅ λ](/img/revistas/ruma/v48n2/2a021012x.png)
![Γ n](/img/revistas/ruma/v48n2/2a021013x.png)
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
![char k = 2](/img/revistas/ruma/v48n2/2a021029x.png)
![A](/img/revistas/ruma/v48n2/2a021030x.png)
![A¯](/img/revistas/ruma/v48n2/2a021031x.png)
Set ,
,
,
. Then
is isomorphic to the algebra
given by the quiver with relations.
![A˜′](/img/revistas/ruma/v48n2/2a021039x.png)
![Δ](/img/revistas/ruma/v48n2/2a021040x.png)
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
![L](/img/revistas/ruma/v48n2/2a021089x.png)
![A](/img/revistas/ruma/v48n2/2a021090x.png)
![L0](/img/revistas/ruma/v48n2/2a021091x.png)
![G](/img/revistas/ruma/v48n2/2a021092x.png)
![L](/img/revistas/ruma/v48n2/2a021093x.png)
- Theorem [5].
- Let
be a Galois covering such that
acts freely on
. Assume that for any weakly
-periodic
-module
,
is a line. Then
- Every module in
is of the form
for some
and some indecomposable
-module
.
, where
is the translation quiver of finite dimensional indecomposable
-modules, consisting of a
-family of stable tubes of rank one.
is tame if and only if so is
.
- Every module in
[1] M. Beattie. A generalization of the smash product of a graded ring. J. Pure Appl. Algebra 52 (1988) 219-229. [ Links ]
[2] K. Bongartz and P. Gabriel. Covering spaces in Representation Theory. Invent. Math. 65 (3) (1982) 331-378. [ Links ]
[3] O. Butscher and P. Gabriel. The standard form of a representation-finite algebra. Bull. Soc. Math. France 111 (1983) 21-40. [ Links ]
[4] C. Cibils and E. Marcos. Skew categories. Galois coverings and smash product of a category over a ring. Preprint (2004). [ Links ]
[5] P. Dowbor and A. Skowronski. Galois coverings of representation infinite algebras. Comment. Math. Helv. 62 (1987) 311-337. [ Links ]
[6] P. Gabriel. Auslander-Reiten sequences and representation-finite algebras. In 'Representation of Algebras', Springer LNM 831 (1975) 1-71. [ Links ]
[7] P. Gabriel. The universal cover of a representation-finite algebra. In 'Representation of Algebras'. Springer LNM 903 (1981) 68-105. [ Links ]
[8] Ch. Geiss and J. A. de la Peña. An interesting family of algebras. Archiv. Math. 60 (1993) 25-35. [ Links ]
[9] R. Martínez-Villa and J. A. de la Peña. The universal cover of a quiver with relations. J. Pure and Appl. Algebra 30 (3) (1983) 277-292. [ Links ]
[10] R. Martínez-Villa and J. A. de la Peña. Automorphisms of representation-finite algebras. Invent. Math. 72 (1983) 359-362. [ Links ]
[11] J. A. de la Peña. Symmetries of quivers and the Coxeter transformation. J. Algebra 169 (1994) 200-215. [ Links ]
[12] J. A. de la Peña. Tame algebras: some fundamental notions. Ergänzungsreihe 95-010 (1995) Universität Bielefeld. [ Links ]
[13] Ch. Riedtmann. Darstellungsköcher, Überlagerungen und zurück. Comment. Math. Helv. 55 (1980) 199-224. [ Links ]
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