SciELO - Scientific Electronic Library Online

 
vol.48 número2Lectures on algebras índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados

Articulo

Indicadores

  • No hay articulos citadosCitado por SciELO

Links relacionados

  • En proceso de indezaciónCitado por Google
  • No hay articulos similaresSimilares en SciELO
  • En proceso de indezaciónSimilares en Google

Bookmark


Revista de la Unión Matemática Argentina

versión 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 k be a field and A a finite dimensional (associative with 1 ) k -algebra. By modA we denote the category of finite dimensional left A -modules. In many important situations we may suppose that A is presented as a quiver with relations (Q,I ) (e.g. if k is algebraically closed, then A is Morita equivalent to kQ ∕I ). We recall that if A is presented by (Q, I) , then Q is a finite quiver and I is an admissible ideal of the path algebra kQ , that is,  m 2 J ⊂ I ⊂ J for some m ≥ 2 , where J is the ideal of kQ generated by the arrows of Q , see [6].

It is convenient to consider A = kQ ∕I as a k -linear category with objects Q0 (= vertices of Q ) and morphisms given by linear maps Q (x,y) = eyAex , where ex is the trivial path at x (for x, y ∈ Q0 ). In this categorical approach we do not need to assume that Q is finite (therefore the k -algebra kQ ∕I may not have unity). Ocasionally we write A0 = Q0 if we do not need to explicit the quiver Q .

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

A symmetry of the quiver Q is a permutation of the set of vertices Q0 inducing an automorphism of Q . We denote by Aut (Q ) the group of all symmetries of Q . Those symmetries g ∈ Aut (Q ) inducing a morphism g : kQ → kQ such that g(I) ⊂ I form the group Aut (Q,I ) . In natural way, any g ∈ Aut (Q,I) induces an automorphism of the module category modA and on the Auslander-Reiten quiver Γ A of A (since the action commutes with the Auslander-Reiten translation τA of Γ A ).

In section 1, we present some basic facts about the actions of groups G ⊂ Aut (Q, I) on A = kQ ∕I (orbits, stabilizers, Burnside's lemma) and show that for a representation-finite standard algebra A , we have Aut (Q, I) = Aut Γ A , where Aut Γ A is formed by the symmetris of Γ A commuting with the translation τA . We recall that A is standard if A is representation-finite and for a choice of representatives of the isoclasses of indecomposables, the induced full subcategory of modA (denoted by  ~ indA ∕= ) is equivalent to k(Γ A ) which is the quotient of the path algebra kΓ A by the ideal generated by the meshes  s ∑ (σ α ) ⋅ α i=1 i i of the almost split sequences  (σαi) ⊕s (αi) 0 - → τAX -→ Yi- → X -→ 0 i=1 . We recall that for chark ⁄= 2 , any representation-finite algebra A = kQ ∕I is standard.

In section 2 we consider relations between the structure of Aut (Q,I ) and the Coxeter polynomial of A (recall that, pA(t) = det(tid - φA ) is the Coxeter polynomial associated to the Coxeter matrix φA which is ℤ -invertible in case gℓ dim A < ∞ ).

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

∙ if G acts freely on A (that is, G ⊂ Aut (Q, I) and g(x) = x for a vertex x , implies g = 1 ), then the Galois covering F : A → A ∕G is a G -invariant functor of k -categories.

∙ if B is a G -graded k -category, the smash product B # G is a k -category accepting the free action of G . Moreover,  ~ (B # G)∕G = B .

Galois coverings were introduced by Bongartz and Gabriel [672] 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 G acts freely on A , then A ∕G is a G -graded category such that  ~ (A ∕G ) # G = A .

In section 4, following [2], we introduce functors relating the module categories of A and A ∕G when G acts freely on A . The main results in these notes (section 5) relate the representation types of A and A ∕G (which was the original purpose of the introduction of Galois coverings). Indeed, given a Galois covering F : A → A ∕G = B with B a finite dimensional k -algebra, then B is representation-finite if and only if A is locally representation-finite (that is, for each i ∈ A0 , there are only finitely many indecomposable A -modules X , up to isomorphism, with X (i) ⁄= 0 ). 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 A is said to be tame if for every d ∈ ℕ there are finitely many A - k[t] -bimodules M1, ...,Ms (d) wich are free finitely generated as right k[t] -modules and such that any indecomposable A -module X with dimension d is of the form Mi ⊗k [t] (k[t]∕(t - λ)) for some 1 ≤ i ≤ s(d) and some λ ∈ k . We say that a tame algebra A is domestic (resp. of polynomial growth) if s(d) can be chosen ≤ c (resp.  m ≤ d for some m ) for all d . It is not hard to show that A is tame if A ∕G is tame for a group acting freely on A . 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 [512] showing that for a Galois covering F : A → A∕G , the category A is tame if and only if A ∕G is tame, provided certain restrictions on the group G or on the category A ∕G are satisfied.

We denote by K0(A ) the Grothendieck group of A , freely generated by representatives S ,...,S 1 n of the simple A -modules, where n = n(Q ) is the number of vertices of Q . We denote Pi (resp. Ii ) the projective cover (resp. injective envelope) of Si . With the categorical approach an A -module is a functor X : A → Modk , and a morphism f : X → Y is a natural transformation. In case gℓ dim A < ∞ , the Coxeter matrix φA : K0 (A ) → K0 (A ) is defined by φ ([P ]) = - [I] A i i .

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 A = kQ ∕I be a finite dimensional k -algebra:

Aut A denotes the group of automorphisms of A . By Aut (Q) we denote the group of symmetries of Q and by Aut (Q, I) the group of symmetries of Q fixing I (that is, g ∈ Aut (Q) induces g : kQ → kQ such that g(I) = I ).

Lemma.
Aut (Q, I) is a subgroup of Aut A . □

Each symmetry g ∈ Aut (Q) gives rise to a matrix g ∈ G ℓℤ (n (Q)) sending Si to Sg (i) . This representation γ : Aut (Q ) → G ℓℤ(n(Q )) is called the canonical representation.

1.2. Let G ⊂ Aut (Q, I) be a subgroup. For a given i ∈ Q0 , Gi denotes the orbit of i and subgroup of G , Gi = {g ∈ G : gi = i} denotes the stabilizer of i . Clearly Gi is a subgroup of G .

Lemma.
The mapping Gi → G ∕G i , gi ↦→ gG i is a bijection from the orbit to the set of left cosets G ∕Gi .

 ∑ -1-- n(Q ) = |G | |Gi |

where the sum runs over representatives of the orbits. □

1.3. Let  n(Q) K0(A ) = ℤ be the Grothendieck group of A . Consider

Inv (A ) = {v ∈ K (A ) ⊗ ℚ : vg = v for g ∈ G} G 0 ℤ

the ℚ -space of G -invariant vectors. Then t0(G) the number of orbits of G in Q0 equals dim ℚInvG (Q ) .

Let S1,...,Sm be a set of representatives of the irreducible ℂ -representations of G (here m is the number of conjugacy classes of G ). Let S 1 be the trivial representation.

Consider χβ the character corresponding to Sβ (that is,  * χ β: G → ℂ , g ↦→ trSβ(g) ). The characters 1 = χ1, ...,χm form an orthonormal basis of the class group X (G) , with the scalar product  ∑ ----- (χ, χ′) = 1|G|- χ(g)χ ′(g) g∈G .

Lemma.
[Burnside's lemma].  ∑ t0(G ) = -1- χ γ(g ) |G |g∈G where χ γ: G → ℚ* , is the character of the canonical representation γ .

Proof. Observe that χγ(g) is the number of fixed points of g .

∑ ∑ ∑ ∑ ∑ ∑ ∑ χγ(g) = 1 = 1 = |Gi | = |G | |Gi|-1 = |G|t0(G) g∈G g∈G i∈Qg i g∈Gi i∈Q0 i∈Q0

where Qg = {j ∈ Q0 : gj = j} . □

1.4. Let G be a subgroup of Aut (Q, I) then G acts on ModA as follows:

 g X ∈ ModA and g ∈ G, then X ∈ ModA

such that for  α i -→ j , we have  X(gα ) Xg (i) = X (gi) -→ X (gj) . Similarly, for f ∈ HomA (X, Y ) , we define f g ∈ HomA (Xg, Y g) .

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

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

For (c), let g ∈ G be an element inducing a trivial action on Aut Γ A ; we shall prove g = 1 . Indeed,  g Pi = Pi = Pgi implies gi = i for every vertex i ∈ Q0 .

Let  α i-→s j , s = 1, ...,m be all arrows between i and j , then g establishes a permutation of the α s . Let X s be the 2 dimensional A -module with Xs (αi) = δis: k → k . Since Xs is indecomposable,  g X s = Xs or gαs = αs . Therefore g = 1 .

Proposition.
Let A = kQ ∕I be an algebra satisfying:

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

then Aut (Q, I) = Aut Γ A .

Proof. We already know that Aut (Q, I) is a subgroup of Aut Γ A . Let g ∈ Aut Γ A and consider the induced automorphism ¯g of the mesh category k(Γ A) . Clearly, ¯g restricts to an automorphism of A (considering the full embedding i ↦→ Pi , i ∈ Q0 ). By definition there is an automorphism h ∈ Aut (Q, I) inducing ¯g , and therefore inducing g . □

2. The canonical representation and the Coxeter matrix

2.1. Let A = kQ ∕I and assume that gℓdim A < ∞ . For example, this happens if A is triangular, that is, Q has no oriented cycles.

Recall that  ~ φA : K0(A ) -→ K0 (A) , dim Pi ↦--→ - dim Ii is the Coxeter matrix if A , which is ℤ -invertible. In case A = kQ (i.e. I = 0 ), then (dim X )φ = dim τ X A A for any non-projective indecomposable X .

The characteristic polynomial pA (t) = det (tId - φA ) is called the Coxeter polynomial.

Examples:
  1. Q Dynkin type, then  1 Spec φA ⊂ 𝕊 \ {1,- 1} .
  2. Q extended Dynkin type, then  1 Spec φA ⊂ 𝕊 and 1 is a root of multiplicity 2 .
  3.  3 4 5 6 7 9 10 pA (t) = 1 + t - t - t - t - t - t + t + t is irreducible (over ℤ [t] ).

Proposition [11].
  1. φA is an automorphism of the representation γ .
  2. If Aut (Q, I) is not trivial, then pA(t) is not irreducible.

Proof. (a): It suffices to observe that gφA = φAg for any g ∈ G .

(b): Let R1, ...,Rm be a set of representatives of the irreducible ℚ -representations of G = Aut (Q, I) . Let R1 be the trivial representation. Up to conjugation (with L )

 m L ⊕ r(α) γ = R α α=1
R : G → GL (dim R ) α ℚ α , then n = m∑ r(α)dim R α=1 α .

By Schur's lemma  ⌊ φ1 0 ⌋ L . φ A = ⌈ .. ⌉ 0 φm , where  r(α) r(α) φα : R α → R α is an automorphism. Hence pA (t) = pφ (t)...pφ (t) 1 m . Therefore if p (t) A is irreducible then r(α) = 0 for α ≥ 2 .

If G ⁄= (1 ) , then t0(G ) < n . Moreover, the characters

 * χ α: G → ℚ , g ↦→ tr Rα(g), α = 1,...,m

form an orthonormal basis of X (G ) with scalar product

 1 ∑ ----- (χ,χ ′) = ---- χ(g)χ′(g). |G |g∈G

Then

 ∑ r(α) = (χ ,χ ) = -1-- χ (g)χ-(g)- γ α |G | γ α g∈G

for α = 1 ,  ∑ r(1) = |G1| χ γ(g) = t0(G ) g∈G .

Finally, n = m∑ r(α)dim R = t (G) + ∑m r(α )dim R α=1 α 0 α=2 α , with implies the existence of α ≥ 2 with r(α ) > 0 . Therefore pA (t) is not irreducible. □

2.2. Example: Q :

Consider G = A ⊂ Aut Q 5 and γ: A → G ℓ(6) 5 the canonical representation. It is not hard to calculate the character table of A5 :

 | conjugacy classes |{1} (123) (12)(34) (12345) (13524) |Gxi | | 1 20 15 12 12 -------------------|---------------------------------------- 1 = χ1 | 1 1 1 1 1 α = (1 + √5-)∕2 χ2 | 4 1 0 - 1 - 1 1 √ -- | α2 = (1 - 5)∕2 χ3 | 5 - 1 1 0 0 χ4 | 3 0 - 1 α1 α2 χ | 3 0 - 1 α α 5 | 2 1

where χi corresponds to Si irreducible ℝ -representation of G (with S1 the trivial representation).

 

Then

 ⊕ s γT = Snβ(β), n(1) = t0(G ) = 2 β=1 1 ∑ ------ 1 n(2) = (χ γ,χ2) = --- χγ(g)χ2(g) = --[24 + 60 - 24] = 1 60 g∈G 60

Since dim γ = 6 , γT = S1 ⊕ S1 ⊕ S2 which is also a ℚ -decomposition. Moreover, p (t) = (1 - 3t + t2)(1 + t)4 A is an irreducible factorization.

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

3.1. Let A be a k -category given as A = kQ ∕I . Let G be a subgroup of Aut (Q, I) ⊂ Aut A . We say that G acts freely on A if gi = i for some i ∈ Q0 implies g = 1 .

Lemma.
Let G be a group acting freely on A = kQ ∕I . Then there exists a k -category B = kQ¯∕¯I and a functor F : A → B satisfying:
  1. (G -invariant): F g = F for every g ∈ G
  2. (universal G -invariant): for any functor F ′: A → B′ which is G -invariant, there exists a unique functor F¯: B → B ′ such that F ′ = F¯F
  3. Q¯0 is formed by the G -orbits of vertices in Q0 and for any a = Gi and b = Gj

     ⊕ B (a,b) = A(i,gj) g∈G

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

 ( ) ( ) ⊕ ⊕ ⊕ ′ B(a,b) ⊗ B (b,c) = A (i,gj) ⊗ A(j,hℓ) → ′ A (i,gj) = B (a,c) g∈G h∈G g( ) ∑ (fg) ⊗ (f ′h) ↦--- ---- ---→ (f ′h)gfg gh=g′

Define F : A → B , f ∈ A (i,j) ⊂ ⊕ A (i,gj) = B (a, b) g in the unique possible way (since gj = j implies g = 1 ). □

3.2. If G acts freely on A = kQ ∕I , the functor F : A → B as in (2.1) is called a Galois covering defined by G and B = A ∕G is a Galois quotient of A .

We say that a k -category B is G -graded if for each pair of objects a,b there is a vector space decomposition  ⊕ g B (a,b) = B (a,b) g∈G such that the composition induces linear maps

 g h gh B (a,b) ⊗ B (b,c) → B (a,c)
Lemma.
B = A∕G is a G -graded k -category. □

3.3. For a G -graded k -category B we define the smash product as the k -category B#G with objects B0 × G and for any pair (a,g),(b,h) ∈ B × G 0 , the morphisms are

 g-1h (B#G )((a, g),(b,h )) = B (a,b)

with composition given by:

(B#G )((a,g),(b,h)) ⊗ (B#G )((b,h),(c,t)) → (B#G )((a,g),(c,t)) ∥ ∥ Bg -1h(a,b)⊗ Bh -1t(b,c) ↦---- ---- -→ Bg -1t(a,c)

Theorem
[4]. The category B#G accepts a free action of G such that (B#G )∕G ~= B .

Moreover, if G acts freely on A , then (A∕G )#G ~= A .

Proof. Clearly there is a G -invariant functor F : B#G → B inducing a functor  ¯ F : (B#G )∕G → B which is the identity on objects. Check that F¯ is an isomorphism.

Assume G acts freely on A and let F : A → A ∕G be the induced Galois covering.

Since G acts freely, there is a bijection  f~ (A ∕G )0 × G -→ A0 which commutes with the G -action. Moreover, if a, b ∈ (A ∕G )0 , f (a,1) = i , f (b,1 ) = j ∈ A 0 then

(A∕G#G )((a, g),(b,h )) = A ∕Gg-1h(a,b) = A (i,g -1hj) = A (gi,hj ) = A (f (a,g),f(b,h))

is compatible with the composition. □

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

3.4. Galois correspondence. Let B be a k -category. There is a 1 - 1 correspondence

(A -F→ B Galois covering defined by G ) ↦→ G group grading B , such that for any

two Galois coverings  F A - → B defined by G and  F′ A ′-→ B defined by H , there exist a commutative diagram

 --F-- A B ¯F ′ F A′

if and only if there exists a normal subgroup H ⊴ G ′ , such that G ′∕H = G .

A Galois triple (A,F, G ) of B is a Galois covering F : A → B defined by the (free) action of a group G . There is a universal object in this set of Galois triples. Indeed, let B = kQ ∕I and consider W the set of all walks in Q starting and ending at a fixed vertex b . Let ~ be the equivalence relation induced by the following elementary relations:

  1. α α- 1 ~ ey and α -1α ~ ex for any arrow  α x - → y ;
  2. if ∑s λiwi ∈ I(x,y ) i=1 with λi ∈ k* , such that for any L ⊆∕ {1,...,s} we have ∑ λi ∕∈ I(x,y ) i∈L , then wi ~ wj for i,j ;
  3. if  ′ w ~ w , then  ′′ ′ ′′ ww ~ w w , whenever the products are defined.

Then  ˜ G = W ∕~ has a group structure such that B is  ˜ G -graded. The group ˜ G is called the fundamental group of B .

Proposition
[9]. Let ˜B = B # G˜ and ˜F : ˜B → B be defined by G˜ . The triple ( ˜B, ˜F,G˜) is a universal Galois covering, that is, for any Galois covering F : A → B defined by the action of a group G , there exists a covering ¯ ˜ F :B → A defined by  ˜ H ⊲ G such that G˜∕H = G . □

4. Actions induced on module categories

4.1. Let F : A → B be a Galois covering of k -categories defined by the action of G . We shall denote by

  • MODA the category of left A -modules;
  • ModA those X ∈ MODA with dimkX (i) < ∞ for every i ∈ A0 ;
  • modA those X ∈ MODA with ∑ dimkX (i) < ∞ i∈A0 .

There are naturally defined functors:

F.: MODB → MODA, (Y : B → Modk ) ↦→ (Y ∘ F : A → Modk ),

called the pull-up functor;

 ⊕ F : MOD → MOD ,(X : A → Mod ) ↦→ F X (a) = X (gi) λ A B k λ g∈G

and such that for  ⊕ f = (fg) ∈ B (a,b) = A (i,gj) g∈G , then F λX (f): FλX (a) → F λX (b) , sends (ag ) to ( ) ∑ h X (fh-g1)(ah) h g , called the push-down functor. We observe that Fλ is a left adjoint to F. . Similarly, there is a right adjoint F ρ: MODA → MODB to F. . For modules X ∈ modA , the modules FλX and F ρX coincide.

Recall that G acts on MODA and X ∈ MODA is G -stable if Xg = X for every g ∈ G . The category of G -stable A -modules is denoted by  G MOD A .

4.2. Proposition [72]. Let F : A → B be a Galois covering defined by the action of G . Then the following happens:

  1. The categories  G MOD A and MODB are equivalent.
  2. For any X ∈ MODA and g ∈ G , we have FλXg ~= FλX . Moreover,
     ~ ⊕ g F.F λX -→ g∈GX as A -modules.
  3. Let H be a subgroup of G and  H X ∈ MOD A . Then FλX has a natural structure as kH -module. If H is a finite group and chark ⁄ | |H | , then FλX decomposes as a direct sum of at least |H | factors.

Proof. (a): clear.

(b): Observe that  ⊕ ~ g ⊕ F λX (a) = h∈GX (hi) -→ F λX (a) = h∈GX (hgi) canonically.

Hence  ⊕ ⊕ h F.F λX (i) = F λX (F i) = h∈G X (hi) = h∈GX (i) and correspondingly in morphisms.

(c): Choose a set W of representatives in G of the right cosets G ∕H . Then  ( ) F λX (a) = ⊕ X (i) = kH ⊗k ⊕ X (wi) g∈G w∈W and correspondingly in morphisms.

Hence we get φ : kH → End (F X ) B λ a group homomorphism such that for each idempotent e of kH , φ(e) is idempotent and there is a factorization of FλX .

In case H is a finite group with char k ⁄ | |H | , by Mashké theorem, the group algebra kH is semisimple (with |H | idempotents). The result follows. □

4.3. Assume G acts freely on A and F : A → B = A∕G is the corresponding Galois covering.

Let X ∈ MODA , the stabilizer GX is the subgroup of G formed by those g ∈ G such that Xg ~= X . That is, X ∈ MODH A if H ⊂ G X .

Proposition [7].
  1. If X ∈ ind A and G is torsion free, then G = (1) X .
  2. If X ∈ indA and GX = (1) , then F λX is indecomposable and for any module Y ∈ modA with FλX ≃ FλY , then Y ≃ Xg for some g ∈ G .

Proof. (a): Let g ∈ GX for some X ∈ indA , then g establishes a permutation of supp X (a finite set). Then for some s ∈ ℕ , 1 = gs on supp X . Since G acts freely on A , then gs = 1 . Since G is torsion free, then g = 1 and G = (1) X .

(b): Assume  ′ FλX ≃ Z ⊕ Z , then ⊕ g ′ X = F.F λX ≃ F.Z ⊕ F.Z g∈G . Assume X is a direct summand of  G F.Z ∈ MOD A , then ⊕ g X ⊂ F.Z g∈G and  ′ F.Z = 0 . Therefore FλX is indecomposable.

If FλX ≃ F λY , then Y is indecomposable and Y ≃ Xg for some g ∈ G . □

5. Coverings and the representation type of an algebra

5.1. Examples:

(a) Let B = k⟨x,y ⟩∕ (x2, y3,xy,yx )

Let A = kQ ∕I be the infinite k -category with quiver

PIC

and I generated by all relations of the form  2 3 x ,y ,xy, yx . Then we get a Galois covering F : A → B defined by the action of ℤ × ℤ on A .

The module

is indecomposable.

It is stable under the action of ℤ `→ ℤ × ℤ , n ↦→ (n,n ) .

The universal Galois covering of B is given by B˜= kQ˜∕˜I

where the free group in two generators F2 acts on  ˜ B . The normal subgroup H = ⟨xy - yx ⟩ of F2 acts on ˜B inducing the covering F : A → B .

Observe that for X ∈ indA , we have GX = (1 ) and therefore F X ∈ ind λ B .

(b) [13] Let B be a standard k -algebra of finite representation type. Then the Auslander-Reiten quiver Γ B is finite and equipped with the mesh relations:  n ∑ βi ∘ σ βi = 0 i=1 , for each almost split sequence  (σβi) (βi) 0 - → τBX -→ ⊕ Yi -→ X -→ 0 .

PIC

There is an universal Galois cover  π ˜Γ B - → Γ B of translation quivers defined by the action of a free group G . Moreover, ˜Γ B is the Auslander-Reiten quiver of a k -category ˜ B such that  ˜ k(Γ B) = indB˜ .

We illustrate the situation in the following example:

PIC

A full subcategory  ′ B of B = kQ ∕I is convex in B if  ′ ′ ′ B = kQ ∕I for a quiver  ′ Q which is path closed in Q (i.e. i0 → i1⋅⋅⋅ → is → is+1 with i0,is+1 ∈ Q ′ implies i1,...,is ∈ Q′ ).

Theorem.
Let B be a representation-finite k -algebra. Then
  1. [13] There exists a universal Galois covering ˜ Γ B → Γ B defined by the action of a free group G . Moreover k(˜Γ B ) = ind ˜B for a k -category ˜B . If B is standard, then B˜ → B is a universal Galois covering.
  2. [3] For every finite convex subcategory C of  ˜ B , we have Γ˜C = Γ C and C is representation-directed (i.e. C is representation-finite and Γ C is a preprojective component). □

5.2. We say that the group G acts freely on indecomposable classes of A -modules if  g X ≃ X for X ∈ indA implies g = 1 . Observe that for the algebra B with quiver

 ∙ k ∙ ∙ X : k k ∙ k

the indecomposable representation X has non-trivial stabilizer.
Theorem.
Let F : A → A ∕G = B be a Galois covering defined by a group G
  1. [2] If G acts freely on indecomposable classes of A -modules, then Fλ induces an injection (indA ∕ ~= )∕G `→ indB ∕ ~= and preserves Auslander-Reiten sequences.
  2. [210] If B is finite (hence a k -algebra), then B is representation-finite if and only if (i) G acts freely on  ~ indA ∕ = and (ii) A is locally representation-finite (i.e. for each i ∈ A0 , {X ∈ indA : X (i) ⁄= 0} is finite). In that case, Γ A∕G ≃ Γ B .

Proof. (a): Let X ∈ indA , then we know GX = (1) and F λX ∈ indB . Moreover F λX ≃ FλY implies Y ≃ Xg for some g ∈ G . For an almost split sequence  α β η: 0 - → τAX -→ E -→ X -→ 0 with X ∈ indA we get an exact sequence

F λη: 0 - → FλτAX - → FλE -→ F λX -→ 0

with FλX , FλτAX indecomposable. For f: Y → F λX a non-invertible epi in modB , we get a map

 F.Y | F.f ⊕ Eg = F.F E F.Fλβ F.F X = ⊕ Xg g∈G λ ----- λ g∈G | | π1| |π1 ---------β--------- E X

There is some g ∈ HomA (F.Y, E ) such that βg = π1F.(f) . From the adjunction (F.,F ρ) we get

 ~ ′ HomA (F.Y, E )- → HomB (Y,F ρE ) = HomB (Y,F λE ), g ↦→ g

Then βg′ = f and F η λ is almost split sequence in mod B .

(b): Assume B is representation-finite and Y1,...,Ys are representatives of  ~ indB ∕ = . Let X ∈ indB and suppose g ∈ G is such that Xg ≃ X . We shall prove that g = 1 . Let  F B˜ -→ B be a covering defined by a free group H 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

Then g induces an automorphism ˜g on ˜Γ B such that gπ = π˜g and  ′ ′ ˜gX = X for some  ′ ˜ X ∈ Γ B with  ′ πX = X . We get an automorphism  ′ g of ˜ B acting freely (since  ′ g = Fg acts freely on B ).

Let C be the full subcategory of B˜ formed by supp X . It is easy to see that C is convex in ˜B and therefore Γ C is a preprojective component. Since g′X ′ = X ′ , then we get h ∈ Aut C with  ′ h ′ (X ) ≃ X . Since h commutes with τC , there is a projective C -module Px with  h P x = Px , which is a contradiction unless h = 1 .

We check that A is locally representation-finite: let i ∈ A0 and let (X α)α∈Δ be all indecomposable A -modules which are direct summands of F.Y j for some 1 ≤ j ≤ s and X (i) ⁄= 0 α . This set is finite [indeed, ⊕ X α = F.Yj Δ ′ , then ⊕ ⊕ (Iℓ) FλX α = F λF.Yj = Yℓ and X α ∈ indA , for all α . Since dimkF.Yj (i) = dimkYj (F i) < ∞ , the set is finite].

Finally, let Y ∈ indB . Then FλF.Y = ⊕F λX α and

HomA (F.Y,F.Y ) ≃ HomB (Y,F ρF.Y ) = HomB (Y,F λF.Y ) 1F.Y ↦---- --- ---- ---- -→ e: Y ∈| F λF.Y

hence Y ≃ F λXα for some α . Therefore Γ B = Γ A ∕G . □

5.3. Let F : A → A ∕G be a Galois covering with group G . In case A ∕G is a representation-finite k -algebra, we have seen that F λ covers all indecomposable modules. The situation is different for representation-infinite algebras. We shall briefly discuss the new occuring phenomena. By ind1A ∕G we denote the full subcategory of indA ∕G formed by all objects isomorphic to FλM for some M ∈ indA (these modules are called A ∕G -modules of the first kind). The remaining indecomposable modules (called of the second kind) form the category ind2A ∕G .

If  H M ∈ mod A is such that supp M is not finite but supp M ∕H is finite, then M is called a weakly G -periodic module.

Lemma [5].
Let F : A → A ∕G be a Galois covering such that G acts freely on  ~ indA∕ = . For X ∈ indA ∕G the following conditions hold:
  1. X ∈ ind1A ∕G if and only if  ~ ⊕ F.X = Zi i∈I , where all Zi ∈ modA .
  2. X ∈ ind2A ∕G if and only if  ⊕ F.X ~= Yi i∈I , where each Yi is weakly G -periodic.

Proof. (1): If X ~= F λM for some M ∈ modA , then  ⊕ F.X ~= M g g∈G . Assume now that Z ∈ indA is a direct summand of F.X . Let j ∈ HomA (Z, F.X ) and p ∈ HomA (F.X, Z ) be such that pj = 1Z . We get morphisms jg: Zg → (F.X )g ~= F.X and pg : (G.X )g → Zg which yield a pair of maps j′: ⊕ Zg → F.X g∈G and p′: F.X → ⊕ Zg g∈G . Since Zg ⁄~= Z for g ⁄= h , then the endomorphism p′ ∘ j′ of ⊕ Zg g∈G is invertible.

Since F. induces an equivalence of categories MODA ∕G ~= MODG A , then F λZ is a direct summand of X . Thus  ~ X = FλZ .

(2): Let X ∈ mod2A ∕G . By the proof of (1),  ⊕ F.X = Yi i∈I where Yi ∈ IndA \ indA . Then we have to show that an indecomposable direct summand Y of F.X is weakly G -periodic. Indeed, let U be a set of representatives of the cosets of G with respect to GY . Then  ⊕ Y g g∈U is a direct summand of F.X . Since supp Y ⊂ supp X is contained in a finite number of G -orbits, then (supp Y )∕GY is finite. The converse follows from (1). □

5.4. A category A = kQ ∕I is called locally support finite if for each x ∈ Q 0 , the full subcategory Ax of A , consisting of the vertices of all supp M with M ∈ indA and M (x) ⁄= 0 , is finite.

Proposition.
Let F : A → A ∕G be a Galois covering. Then
  1. If G acts freely on  ~ IndA ∕ = , then F λ: modA → modA ∕G induces a bijection  ~ ~ F λ: (indA ∕ =)∕G → (indA ∕G∕) = .
  2. If A is locally support finite, then IndA = indA . In particular if G acts freely on indA ∕ ~= , then F λ induces a bijection between (ind ∕ ~= )∕G A and (ind ∕G )∕ ~= A . Moreover, in this case A ∕G is tame (resp. domestic, poynomial growth) if and only if so is A . For the Auslander-Reiten quivers we have Γ A∕G = Γ A∕G .

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

(b): Let Y ∈ IndA and consider x ∈ Q0 with Y (x) ⁄= 0 . Let ˆ Ax be the (finite) full subcategory of A consisting in the objects y ∈ Q0 such that A (y,z) ⁄= 0 or A (z,y) ⁄= 0 for some z ∈ Ax . Let Z be an indecomposable direct summand of the restriction Y |Aˆx such that Z (x) ⁄= 0 . Hence supp Z ⊂ A x and it is easy to check that Z is a direct summand of Y in modA .

Thus Y = Z ∈ indA . The last claim is easy to prove. □

Example:
The category Aα,β given by the quiver with relations

------ -ρ-1- --ρ0- --ρ1- ----- ------ ) a -1 a0 a1 a2 σi+1σi = α νi+1 γi||| γ-1 γ0 γ1 ν ρ = β σ ν ||} i+1 i i+1 i γ ν = ρ ρ || ν-1 ν0 ν1 i+1 i i+1 i ||| ------ ----- ----- ----- ----- ------ ρi+1γi = γi+1σi ) b-1 σ-1 b0 σ0 b1 σ1 b2

with (α, β) ⁄= (1,1) , is locally support finite (why?). Moreover the group ℤ generated by the action (ai ↦→ ai+1,bi ↦→ bi+1) acts freely on A α,β and on indA α,β∕ ~= . Hence the Galois covering F : A α,β → ¯Aα,β∕ℤ yields a bijection Fλ : (indA ∕ ~= )∕ℤ → (indA¯ )∕ ~= α,β α,β . The algebra ¯Aα,β is given by the quiver with relations

 ) σ2 = α γν | --- --ν--- --- ρν = β νσ |} ρ---a -----b ---σ 2 γ νγ = ρ ||) γ ρ = σ γ

Since Aα,β is tame (resp. polynomial growth for α β ⁄= 1 ), so is A¯ α,β .

5.5. Given a natural number p ≥ 0 , a group G is said to be p -residually finite if for each finite subset S ⊂ G \ {1} there is a normal subgroup of finite index H ⊲ G such that H ∩ S = ∅ and p ⁄ | (G ∕H ) . For example, free groups are p -residually finite.

Theorem [12].
Let F : ( ˜Q, ˜I) → (Q, I) be a Galois covering given by the action of a p -residually finite group G , where p = chark . Assume that A˜ = kQ˜∕˜I is locally support finite. Then A˜ is tame if and only if A = kQ ∕I is tame.

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

(1) Let Fλ : mod A˜→ modA be the push-down functor. Consider a sequence (Γ n)n∈ℕ of finite full subcategories of ˜A such that ⋃ Γ = A˜ n n and if ˜A(x,Γ ) ⁄= 0 n or A˜(Γ ,x ) ⁄= 0 n , then x ∈ Γ n+1 . The restriciton functor ɛ.n: mod ˜A → mod Γ n has a left adjoint ɛn : mod Γ n → mod ˜ λ A such that ɛ.nɛn = idmod λ Γn . Therefore, the functor

 n Fn = F λɛλ: mod Γ n → modA

is right exact, n ∈ ℕ .

(2) Let Y ∈ indA , we shall prove that Y is a direct summand of F λF.Y . Indeed, since ˜ A is locally support-finite, by (5.3), the pull-up F.Y decomposes as  ⊕ F.Y ≃ Xi i∈L , where Xi ∈ ind ˜A . Thus  ⊕ F λF.Y ≃ i∈L FλXi . We proceed in two steps.

(2.1) There exists a normal subgroup H of G with finite index not divisible by p , such that the Galois covering ¯F :A˜ → A˜∕H induces a bijection F¯λ: (ind ˜∕ ≃ )∕H → (ind ˜∕H )∕ ≃ A A .

Since A is finite, the set of vertices of ˜ Q is a disjoint union of a finite number of G -orbits Gxi , i = 1,...,s . With the notation introduced above, we consider the full subcategory Ri = ˜Ax , of A˜ . Let S be the set of all elements g ∈ G ∕{1 } such that  ˆˆ ˆˆ g(Ri ) ∩ Ri ⁄= ∅ , for some i ∈ {1,...,s} . Since G acts freely on ˜ Q , then S is finite and there is a normal subgroup H of G such that G ∕H is finite of order not divisible by p and H ∩ S = ∅ . Hence  ˆˆ ˆˆ h (Ri ) ∩ Ri = ∅ , i = 1, ...,s . By (5.1), the induced Galois covering F¯: ˜A → ˜A ∕H yields an injection ¯Fλ(indA˜∕ ≃)∕H → (ind ˜A∕H )∕ ≃ . We show that this map is also surjective. Let M ∈ ind˜A∕H and b = F¯(xj) a vertex of Q˜∕H such that M (b) ⁄= 0 . The restriction  ′ ¯ ˆ ˆ F = F| :Rj → F (Rj) is a bijection and there is an indecomposable Rˆj -module  ′ Y = F N with Y (xj ) ⁄= 0 and such that N is an indecomposable direct summand of the restriction M ′ of M to F (ˆRj) . We conclude that Y ∈ ind ˜ A and N ∈ ind ˜∕H A , showing that M = N = F Y λ .

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

 ------F-- ˜A| A = A˜∕G | F | F′ ˜ ˜ A = A∕H

Then F λ = Fλ′F¯λ and F.= F¯.F. ′ . For an indecomposable Y ∈ indA , we get

 ′ ¯ ¯ ′ FλF.Y = F λ(F λF .)F. Y

Consider the indecomposable decomposition  ′ ⊕t F.Y = Yi i=1 with  ¯ ¯ F λF.Yi ≃ Yi for i = 1,...,t . Then F F.Y ≃ F ′F.′Y λ λ and the result follows from (4.2). In particular, Y is a direct summand of F λXj for some j ∈ L .

(3) There is some n ∈ ℕ and a function f : ℕ → ℕ such that

Fn : mod Γ n → modA

has the property that for every Y ∈ indA with dimkY = d , there exists some X ∈ indΓ n with dimkX ≤ f(d) and such that Y is a direct summand of F.X .

Indeed, the set of vertices Q˜0 is the disjoint union ⋃s Gxi i=1 , then there is some n ∈ ℕ such that Γ n contains all the categories Rˆi for Ri = A˜xi , i = 1,...,s . By (2), Y is a direct summand of FλZ for some Z ∈ ind ˜A . For some g ∈ G , supp Zg ⊂ Γ n and X = ɛ.nZg ∈ ind Γ n satisfies that Y is a direct summand of FnX . Clearly, f(d) = |Γ |d n (where |Γ | n denotes the number of objects of Γ n ) satisfies the desired property.

(4) Assume that A˜ is tame, we show that A is tame. Indeed, let n ∈ ℕ and f : ℕ → ℕ be as in (3). Since Fn is right exact (1), there is a A - F n -bimodule N such that F = N ⊗ (- ) n Γ n . Let z ∈ ℕQ0 .

Since Γ n is tame, there is a family M1, ...,Mt of Γ n - k[T ] -bimodules which are finitely generated free as k[T] -right modules and such that any indecomposable Γ n -module X with dimkX ≤ f(d) is isomorphic to Mi ⊗k[T] S for some 1 ≤ i ≤ t and S a simple k[T] -module. By (3), for every Y ∈ ind A with dim Y = d k , there exists some 1 ≤ i ≤ t and S a simple k [T ] -module such that Y is a direct summand of (N ⊗ Γ n Mi ) ⊗k[T] S . It is easy to see that this (aparently) weaker property is equivalent to the tameness of A .

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

 n Hn = ɛ. F.: modA → mod Γ n.

Let Y ∈ ind Γ n . Thus X = F λɛnλY ∈ modA and

 ( ) ( ) n ⊕ n g ~ ⊕ n n g ( ⊕ n n g) HnX = ɛ (ɛ λY) = ɛ. (ɛλY) = Y ⊕ ɛ. (ɛλY ) , g∈G g∈S g∈S\{1}

where S is the finite set of g ∈ G such that supp (ɛnY )g ∩ Γ n ⁄= ∅ λ . As in (3), (4) we get that Γ n is tame. □

5.6. We consider several examples:

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

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

(b) Consider example (5.4). For (α,β) ⁄= (1,1) , the category A α,β is tame and locally support finite. Hence the algebra A¯α,β is tame.

(c) [8] Consider the Galois covering

F : A = A (2)→ ¯A = A(2)∕ ℤ 1,1 1,1 2

given in (2.2.c) and assume that char k = 2 . We know that A is tame. We show that A¯ is a wild algebra.

Set x0 = α0 + β0 , y0 = β0 , x1 = α1 + β1 , y1 = β1 . Then A¯ is isomorphic to the algebra A ′ given by the quiver with relations.

 x x } A ′: ∙----0------∙------1---- x1x0 = 0 y0 y1 y1x0 = x1y0

The universal covering A˜′ constructed as in (2.3) admits a full convex subcategory Δ as follows

PIC

Since Δ is wild (observe that the Tits form qΔ takes value - 1 in the indicated vector), then A˜′ is wild. Then A is also wild.

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

 2 qA¯(a,b,c) = (a - b + c) .

5.7. We consider briefly the situation of coverings F : A → A∕G where A is not necessarily locally support-finite.

Let F : A → A ∕G be a Galois covering. A line in A is a full convex subcategory isomorphic to the path category of a linear quiver (of type 𝔸n , 𝔸 ∞ or 𝔸 ∞∞ ). A line L is G -periodic if its stabilizer GL = {g ∈ G : gL = L } is non-trivial (then L is of type 𝔸 ∞∞ ).

Let L be a G -periodic line in A , we construct an indecomposable weakly G -periodic A -module B L by setting B (x) = k L for x ∈ L and BL (x) = 0 for x ∕∈ L and BL (α) = id for any arrow α in L . Then GBL ≃ GL is isomorhic to ℤ . Let Wx be a set of representatives of the GBL -orbits in Gx , for any object x in A . Then FλBL is a A ∕G - k[T, T -1] -bimodule such that for each x ∈ A , F λBL (x) is a free k[T, T -1] -module of rank  ∑ dimkBL (y) y∈Wx . We consider the functor

φL = FλBL ⊗k[T,T- 1] (- ): modk [T,T-1] → modA ∕G.

Let L be the set of all lines in A and L0 be the set of representatives of the G -orbits in L .
Theorem [5].
Let F : A → A ∕G be a Galois covering such that G acts freely on (indA )∕ ≃ . Assume that for any weakly G -periodic A -module X , supp X is a line. Then
  1. Every module in ind2A ∕G is of the form φL (V ) for some L ∈ L0 and some indecomposable k [T, T- 1] -module V .
  2. Γ = Γ ∕G ∨ ∨ Γ -1 A∕G A L∈L0 k[T,T ] , where Γ k[T,T-1] is the translation quiver of finite dimensional indecomposable k[T,T -1] -modules, consisting of a k * -family of stable tubes of rank one.
  3. A is tame if and only if so is A ∕G . □

References

[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