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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