Group Actions on Algebras and Module Categories

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, 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 ΓA of A (since the action commutes with the Auslander-Reiten translation τA of ΓA). 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 ΓA, where AutΓA is formed by the symmetris of ΓA 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(ΓA) which is the quotient of the path algebra kΓA by the ideal generated by the meshes s