Quivers with potentials and actions of finite abelian groups

Abstract

Let G be a finite abelian group acting on a path algebra kQ by permuting the vertices and preserving the arrowspans. Let W be a potential on the quiver Q which is fixed by the action. We study the skew group dg algebra Γ Q,W G of the Ginzburg dg algebra of (Q, W ). It is known that Γ Q,W G is Morita equivalent to another Ginzburg dg algebra Γ Q G ,W G , whose quiver Q G was constructed by Demonet. In this article we give an explicit construction of the potential W G as a linear combination of cycles in Q G , and write the Morita equivalence explicitly. As a corollary, we obtain functors between the cluster categories corresponding to the two quivers with potentials.

Gluing of derived equivalences of dg categories

“…which is also equivalent to the skew group dg algebra Γ(Q, W ) * G, and is calculated as Γ(Q G , W G ) up to Morita equivaleces in the case that G is a finite group in [37] (see also [25] for the finite abelian case). Therefore in this case note that Gr(X Q,W ) is calculated as Γ(Q G , W G ) up to Morita equivalences.…”

“…The orbit category (the skew group algebra) J(Q, W )/G was computed up to Morita equivalence as the form J(Q G , W G ) for some quiver with potentials (Q G , W G ) by Paquette-Schiffler in [36] in the case that G is a finite subgroup of the automorphisms group of J(Q G , W G ) acting freely on vertices. On the other hand, the orbit dg category (the skew group dg algebra) Γ(Q, W )/G was computed up to Morita equivalence as the form Γ(Q G , W G ) for some quiver with potentials (Q G , W G ) by Le Meur in [37] in the case that G is a finite group (see also Amiot-Plamondon [1] for the case that G = Z/2Z, Giovannini and Pasquali [24] for the cyclic case, and Giovannini, Pasquali and Plamondon [25] for the finite abelian case). We remark that for both J(Q, W ) and Γ(Q, W ), the quiver Q G can be computed by using a result by Demonet in [20] on the computation of the skew group algebra of the path algebra of a quiver with an action of a finite group, and in the arbitrary group case, Q G can be computed from a non-admissible presentation given in [9] by making it as an admissible presentation.…”