Skew group algebras of Jacobian algebras

Abstract

For a quiver with potential (Q, W ) with an action of a finite cyclic group G, we study the skew group algebra ΛG of the Jacobian algebra Λ = P(Q, W ). By a result of Reiten and Riedtmann, the quiver Q G of a basic algebra η(ΛG)η Morita equivalent to ΛG is known. Under some assumptions on the action of G, we explicitly construct a potential W G on Q G such that η(ΛG)η ∼ = P(Q G , W G ). The original quiver with potential can then be recovered by the skew group algebra construction with a natural action of the dual group of G. If Λ is self-injective, then ΛG is as well, and we investigate this case. Motivated by Herschend and Iyama's characterisation of 2-representation finite algebras, we study how cuts on (Q, W ) behave with respect to our construction.2010 Mathematics Subject Classification. 16G20, 16S35.

Orbifold diagrams

“…The fact that Q G = Q O is clear, as the two constructions both agree with the general construction presented in [RR85,Section 2]. This is also illustrated in Examples 8.1 and 8.3 of [GP19].…”

“…Moreover, the G-orbits of the boundary arrows for A(sym d (O)) correspond exactly (in the sense of [GP19, Notation 3.13]) to the boundary arrows of A(O). It follows that even in our case, it is enough to show that the QP (Q G , W G ) of [GP19] is equal to (Q O , W O ), if we make appropriate choices. The fact that Q G = Q O is clear, as the two constructions both agree with the general construction presented in [RR85,Section 2].…”

“…We will use the construction of the quiver with potential of a skew group algebra given in [GP19]. For convenience of the reader, we summarize here some points about this construction.…”

“…Proof. We will use Theorem 3.20 of [GP19], with Λ being A(sym d (O)). We remark this theorem still works if we replace the usual definition of the Jacobian ideal by any ideal generated by cyclic derivatives with respect to arrows (see Definition 4.3) provided that these arrows are closed under the G-action.…”

On the Morita Reduced Versions of Skew Group Algebras of Path Algebras

“…The problem here is that describing that linear combination explicitly is not easy. Up to now, the only fairly general explicit description of W G is due to Giovannini and Pasquali ( [5]) when G is cyclic, the stabiliser of each vertex is either trivial or the whole group, and every cycle appearing in W goes through fixed vertices only (or, through vertices with trivial stabilisers only, respectively) as soon as it goes through two of them.…”