We study a finite-dimensional algebra Λ constructed from a Postnikov diagram D in a disk, obtained from the dimer algebra of Baur-King-Marsh by factoring out the ideal generated by the boundary idempotent. Thus, Λ is isomorphic to the stable endomorphism algebra of a cluster tilting module T ∈ CM(B) introduced by Jensen-King-Su in order to categorify the cluster algebra structure of C[Gr k (C n )]. We show that Λ is self-injective if and only if D has a certain rotational symmetry. In this case, Λ is the Jacobian algebra of a self-injective quiver with potential, which implies that its truncated Jacobian algebras in the sense of Herschend-Iyama are 2-representation finite. We study cuts and mutations of such quivers with potential leading to some new 2-representation finite algebras. Proposition 2.3. If (Q, W ) is a quiver with finite potential such that ∂ a W | a ∈ Q 1 is an admissible ideal of CQ, then the canonical map ℘(Q, W ) →℘(Q, W ) is an isomorphism.Proof. Call I = ∂ a W | a ∈ Q 1 ⊆ CQ andÎ = ∂ a W | a ∈ Q 1 ⊆ CQ. Call J andĴ the arrow ideals of CQ and CQ respectively. By assumption we have that there exists N ≫ 0 such that J N ⊆ I and then J N ⊆Î. Observe that we have that CQ = CQ +Ĵ N , and that J N = CQ ∩Ĵ N . There is a commutative diagram J N

Abstract. We investigate how the higher almost split sequences over a tensor product of algebras are related to those over each factor. Herschend and Iyama give in [HI11] a criterion for when the tensor product of an n-representation finite algebra and an m-representation finite algebra is (n + m)-representation finite. In this case we give a complete description of the higher almost split sequences over the tensor product by expressing every higher almost split sequence as the mapping cone of a suitable chain map and using a natural notion of tensor product for chain maps.

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.

No abstract

If A and B are n-and m-representation finite k-algebras, then their tensor product Λ = A ⊗ k B is not in general (n + m)-representation finite. However, we prove that if A and B are acyclic and satisfy the weaker assumption of n-and m-completeness, then Λ is (n + m)-complete. This mirrors the fact that taking higher Auslander algebra does not preserve drepresentation finiteness in general, but it does preserve d-completeness. As a corollary, we get the necessary condition for Λ to be (n + m)-representation finite which was found by Herschend and Iyama by using a certain twisted fractionally Calabi-Yau property.