Constructions in higher-dimensional Auslander-Reiten theory

Abstract: This thesis consists of an introduction and five research articles about representation theory of algebras.Papers I and II focus on the tensor product of algebras from the point of view of higher-dimensional Auslander-Reiten theory. In Paper I we consider the tensor product Λ of two algebras which are n- respectively m-representation finite. In the case when Λ itself is (n+m)-representation finite, we construct its (n+m)-almost split sequences explicitly in function of the n- and m-almost split sequences of the factors. In Paper II we use the constructions of Paper I to prove the following result: the tensor product of an n- and an m-complete acyclic algebras (in the sense of Iyama) is (n+m)-complete and acyclic.Papers III and IV deal with the combinatorics of Postnikov diagrams, or equivalently of the Grassmannian cluster category. This is motivated by 2-dimensional Auslander-Reiten theory: we are interested in constructing self-injective Jacobian algebras as they are the 3-preprojective algebras of 2-representation finite algebras. In Paper III we investigate when the stable Jacobian algebra associated to a (k,n)-Postnikov diagram is self-injective. We prove that this happens if and only if the Postnikov diagram is invariant under rotation by 2πk ⁄ n. In Paper IV (joint with Thörnblad and Zimmermann) we determine a necessary and sufficient condition on (k,n) for such a symmetric Postnikov diagram to exist, namely k ≡ -1, 0 or 1 modulo n ⁄ GCD(k,n). As a corollary, we prove that there exist self-injective planar quivers with potential with Nakayama automorphism of any prescribed order, answering a question by Herschend and Iyama.Paper V (joint with Giovannini) is about skew group algebras. Let G be a finite group acting on a quiver with potential (Q, W), such that certain assumptions hold. We construct a quiver with potential (QG, WG) such that the skew group algebra of the Jacobian algebra of (Q, W) is Morita equivalent to the Jacobian algebra of (QG, WG). Moreover, we show that this construction is a duality if G is abelian. We also apply our results to quivers with potential associated to Postnikov diagrams.