Darius Dramburg: Higher representation infinite algebras from skew-group algebras: Higher preprojective gradings, Koszul gradings, and McKay quivers

Abstract

This thesis consist of five papers in the area of representation theory of algebras. The focus lies on higher representation infinite algebras and their higher preprojective algebras. We consider the case of a polynomial ring skewed by the action of a finite subgroup of the special linear group, and construct and classify higher preprojective structures on this skew-group algebra. This involves the construction of McKay quivers and gradings of these quivers.

Paper I covers the easiest previously unknown case. We classify the finite abelian subgroups of the special linear group in dimension 3 such that the corresponding skew-group algebra is the higher preprojective algebra of a higher representation infinite algebra. We also describe the involved mutations of these algebras, and give a combinatorial description.

Paper II is a generalisation of paper I to arbitrary dimensions. For each finite abelian subgroup of the special linear group in dimension n+1, we consider the possible higher preprojective structures on the corresponding skew-group algebra. These structures naturally fall into several mutation classes, and we identify these classes with the internal points of a lattice simplex. We show that this lattice simplex is the junior simplex of the group. Furthermore, we equip each mutation class with the structure of a finite distributive lattice and construct the minimal and maximal elements of these lattices.

Papers III and IV deal with the case of finite non-abelian subgroups of the special linear group in dimension 3. For each group, we decide whether the skew-group algebra can be endowed the the structure of a higher preprojective algebra, and describe the resulting higher representation infinite algebras. We give detailed descriptions of the relevant McKay quivers and provide numerous examples and computations.

Paper V investigates the interaction between a Koszul- and a higher preprojective grading on the same algebra. While the two gradings need not be related, we show that in most cases, the Koszul grading can be moved by an automorphism to another Koszul grading that forms a bigrading together with the higher preprojective grading. As a consequence we show that a basic higher hereditary algebra can be endowed with an (almost) Koszul grading if and only if its higher preprojective algebra can be endowed with a Koszul grading. We also show that higher Auslander-Platzeck-Reiten tilting preserves Koszulity.