Markus Thuresson: Ext-algebras of standard modules over quasi-hereditary algebras

Abstract

This thesis concerns the representation theory of a particular class of finite-dimensional algebras, called quasi-hereditary algebras. In particular, it considers two subclasses of quasi-hereditary algebras: dual extension algebras and hereditary algebras. It consists of three papers.

Paper I describes the Ext-algebra of standard modules over the dual extension algebra of two arbitrary directed algebras up to isomorphism of graded associative algebras. This Ext-algebra carries the natural structure of an A-infinity algebra, and under certain technical assumptions, we provide explicit formulae for its A-infinity structure in terms of the directed algebras used to construct the dual extension algebra.

Paper II classifies all (basic) generalized tilting modules and all full exceptional sequences of a certain family of quasi-hereditary algebras, which are examples of the dual extension algebras studied in Paper I. This is achieved by constructing a combinatorial model for the poset of self-orthogonal indecomposable modules with standard filtration, where the order relation arises from extensions of positive degree.

Paper III describes the quiver and relations of the Ext-algebra of standard modules over the path algebra of a uniformly oriented linear quiver. For a deconcatenation of a quiver at a sink or a source, we describe the Ext-algebra of standard modules over the path algebra of the quiver in terms of that over the path algebras of the parts of the deconcatenation. Moreover, we characterize the deconcatenations for which the path algebra of the quiver admits a regular exact Borel subalgebra in the sense of König. We apply these results to determine a necessary and sufficient condition for the path algebra of a linear quiver of arbitrary orientation to admit a regular exact Borel subalgebra.