A generalized Koszul theory and its application in representation theory

TL;DR

This paper develops a generalized Koszul theory for graded algebras with non-semisimple degree 0 parts, extending classical results and applying to various algebraic structures in representation theory.

Contribution

It introduces a new generalized Koszul theory that applies to non-semisimple degree 0 components, broadening the scope of classical Koszul duality.

Findings

Preserves classical Koszul duality under certain conditions

Applies to finite EI categories and directed categories

Classifies standardly stratified algebras for all linear orders

Abstract

There are many structures (algebras, categories, etc) with natural gradings such that the degree 0 components are not semisimple. Particular examples include tensor algebras with non-semisimple degree 0 parts, extension algebras of standard modules of standardly stratified algebras. In this thesis we develop a generalized Koszul theory for graded algebras (categories) whose degree 0 parts may be non-semisimple. Under some extra assumption, we show that this generalized Koszul theory preserves many classical results such as the Koszul duality. Moreover, it has some close relation to the classical theory. Applications of this generalized theory to finite EI categories, directed categories, and extension algebras of standard modules of standardly stratified algebras are described. We also study the stratification property of standardly stratified algebras, and classify algebras standardly (resp., properly) stratified for all linear orders.