Extension algebras of standard modules

TL;DR

This paper investigates the extension algebra of standard modules over a finite dimensional algebra, characterizing its stratification properties and providing conditions under which it becomes a generalized Koszul algebra.

Contribution

It offers a new analysis of the extension algebra's stratification and introduces criteria for its generalized Koszulity, expanding understanding of algebraic structures.

Findings

Characterization of the stratification property of the extension algebra

Sufficient conditions for the extension algebra to be generalized Koszul

Insights into the structure of extension algebras of standard modules

Abstract

Let $A$ be a finite dimensional $k$-algebra standardly stratified for a partial order $\leqslant$ and $\Delta$ be the direct sum of all standard modules. In this paper we study the extension algebra $E= \text{Ext}_A^{\ast} (\Delta, \Delta)$ of standard modules, characterize the stratification property of $E$ for $\leqslant$ and $\leqslant ^{op}$, and obtain a sufficient condition for $E$ to be a generalized Koszul algebra (in a sense which we define).