Extension Algebras of Standard Modules

Abstract

Let A be a basic finite-dimensional k-algebra standardly stratified for a partial order ≤and Δ be the direct sum of all standard modules. In this article, we study the extension algebra of standard modules, characterize the stratification property of Γ for ≤and ≤ ^op , and obtain a sufficient condition for Γ to be a generalized Koszul algebra (in a sense which we define).

Key Words:

• Extension algebras
• Koszul
• Quasi-hereditary
• Standardly stratified

2010 Mathematics Subject Classification:

• 16G10
• 16E40

ACKNOWLEDGMENTS

The author would like to thank his thesis advisor, Professor Peter Webb, for proposing to study the the extension algebras of standard modules, and carefully checking the manuscript. He also thanks Prof. Mazorchuk for pointing out many related works, which are unknown to the author before.

Notes

In [2, 3] standard modules are defined as . Note that in their setup ≤is a linear order, so this description of standard modules coincides with ours.

In [12] we generalized these results to the situation that A ₀ is a self-injective algebra.

Communicated by D. Zacharia.