The $\operatorname{Ext}$-algebra of standard modules over dual extension algebras

TL;DR

This paper establishes an isomorphism between the Ext-algebra of standard modules over a dual extension algebra and the dual extension algebra of the Ext-algebra of a directed algebra, including explicit computations of associated $A_ abla$-structures.

Contribution

It introduces a novel isomorphism linking Ext-algebras over dual extension algebras and describes their $A_ abla$-structures explicitly in certain cases.

Findings

Established an algebra isomorphism between Ext-algebras over dual extension algebras.

Described $A_ abla$-structures on Ext-algebras under specific conditions.

Provided explicit computations for the case of path algebras of type A.

Abstract

We exhibit an isomorphism of associative algebras between the $\operatorname{Ext}$-algebra $\operatorname{Ext}_\Lambda^\ast(\Delta,\Delta)$ of standard modules over the dual extension algebra $\Lambda$ of two directed algebras $B$ and $A$ and the dual extension algebra of the $\operatorname{Ext}$-algebra $\operatorname{Ext}_B^\ast(\mathbb{L},\mathbb{L})$ with $A$. There are natural $A_\infty$-structures on these $\operatorname{Ext}$-algebras, and, under certain technical assumptions on $B$, we describe that on $\operatorname{Ext}_\Lambda^\ast(\Delta,\Delta)$ completely in terms of that on $\operatorname{Ext}_B^\ast(\mathbb{L},\mathbb{L})$. As an example, we compute these $A_\infty$-structures explicitly in the case where $B=A=K\mathbb{A}_n /(\operatorname{rad}K\mathbb{A}_n)^\ell$.