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

Abstract

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

35 pages

References in corpus (3)

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• On the Ext algebras of parabolic Verma modules and A infinity-structures