Hopf algebra actions on differential graded algebras and applications

TL;DR

This paper explores the actions of finite dimensional semisimple Hopf algebras on differential graded algebras, establishing a key quasi-isomorphism and applying it to various algebraic structures such as Koszul, Calabi-Yau, and Gorenstein dg algebras.

Contribution

It proves a fundamental quasi-isomorphism relating Hochschild cohomology of dg algebras under Hopf algebra actions and applies it to important classes of dg algebras.

Findings

Established a quasi-isomorphism for dg $A\#H$-modules

Applied results to $d$-Koszul, Calabi-Yau, and AS-Gorenstein dg algebras

Extended understanding of Hopf algebra actions on differential graded structures

Abstract

Let $H$ be a finite dimensional semisimple Hopf algebra, $A$ a differential graded (dg for short) $H$-module algebra. Then the smash product algebra $A\#H$ is a dg algebra. For any dg $A\#H$-module $M$, there is a quasi-isomorphism of dg algebras: $\mathrm{RHom}_A(M,M)\#H\longrightarrow \mathrm{RHom}_{A\#H}(M\ot H,M\ot H)$. This result is applied to $d$-Koszul algebras, Calabi-Yau algebras and AS-Gorenstein dg algebras