Cohomology for partial actions of Hopf algebras

TL;DR

This paper develops a cohomology theory for partial actions of co-commutative Hopf algebras on commutative algebras, generalizing existing theories and providing new examples and structural insights.

Contribution

It introduces a generalized cohomology framework for partial Hopf algebra actions, constructs new examples, and links partially cleft extensions to Hopf algebroids.

Findings

Existence of a new Hopf algebra $ ilde{A}$ with equivalent cochain complex.

Construction of nontrivial examples not derived from groups.

Partial cleft extensions correspond to cleft extensions by Hopf algebroids.

Abstract

In this work, the cohomology theory for partial actions of co-commutative Hopf algebras over commutative algebras is formulated. This theory generalizes the cohomology theory for Hopf algebras introduced by Sweedler and the cohomology theory for partial group actions, introduced by Dokuchaev and Khrypchenko. Some nontrivial examples, not coming from groups are constructed. Given a partial action of a co-commutative Hopf algebra $H$ over a commutative algebra $A$, we prove that there exists a new Hopf algebra $\widetilde{A}$, over a commutative ring $E(A)$, upon which $H$ still acts partially and which gives rise to the same cochain complex as the original algebra $A$. We also study the partially cleft extensions of commutative algebras by partial actions of cocommutative Hopf algebras and prove that these partially cleft extensions can be viewed as a cleft extensions by Hopf algebroids.