Symmetric cohomology and symmetric Hochschild cohomology of cocommutative Hopf algebras

TL;DR

This paper introduces symmetric cohomology and symmetric Hochschild cohomology for cocommutative Hopf algebras, establishing an isomorphism and exploring conditions for their equivalence with classical cohomology.

Contribution

It generalizes symmetric cohomology from groups to cocommutative Hopf algebras and proves an isomorphism between symmetric Hochschild and symmetric cohomology.

Findings

Established an isomorphism between symmetric cohomology and symmetric Hochschild cohomology.

Extended symmetric cohomology concepts from groups to cocommutative Hopf algebras.

Investigated conditions under which symmetric cohomology coincides with classical cohomology.

Abstract

Staic defined symmetric cohomology of groups and studied that the secondary symmetric cohomology group is corresponding to group extensions and the injectivity of the canonical map from symmetric cohomology to classical cohomology. In this paper, we define symmetric cohomology and symmetric Hochschild cohomology for cocommutative Hopf algebras. The first one is a generalization of symmetric cohomology of groups. We give an isomorphism between symmetric cohomology and symmetric Hochschild cohomology, which is a symmetric version of the classical result about cohomology of groups by Eilenberg-MacLane and cohomology of Hopf algebras by Ginzburg-Kumar. Moreover, to consider the condition that symmetric cohomology coincides with classical cohomology, we investigate the projectivity of a resolution which gives symmetric cohomology.