Cohomology of the adjoint of Hopf algebras

TL;DR

This paper develops a cohomology theory for the adjoint of Hopf algebras using diagrammatic methods, providing explicit calculations and applications to the Yang-Baxter equation and quandle cocycles.

Contribution

It introduces a new cohomology framework for Hopf algebra adjoints via deformations and diagrammatic techniques, with concrete examples and applications.

Findings

Explicit cohomology calculations for specific Hopf algebras

Solutions to the Yang-Baxter equation derived from the theory

Construction of quandle cocycles from groupoid cocycles

Abstract

A cohomology theory of the adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic techniques. Explicit calculations are provided in the cases of group algebras, function algebras on groups, and the bosonization of the super line. As applications, solutions to the YBE are given and quandle cocycles are constructed from groupoid cocycles.