Cohomological Properties of Differential Calculi on Hopf Algebras

TL;DR

This paper explores the cohomological aspects of differential calculi on Hopf algebras, establishing connections with the Drinfeld double and Hochschild cohomology, and characterizing differential calculi via 1-cocycles.

Contribution

It provides an intrinsic framework linking differential calculi on Hopf algebras to Hochschild cohomology and Drinfeld double representations, extending previous results.

Findings

Bicovariant bimodules correspond to Drinfeld double representations.

Differential calculi are characterized by specific 1-cocycles with invariance conditions.

Hochschild cohomology of the double incorporates additional invariance constraints.

Abstract

In this report we give an intrinsic treatment of the results we developed in a previous work connecting the differential calculi on Hopf algebras to the Drinfeld double. In the first place we recover that bicovariant bimodules are in one to one correspondence with the Drinfeld double representations; we then introduce a Hochschild cohomology of the algebra of functions and discuss the main result stating that each differential calculus is associated to a 1-cocycle satisfying an additional invariance condition with respect to a natural action. Defining a Hochschild cohomology of the double, the above invariance becomes a condition with respect to the enveloping algebra component of the double that must be added to the 1-cocycle relation.