Hochschild Cohomology and the Modular Group

TL;DR

This paper extends the known projective action of the modular group from the center to all Hochschild cohomology groups and the entire Hochschild cochain complex of a factorizable ribbon Hopf algebra, up to homotopy.

Contribution

It introduces a new framework for the modular group's action on Hochschild cohomology and complexes of factorizable ribbon Hopf algebras, broadening previous results.

Findings

Extended the modular group's action to all Hochschild cohomology groups.

Established a projective action on the entire Hochschild cochain complex.

Demonstrated the action up to homotopy.

Abstract

It has been shown in previous work that the modular group acts projectively on the center of a factorizable ribbon Hopf algebra. The center is the zeroth Hochschild cohomology group. In this article, we extend this projective action of the modular group to an arbitrary Hochschild cohomology group of a factorizable ribbon Hopf algebra, in fact up to homotopy even to a projective action on the entire Hochschild cochain complex.