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Hochschild Cohomology and the Modular Group

Simon Lentner       Svea Nora Mierach       Christoph Schweigert       Yorck Sommerhäuser

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.

Introduction

An important idea coming from conformal field theory is that modular categories lead to projective representations of mapping class groups of surfaces (see [BK], [G], [T] and the references cited therein). At least for certain aspects of this construction, it is not necessary that the category under consideration is semisimple. For a particularly simple surface, the torus, the mapping class group is the homogeneous modular group of two-times-two matrices with integer entries and determinant one. By applying these ideas in the case of the representation category of a factorizable ribbon Hopf algebra, which is not required to be semisimple, we obtain a projective representation of the homogeneous modular group on the center of this Hopf algebra (see for example [CW1], [CW2], [Ke], [KL], [LM] and [T]). As the center is the zeroth Hochschild cohomology group of the Hopf algebra, it is natural to ask whether there is a corresponding action on the higher cohomology groups. In this article, we answer this question affirmatively by showing that the modular group acts, projectively and up to homotopy, even on the entire Hochschild cochain complex.

The article is organized as follows: In the first section, we briefly review the Hochschild cohomology of an algebra A with coefficients in an A-bimodule M, as found for example in [W]. We then construct in Proposition 1.3 a particular homotopy between two cochain maps that will be important later for the verification of the defining relations of the modular group. In the second section, we turn to the case where the algebra A is a Hopf algebra and introduce a way to modify the bimodule structure of M while leaving the Hochschild cohomology groups essentially unchanged. In the third section, we turn to the case where A is a factorizable ribbon Hopf algebra and recall the action of the modular group on its center. In particular, we introduce the Radford and the Drinfel'd map. Our treatment here follows largely the exposition in [SZ], to which the reader is referred for references to the original work. In the fourth section, we take advantage of our modification of the bimodule structure introduced in the second section to generalize the Radford and the Drinfel'd map to the Hochschild cochain complex. In the fifth and final section, we use these maps to generalize the action of the modular group on the center to an action on all Hochschild cohomology groups of our factorizable ribbon Hopf algebra.

We will always work over a base field that is denoted by K, and all unadorned tensor products are taken over K. The dual of a vector space V is denoted by V*:=HomK(V,K).

The authors would like to thank Sarah Witherspoon for pointing out References [FS], [GK], [PW] and [SS] as well as for further helpful discussions. During the work on this article, the first and the third author were partially supported by SFB 676 and RTG 1670.