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arXiv · 1910.00559

The Hochschild Complex of a Finite Tensor Category

Christoph Schweigert·Lukas Woike
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Abstract

Modular functors, i.e. consistent systems of projective representations of mapping class groups of surfaces, have been constructed for non-semisimple modular categories already decades ago. Concepts from homological algebra have not been used in this construction although it is an obvious question how they should enter in the non-semisimple case. In the present paper, we elucidate the interplay between the structures from topological field theory and from homological algebra by constructing a homotopy coherent projective action of the mapping class group $\text{SL}(2,\mathbb{Z})$ of the torus on the Hochschild complex of a modular category. This is a further step towards understanding the Hochschild complex of a modular category as a differential graded conformal block for the torus. Moreover, we describe a differential graded version of the Verlinde algebra.

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BibTeXRIS

Christoph Schweigert, Lukas Woike. 2019-10-01. The Hochschild Complex of a Finite Tensor Category. https://doi.org/10.2140/agt.2021.21.3689

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