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The Hochschild complex
of a finite tensor category
Christoph Schweigert and Lukas Woike |
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Algebraic & Geometric Topology 21 (2021) 3689–3734
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Abstract |
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Modular functors, ie consistent systems of projective representations of mapping class groups of surfaces, were constructed for nonsemisimple modular categories 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 nonsemisimple case. 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 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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Keywords
modular functor, Hochschild complex, finite tensor
category, mapping class group
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Mathematical Subject Classification
Primary: 13D03, 18M15, 57K16
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References |
Publication
Received: 4 October 2020
Revised: 7 January 2021
Accepted: 30 January 2021
Published: 28 December 2021
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Authors |
| Christoph Schweigert | |
| Fachbereich Mathematik Universität Hamburg Hamburg Germany |
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| Lukas Woike | |
| Institut for Matematiske Fag Københavns Universitet København Denmark |
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