Homotopy invariants of braided commutative algebras and the Deligne conjecture for finite tensor categories
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It is easy to find algebras T∈C in a finite tensor category C that naturally come with a lift to a braided commutative algebra T∈Z(C) in the Drinfeld center of C. In fact, any finite tensor category has at least two such algebras, namely the monoidal unit I and the canonical end ∫X∈CX⊗X∨. Using the theory of braided operads, we prove that for any such algebra T the homotopy invariants, i.e. the derived morphism space from I to T, naturally come with the structure of a differential graded E2-algebra. This way, we obtain a rich source of differential graded E2-algebras in the homological algebra of finite tensor categories. We use this result to prove that Deligne's E2-structure on the Hochschild cochain complex of a finite tensor category is induced by the canonical end, its multiplication and its non-crossing half braiding. With this new and more explicit description of Deligne's E2-structure, we can lift the Farinati-Solotar bracket on the Ext algebra of a finite tensor category to an E2-structure at cochain level. Moreover, we prove that, for a unimodular pivotal finite tensor category, the inclusion of the Ext algebra into the Hochschild cochains is a monomorphism of framed E2-algebras, thereby refining a result of Menichi.
|Advances in Mathematics
|Number of pages
|Published - 2023
CS is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306 . LW gratefully acknowledges support by the Danish National Research Foundation through the Copenhagen Centre for Geometry and Topology ( DNRF151 ) and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 772960 ). This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101022691 .
© 2023 The Authors
- Deligne's conjecture, E-algebra, Ext algebra, Hochschild homology, Tensor category