Braided Hochschild cohomology and Hopf actions

  • Cris Negron

    Massachusetts Institute of Technology, Cambridge, USA
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We showt hat the braided Hochschild cohomology, of an algebra in a suitably algebraic braided monoidal category, admits a graded ring structure under which it is braided commutative. We then give a canonical identification between the usual Hochschild cohomology ring of a smash product and the (derived) invariants of its braided Hochschild cohomology ring. We apply our results to identify the associative formal deformation theory of a smash product with its formal deformation theory as a module algebra over the given Hopf algebra (when the Hopf algebra is sufficiently semisimple). As a second application we deduce some structural results for the usual Hochschild cohomology of a smash product, and discuss specific implications for finite group actions on smooth affine schemes.

Cite this article

Cris Negron, Braided Hochschild cohomology and Hopf actions. J. Noncommut. Geom. 13 (2019), no. 1, pp. 1–33

DOI 10.4171/JNCG/308