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Braided Picard groups and graded extensions of braided tensor categories

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arxiv 2006.08022 v2 pith:LUNMKEI6 submitted 2020-06-14 math.QA math.CT

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keywords braidedcategoriespicardcategoricalextensionsgroupsfinitefunctors
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abstract

We classify various types of graded extensions of a finite braided tensor category $\cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $\cal B$ by a finite group $A$ correspond to braided monoidal $2$-functors from $A$ to the braided $2$-categorical Picard group of $\cal B$ (consisting of invertible central $\cal B$-module categories). Such functors can be expressed in terms of the Eilnberg-Mac~Lane cohomology. We describe in detail braided $2$-categorical Picard groups of symmetric fusion categories and of pointed braided fusion categories.

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  1. Clifford and Weyl algebras in symmetric tensor categories

    math.RT 2026-07 accept novelty 8.0 of 10

    For a symplectic object V in a Frobenius exact symmetric tensor category with finite symmetric algebra, the Weyl algebra A(V) is Azumaya, and the resulting symplectic Witt group is described by Stiefel-Whitney classes...

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