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We study deformation of tube algebra under twisting of graded monoidal categories. When a tensor category $mathcal{C}$ is graded over a group $Gamma$, a torus-valued 3-cocycle on $Gamma$ can be used to deform the associator of $mathcal{C}$. Based on a natural Fell bundle structure of the tube algebra over the action groupoid of the adjoint action of $Gamma$, we show that the tube algebra of the twisted category is a 2-cocycle twisting of the original one.
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Let C be a fusion category faithfully graded by a finite group G and let D be the trivial component of this grading. The center Z(C) of C is shown to be canonically equivalent to a G-equivariantization of the relative center Z_D(C). We use this result to obtain a criterion for C to be group-theoretical and apply it to Tambara-Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara-Yamagami categories. Finally, we prove a general result about existence of zeroes in S-matrices of weakly integral modular categories.
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.
In our recent paper [Sh1] a version of the generalized Deligne conjecture for abelian $n$-fold monoidal categories is proven. For $n=1$ this result says that, given an abelian monoidal $k$-linear category $mathscr{A}$ with unit $e$, $k$ a field of characteristic 0, the dg vector space $mathrm{RHom}_{mathscr{A}}(e,e)$ is the first component of a Leinster 1-monoid in $mathscr{A}lg(k)$ (provided a rather mild condition on the monoidal and the abelian structures in $mathscr{A}$, called homotopy compatibility, is fulfilled). In the present paper, we introduce a new concept of a ${it graded}$ Leinster monoid. We show that the Leinster monoid in $mathscr{A}lg(k)$, constructed by a monoidal $k$-linear abelian category in [Sh1], is graded. We construct a functor, assigning an algebra over the chain operad $C(E_2,k)$, to a graded Leinster 1-monoid in $mathscr{A}lg(k)$, which respects the weak equivalences. Consequently, this paper together with loc.cit. provides a complete proof of the generalized Deligne conjecture for 1-monoidal abelian categories, in the form most accessible for applications to deformation theory (such as Tamarkins proof of the Kontsevich formality).
This paper provides a conceptual study of the twisting procedure, which amounts to create functorially new differential graded Lie algebras, associative algebras or operads (as well as their homoto
We prove that representations of the braid groups coming from weakly group-theoretical braided fusion categories have finite images.
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