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Register a new userExtensions of tensor categories by finite group fusion categories
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Authors
Sonia Natale
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We study exact sequences of finite tensor categories of the form $Rep G to C to D$, where $G$ is a finite group. We show that, under suitable assumptions, there exists a group $Gamma$ and mutual actions by permutations $rhd: Gamma times G to G$ and $lhd: Gamma times G to Gamma$ that make $(G, Gamma)$ into matched pair of groups endowed with a natural crossed action on $D$ such that $C$ is equivalent to a certain associated crossed extension $D^{(G, Gamma)}$ of $D$. Dually, we show that an exact sequence of finite tensor categories $vect_G to C to D$ induces an $Aut(G)$-grading on $C$ whose neutral homogeneous component is a $(Z(G), Gamma)$-crossed extension of a tensor subcategory of $D$. As an application we prove that such extensions $C$ of $D$ are weakly group-theoretical fusion categories if and only if $D$ is a weakly group-theoretical fusion category. In particular, we conclude that every semisolvable semisimple Hopf algebra is weakly group-theoretical.
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This is a study of weakly integral braided fusion categories with elementary fusion rules to determine which possess nondegenerately braided extensions of theoretically minimal dimension, or equivalently in this case, which satisfy the minimal modular extension conjecture. We classify near-group braided fusion categories satisfying the minimal modular extension conjecture; the remaining Tambara-Yamagami braided fusion categories provide arbitrarily large families of braided fusion categories with identical fusion rules violating the minimal modular extension conjecture. These examples generalize to braided fusion categories with the fusion rules of the representation categories of extraspecial $p$-groups for any prime $p$, which possess a minimal modular extension only if they arise as the adjoint subcategory of a twisted double of an extraspecial $p$-group.
We introduce and study the new notion of an {em exact factorization} $mathcal{B}=mathcal{A}bullet mathcal{C}$ of a fusion category $mathcal{B}$ into a product of two fusion subcategories $mathcal{A},mathcal{C}subseteq mathcal{B}$ of $mathcal{B}$. This is a categorical generalization of the well known notion of an exact factorization of a finite group into a product of two subgroups. We then relate exact factorizations of fusion categories to exact sequences of fusion categories with respect to an indecomposable module category, which was introduced and studied by P. Etingof and the author in cite{EG}. We also apply our results to study extensions of a group-theoretical fusion category by another one, provide some examples, and propose a few natural questions.
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.
We consider the finite generation property for cohomology of a finite tensor category C, which requires that the self-extension algebra of the unit Ext*_C(1,1) is a finitely generated algebra and that, for each object V in C, the graded extension group Ext*_C(1,V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C. For example, the stated result holds when C is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0, we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes.
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.
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