Group-theoretical Fusion Categories Research Articles

Non-invertible symmetries and higher representation theory II

In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the construction of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. We propose that the symmetry categories obtained by gauging higher subgroups may be defined as higher group-theoretical fusion categories, which are built from the projective higher representations of higher groups. As concrete applications we provide a unified description of the symmetry categories of gauge theories in three and four dimensions based on the Lie algebra \mathfrak{so}(N)𝔰𝔬(N), and a fully categorical description of non-invertible symmetries obtained by gauging a 1-form symmetry with a mixed ’t Hooft anomaly. We also discuss the effect of discrete torsion on symmetry categories, based a series of obstructions determined by spectral sequence arguments.

On the minimal extension and structure of braided weakly group-theoretical fusion categories

On the minimal extension and structure of braided weakly group-theoretical fusion categories

On triality defects in 2d CFT

We consider the triality fusion category discovered in the c = 1 Kosterlitz-Thouless theory [1]. We analyze this fusion category using the tools from the group theoretical fusion category and compute the simple lines, fusion rules and F-symbols. We then studied the physical implication of this fusion category including deriving the spin selection rule, computing the asymptotic density of states of irreducible representations of the fusion category symmetries, and analyzing its anomaly and constraints under the renormalization group flow. There is another set of F-symbols for the fusion categories with the same fusion rule known in the literature [2]. We find these two solutions are different as they lead to different spin selection rules. This gives a complete list of the fusion categories with the same fusion rule by the classification result in [3].

Algebraic Structures in Group-theoretical Fusion Categories

It was shown by Ostrik (Int. Math. Res. Not. 2003(27), 1507–1520 2003) and Natale (SIGMA Symmetry Integrability Geom. Methods Appl. 13, 042 2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We generalize this result by constructing explicit Morita equivalence class representatives of indecomposable, separable algebras in group-theoretical fusion categories. This is achieved by providing the free functor Φ from fusion category to a category of bimodules in the original category with a (Frobenius) monoidal structure. Our algebras of interest are then constructed as the image of twisted group algebras under Φ. We also show that twisted group algebras admit the structure of Frobenius algebras in a pointed fusion category, and as a consequence, our algebras are Frobenius algebras in a group-theoretical fusion category. They also enjoy several good algebraic properties.

Q-system completion for C⁎ 2-categories

A Q-system in a C⁎ 2-category is a unitary version of a separable Frobenius algebra object and can be viewed as a unitary version of a higher idempotent. We define a higher unitary idempotent completion for C⁎ 2-categories called Q-system completion and study its properties. We show that the C⁎ 2-category of right correspondences of unital C⁎-algebras is Q-system complete by constructing an inverse realization †-2-functor. We use this result to construct induced actions of group theoretical unitary fusion categories on continuous trace C⁎-algebras with connected spectra.

On the braid group representations coming from weakly group-theoretical fusion categories

We prove that representations of the braid groups coming from weakly group-theoretical braided fusion categories have finite images.

The character algebra formodule categories over Hopf algebras

Given a finite dimensional Hopf algebra H and an exact indecomposable module category M over Rep(H), we explicitly compute the adjoint algebra A_M as an object in the category of Yetter-Drinfeld modules over H, and the space of class functions CF(M) associated to M, as introduced by K. Shimizu [ Further results on the structure of (Co)ends in fintite tensor categories, preprint arXiv:1801.02493]. We use our construction to describe these algebras when H is a group algebra and a dual group algebra. This result allows us to compute the adjoint algebra for certain group-theoretical fusion categories.

Tensor Algebras in Finite Tensor Categories

Abstract This paper introduces methods for classifying actions of finite-dimensional Hopf algebras on path algebras of quivers and more generally on tensor algebras $T_B(V)$ where $B$ is semisimple. We work within the broader framework of finite (multi-)tensor categories $\mathcal{C}$, classifying tensor algebras in $\mathcal{C}$ in terms of $\mathcal{C}$-module categories. We obtain two classification results for actions of semisimple Hopf algebras: the first for actions that preserve the ascending filtration on tensor algebras and the second for actions that preserve the descending filtration on completed tensor algebras. Extending to more general fusion categories, we illustrate our classification result for tensor algebras in the pointed fusion categories $\textsf{Vec}_{G}^{\omega }$ and in group-theoretical fusion categories, especially for the representation category of the Kac–Paljutkin Hopf algebra.

Extensions of tensor categories by finite group fusion categories

AbstractWe study exact sequences of finite tensor categories of the form Rep G → 𝒞 → 𝒟, where G is a finite group. We show that, under suitable assumptions, there exists a group Γ and mutual actions by permutations ⊳ : Γ × G → G and ⊲ : Γ × G→ Γ that make (G, Γ) 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,Γ) of 𝒟. Dually, we show that an exact sequence of finite tensor categories VecG → 𝒞 → 𝒟 induces an Aut(G)-grading on 𝒞 whose neutral homogeneous component is a (Z(G), Γ)-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.

On the Equivalence of Module Categories over a Group-Theoretical Fusion Category

We give a necessary and sufficient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category ${\mathcal C}$ to be equivalent. This concludes the classification of such module categories.

Exact factorizations and extensions of fusion categories

We introduce and study the new notion of an exact factorizationB=A•C of a fusion category B into a product of two fusion subcategories A,C⊆B of 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 in [7]. 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.

Erratum to: A Higher Frobenius–Schur Indicator Formula for Group-Theoretical Fusion Categories

Erratum to: A Higher Frobenius–Schur Indicator Formula for Group-Theoretical Fusion Categories

On Müger's centralizer in braided equivariantized fusion categories

We provide a general formula for Müger's centralizer of any fusion subcategory of a braided fusion category containing a Tannakian subcategory. This entails a description for Müger's centralizer of all fusion subcategories of a group theoretical braided fusion category.

Computing higher Frobenius–Schur indicators in fusion categories constructed from inclusions of finite groups

We consider a subclass of the class of group-theoretical fusion categories: To every finite group G and subgroup H one can associate the category of G-graded vector spaces with a two-sided H-action compatible with the grading. We derive a formula that computes higher Frobenius-Schur indicators for the objects in such a category using the combinatorics and representation theory of the groups involved in their construction. We calculate some explicit examples for inclusions of symmetric groups.

A Jordan–Hölder theorem for weakly group-theoretical fusion categories

We prove a version of the Jordan–Holder theorem in the context of weakly group-theoretical fusion categories. This allows us to introduce the composition factors and the length of such a fusion category \({\mathcal C}\), which are in fact Morita invariants of \({\mathcal C}\).

A Higher Frobenius–Schur Indicator Formula for Group-Theoretical Fusion Categories

Group-theoretical fusion categories are defined by data concerning finite groups and their cohomology: A finite group $G$ endowed with a three-cocycle $\omega$, and a subgroup $H\subset G$ endowed with a two-cochain whose coboundary is the restriction of $\omega$. The objects of the category are $G$-graded vector spaces with suitably twisted $H$-actions; the associativity of tensor products is controlled by $\omega$. Simple objects are parametrized in terms of projective representations of finite groups, namely of the stabilizers in $H$ of right $H$-cosets in $G$, with respect to two-cocycles defined by the initial data. We derive and study general formulas that express the higher Frobenius-Schur indicators of simple objects in a group-theoretical fusion category in terms of the group-theoretical and cohomological data defining the category and describing its simples.

Some quasitensor autoequivalences of Drinfeld doubles of finite groups

We report on two classes of autoequivalences of the category of Yetter-Drinfeld modules over a finite group, or, equiv- alently the Drinfeld center of the category of representations of a finite group. Both operations are related to the r-th power opera- tion, with r relatively prime to the exponent of the group. One is defined more generally for the group-theoretical fusion category de- fined by a finite group and an arbitrary subgroup, while the other seems particular to the case of Yetter-Drinfeld modules. Both au- toequivalences preserve higher Frobenius-Schur indicators up to Galois conjugation, and they preserve tensor products, although neither of them can in general be endowed with the structure of a monoidal functor.

Categorical Lagrangian Grassmannians and Brauer–Picard groups of pointed fusion categories

We analyze the action of the Brauer–Picard group of a pointed fusion category on the set of Lagrangian subcategories of its center. Using this action we compute the Brauer–Picard groups of pointed fusion categories associated to several classical finite groups. As an application, we construct examples of weakly group-theoretical fusion categories.

Fusion rules of equivariantizations of fusion categories

We determine the fusion rules of the equivariantization of a fusion category \documentclass[12pt]{minimal}\begin{document}${\mathcal {C}}$\end{document}C under the action of a finite group G in terms of the fusion rules of \documentclass[12pt]{minimal}\begin{document}${\mathcal {C}}$\end{document}C and group-theoretical data associated to the group action. As an application we obtain a formula for the fusion rules in an equivariantization of a pointed fusion category in terms of group-theoretical data. This entails a description of the fusion rules in any braided group-theoretical fusion category.

Clifford theory for graded fusion categories

We develop a categorical analogue of Clifford theory for strongly graded rings over graded fusion categories. We describe module categories over a fusion category graded by a group G as induced from module categories over fusion subcategories associated with the subgroups of G. We define invariant C e -module categories and extensions of C e -module categories. The construction of module categories over C is reduced to determining invariant module categories for subgroups of G and the indecomposable extensions of these module categories. We associate a G-crossed product fusion category to each G-invariant C e -module category and give a criterion for a graded fusion category to be a group-theoretical fusion category. We give necessary and sufficient conditions for an indecomposable module category to be extendable.