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Dmitri Nikshych

Publications and source records attributed to Dmitri Nikshych.

At least 19 recordsLinked to original sources
arXiv2026

Grothendieck Rings of Module Categories over Drinfeld Doubles

Let $k$ be an algebraically closed field of characteristic zero and $G$ a finite group. We realize the based Grothendieck ring $R_G$ of ${\mathcal{R}\mathit{ep}}(D(G))$-module categories as the degree-two cocycle-decorated double Burnside ring and derive an explicit Clifford formula for multiplication and for the action of $R_G$ on the Grothendieck group of ${\operatorname{\operatorname{\mathsf{Vec}}}}_G$-module categories. We determine the extremal based ideals, study factorization through smaller groups, and prove a Mackey theorem for standard two-sided subgroup inductions. We prove that $\mathbb C\otimes_{\mathbb Z}R_G$ is semisimple exactly when $G$ is cyclic. We study the Brauer--Picard action on indecomposable ${\operatorname{\operatorname{\mathsf{Vec}}}}_G$-module categories and show that it is transitive exactly when $G$ is abelian of square-free exponent. For abelian $G$, we determine the possible abstract group types of Lagrangian subgroups of $G\oplus\widehat G$, apply the known orthogonal classification in the homocyclic case, and exhibit same-type nonconjugate Lagrangians for mixed exponents.

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arXiv2026

Twisted Deligne products of semisimple tensor categories

We discuss the classification of twisted Deligne products of two semisimple tensor categories $\mathcal C,\mathcal D$, i.e., categorifications of the tensor product of their Grothendieck rings in which the factors are categorified by $\mathcal C$ and $\mathcal D$. In particular, we show that if both factors have no non-trivial gradings, or if one factor has neither non-trivial gradings nor tensor structures on the identity functor, then the only twisted Deligne product is the ordinary one. Using the work arXiv:2405.10207 by Müller, Peña Pollastri and Plavnik, this gives, in principle, a group-theoretical classification of twisted Deligne products and, more generally, exact factorizations of arbitrary fusion categories. In the Appendix we introduce the notion of categorical $n$-cocycles for $n=2,3,4$ and show that they are all pullbacks of group $n$-cocycles from the universal grading group of the underlying based ring. In the case of $4$-cocycles, this answers a question of Johnson-Freyd, Ostrik and Yu from arXiv:2601.09060.

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arXiv2026

Pivotal Brauer-Picard groupoids and graded extensions

We develop pivotal and spherical versions of graded extension theory. We define the corresponding analogues of Brauer-Picard $2$-categorical groups and realize them as fixed points of natural $\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$ $2$-categorical actions. We classify graded extensions of a pivotal tensor category by monoidal $2$-functors into the pivotal Brauer-Picard $2$-categorical group. A similar statement is proven for spherical (unimodular) tensor categories. We also develop an obstruction theory for determining when pivotal and spherical structures can be extended.

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arXiv2025

The Tannakian radical and the mantle of a braided fusion category

We define the Tannakian radical of a braided fusion category $\mathcal{C}$ as the intersection of its maximal Tannakian subcategories. The localization of $\mathcal{C}$ corresponding to the Tannakian radical, termed the mantle of $\mathcal{C}$, admits a canonical central extension that serves as a complete invariant of $\mathcal{C}$. The mantle has a trivial Tannakian radical, and we refer to braided fusion categories with this property as reductive. We investigate the properties and structure of reductive categories and prove several classification results.

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arXiv2024

The Classification of Fusion 2-Categories

We classify (multi)fusion 2-categories in terms of braided fusion categories and group cohomological data. This classification is homotopy coherent -- we provide an equivalence between the 3-groupoid of (multi)fusion 2-categories up to monoidal equivalences and a certain 3-groupoid of commuting squares of $\mathrm{B}\mathbb{Z}/2$-equivariant spaces. Rank finiteness and Ocneanu rigidity for fusion 2-categories are immediate corollaries of our classification.

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arXiv2023

On a necessary condition for unitary categorification of fusion rings

In arXiv:1910.12059 Liu, Palcoux and Wu proved a remarkable necessary condition for a fusion ring to admit a unitary categorification, by constructing invariants of the fusion ring that have to be positive if it is unitarily categorifiable. The main goal of this note is to provide a somewhat more direct proof of this result. In the last subsection we discuss integrality properties of the Liu-Palcoux-Wu invariants.

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arXiv2022

Computing the group of minimal non-degenerate extensions of a super-Tannakian category

We prove an analog of the Künneth formula for the groups of minimal non-degenerate extensions arXiv:1602.05936 of symmetric fusion categories. We describe in detail the structure of the group of minimal extensions of a pointed super-Tannakian fusion category. This description resembles that of the third cohomology group of a finite abelian group. We explicitly compute this group in several concrete examples.

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arXiv2021

Braided Picard groups and graded extensions of braided tensor 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.

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arXiv2019

Rank-finiteness for G-crossed braided fusion categories

We establish rank-finiteness for the class of $G$-crossed braided fusion categories, generalizing the recent result for modular categories and including the important case of braided fusion categories. This necessitates a study of slightly degenerate braided fusion categories and their centers, which are interesting for their own sake.

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arXiv2018

Classifying braidings on fusion categories

We show that braidings on a fusion category $\mathcal{C}$ correspond to certain fusion subcategories of the center of $\mathcal{C}$ transversal to the canonical Lagrangian algebra. This allows to classify braidings on non-degenerate and group-theoretical fusion categories.

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arXiv2017

On fusion categories

Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any modular category (not necessarily hermitian) is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category and classify categories of prime dimension.

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arXiv2017

Pointed braided tensor categories

We classify finite pointed braided tensor categories admitting a fiber functor in terms of bilinear forms on symmetric Yetter-Drinfeld modules over abelian groups. We describe the groupoid formed by braided equivalences of such categories in terms of certain metric data, generalizing the well-known result of Joyal and Street for fusion categories. We study symmetric centers and ribbon structures of pointed braided tensor categories and examine their Drinfeld centers.

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arXiv2016

On the Brauer-Picard groups of fusion categories

We develop methods of computation of the Brauer-Picard groups of fusion categories and apply them to compute such groups for several classes of fusion categories of prime power dimension: representation categories of elementary abelian groups with twisted associativity constraint, extra special p-groups, and the Kac-Paljutkin Hopf algebra. We conclude that many finite groups of Lie type occur as composition factors of the Brauer-Picard groups of pointed fusion categories.

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arXiv2015

On the Brauer-Picard group of a finite symmetric tensor category

Let C_n denote the representation category of a finite supergroup generated by purely odd n-dimensional vector space. We compute the Brauer-Picard group BrPic(C_n) of C_n. This is done by identifying BrPic(C_n) with the group of braided tensor autoequivalences of the Drinfeld center of C_n and studying the action of the latter group on the categorical Lagrangian Grassmannian of C_n. We show that this action corresponds to the action of a projective symplectic group on a classical Lagrangian Grassmannian.

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