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Dmitri Nikshych

Publications and source records attributed to Dmitri Nikshych.

At least 19 recordsLinked to original sources
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.

math.CT↗
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.

math.QA↗
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.

math.QA↗
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.

math.CT↗
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.

math.QA↗
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.

math.QA↗
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.

math.QA↗
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.

math.QA↗
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.

math.QA↗
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.

math.QA↗
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.

math.QA↗
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.

math.QA↗
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.

math.QA↗
arXiv2012

The Picard crossed module of a braided tensor category

For a finite braided tensor category we introduce its Picard crossed module consisting of the group of invertible module categories and the group of braided tensor autoequivalences. We describe the Picard crossed module in terms of braided autoequivalences of the Drinfeld center of the braided tensor category. As an illustration, we compute the Picard crossed module of a braided pointed fusion category.

math.QA↗