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Sebastian Burciu

Publications and source records attributed to Sebastian Burciu.

At least 19 recordsLinked to original sources
arXiv2023

Burnside type results for fusion categories

In this paper, we extend a classical vanishing result of Burnside from the character tables of finite groups to the character tables of commutative fusion rings, or more generally to a certain class of abelian normalizable hypergroups. We also treat the dual vanishing result. We show that any nilpotent unitary fusion categories satisfy both Burnside's property and its dual. Using Drinfeld's map, we obtain that the Grothendieck ring of any weakly-integral modular fusion category satisfies both properties. As applications, we prove new identities that hold in the Grothendieck ring of any weakly-integral fusion category satisfying the dual-Burnside's property, thus providing new categorification criteria. In particular we improve [OY23, Theorem 4.5] as follows: A weakly integral modular fusion category of FPdim md with d square-free coprime with m and FPdim(X)^2 for every simple object X, has a pointed modular fusion subcategory of FPdim d. We also present new results on perfect modular fusion categories, including a Cauchy-type theorem.

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arXiv2022

Structure constants, Isaacs property and Extended Haagerup fusion categories

This paper presents an abstract Isaacs property that involves the Fourier transform for fusion rings, which may be non-commutative, thus expanding upon the commutative version described in [12]. A categorical version of this property was subsequently introduced in [8] for any spherical fusion category, matching with our abstract version in the pseudo-unitary case. We demonstrate that the Isaacs property occupies a distinct position, falling between the integrality of structure constants and the 1-Frobenius properties, in the commutative case. We show that the Extended Haagerup fusion categories, denoted as EHi, do not satisfy the Isaacs property. This finding provides a negative response to [8, Question 5.8], refutes [12, Conjecture 2.5], and recovers that EH1 lacks a braiding structure.

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arXiv2022

Subalgebras of etale algebras and fusion subcategories

In \cite[Rem. 3.4]{DNO} the authors asked the question if any étale subalgebra of an étale algebra in a braided fusion category is also étale. We give a positive answer to this question if the braided fusion category $\cc$ is pseudo-unitary and non-degenerate. In the case of a pseudo-unitary fusion category we also give a new description of the lattice correspondence from \cite[Theorem 4.10]{DMNO}. This new description enables us to describe the two binary operations on the lattice of fusion subcategories.

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arXiv2020

On the Galois symmetries for the character table of an integral fusion category

In this paper we show that integral fusion categories with rational structure constants admit a natural group of symmetries given by the Galois group of their character tables. We also generalize a well known result of Burnside from representation theory of finite groups. More precisely, we show that any row corresponding to a non invertible object in the character table of a weakly integral fusion category contains a zero entry.

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arXiv2020

Representations and Conjugacy Classes of Semisimple Quasitriangular Hopf Algebras

In this paper we give two general formulae for the Müger centralizers in the category of representations of a semisimple quasitriangular Hopf algebra. The first formula is given in the terms of the Drinfeld map associated to the quasitriangular Hopf algebra. The second formula for the Müger centralizer is given in the terms of the conjugacy classes introduced by Cohen and Westreich in [J. Algebra 283 (2005), 42-62]. In the case of a factorizable Hopf algebra these formulae extend some particular cases obtained by the author in [Math. Z. 279 (2015), 227-240].

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arXiv2020

Structure constants for pre-modular categories

In this paper we study conjugacy classes for pivotal fusion categories. In particular we prove a Burnside type formula for the structure constants concerning the product of two conjugacy class sums of a such fusion category. For a braided weakly integral fusion category $\mathcal C$ we show that these structure constants multiplied by $\mathrm{dim}(\mathcal C)$ are non-negative integers, extending some results obtained by Zhou and Zhu (see \cite{zz}) in the settings of semisimple quasitriangular Hopf algebras.

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arXiv2019

Fusion rules of equivariantizations of fusion categories

We determine the fusion rules of the equivariantization of a fusion category $\mathcal{C}$ under the action of a finite group $G$ in terms of the fusion rules of $\mathcal{C}$ 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.

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arXiv2019

Conjugacy classes and centralizers for pivotal fusion categories

A criterion for Müger centralizer of a fusion subcategory of a braided non-degenerate fusion category is given. Along the way we extend some identities on the space of class functions of a fusion category introduced by Shimizu in \cite{scalg}. We also show that in a modular tensor category the product of two conjugacy class sums is a linear combination of conjugacy class sums with rational coefficients.

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arXiv2015

On an analogue of a Brauer theorem for fusion categories

In this paper we prove an analogue of Brauer's theorem for faithful objects in fusion categories. Other notions, such as the order and the index associated to faithful objects of fusion categories are also discussed. We show that the index of a faithful simple object of a fusion categories coincides with the order of the universal grading group of the fusion category.

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arXiv2014

Categorical Green functors arising from group actions on categories

In this paper we introduce the notion of a categorical Mackey functor. This categorical notion allows us to obtain new Mackey functors by passing to Quillen's $K$-theory of the corresponding abelian categories. In the case of an action by monoidal autoequivalences on a monoidal category the Mackey functor obtained at the level of Grothendieck rings has in fact a Green functor structure.

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arXiv2013

On a symmetry of Müger's centralizer for the Drinfeld double of a semisimple Hopf algebra

In this paper we prove a formula that relates Müger's centralizer in the category of representations of a factorizable Hopf algebra to the notion of Hopf kernel of a representation of the dual Hopf algebra. Using this relation we obtain a complete description for Müger's centralizer of some fusion subcategories of the fusion category of finite dimensional representations of a Drinfeld double of a semisimple Hopf algebra.

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arXiv2013

New examples of Green functors arising from representation theory of semisimple Hopf algebras

A general Mackey type decomposition for representations of semisimple Hopf algebras is investigated. We show that such a decomposition occurs in the case that the module is induced from an arbitrary Hopf subalgebra and it is restricted back to a group subalgebra. Some other examples when such a decomposition occurs are also constructed. They arise from gradings on the category of corepresentations of a semisimple Hopf algebra and provide new examples of Green functors in the literature.

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arXiv2013

Normal Hopf subalgebras of semisimple Drinfeld doubles

A description of all normal Hopf subalgebras of a semisimple Drinfeld double is given. This is obtained by considering an analogue of Goursat's lemma concerning fusion subcategories of Deligne products of two fusion categories. As an application we show that the Drinfeld double of any abelian extension is also an abelian extension.

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arXiv2012

On the irreducible representations of generalized quantum doubles

A description of all the irreducible representations of generalized quantum doubles associated to skew pairings of semisimple Hopf algebras is given. In particular a description of the irreducible representations of semisimple Drinfeld doubles is obtained. It is shown that the Grothendieck ring of these generalized quantum doubles have a structure similar to the rings that arise from Green functors. In order to do this we give a formula for the tensor product of any two such irreducible representations.

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arXiv2010

Kernels of representations and coideal subalgebras of Hopf algebras

We define left and right kernels of representations of Hopf algebras. In the case of group algebras, left and right kernels coincide and they are the usual kernels of modules. In the general case we show that these kernels coincide with the categorical left and right Hopf kernels of morphisms of Hopf algebras defined in \cite{AD}. Brauer's theorem for kernels over group algebras is generalized to Hopf algebras.

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arXiv2010

On complements and the factorization problem for Hopf algebras

Two new results concerning complements in a semisimple Hopf algebra are proved. They extend some well known results from group theory. The uniqueness of Krull Schmidt Remak type decomposition is proved for semisimple completely reducible Hopf algebras.

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