Sebastian Burciu

Institute of Mathematics of the Romanian Academy | IMAR · Algebra

The question of how certain arithmetical conditions on the sizes of conjugacy classes of a finite group influence the group structure was extensively studied in the last decade. In this paper we study some analogue of these properties for fusion categories.

In this short note, we prove some consequences of Burnside's vanishing property \cite{b-pa}. It is known that Harada's identity concerning the product of all conjugacy classes of a finite group is a consequence of Burnside's vanishing property of characters. We prove a similar formula for any weakly-integral fusion category. In the second part, we...

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 fusion categories satisfy both Burnside's prop...

This paper introduces an abstract Isaacs property involving the Fourier transform for (possibly non-commutative) fusion rings, extending the one introduced in \cite{lpr1} in the commutative case. A categorical version was also defined in \cite{eno-nec} for any spherical fusion category, and we prove that it matches with our abstract version in the...

In this note some new Frobenius type divisibility results are obtained for premodular categories. In particular, we extend Corollary 3.4 of [Yu20] from the settings of super-modular categories to arbitrary pseudo-unitary premodular categories.

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. Based on these symmetries, we generalize a well-known result of Burnside from representation theory of finite groups. More precisely, we show that any row corresponding to...

Let $\mathcal C$ be a pseudo-unitary fusion category and $R:\mathcal{C}\rightarrow \mathcal Z(\mathcal C)$ the right adjoint of the forgetful functor. It is well known, see [DMNO13] that $\mathbb A:=R(\mathbb{1})$ is an etale algebra in $\mathcal Z(\mathcal C)$. We show that any unitary subalgebra of the adjoint algebra $\mathbb A$ associated to a...

A criterion for the 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 (J Pure Appl Algebra 221(9):2338–2371, 2017). We also show that in a modular tensor category the product of two conjuga...

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 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}...

A criterion for M\"uger 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 combi...

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 h...

In this paper we give two general formulae for the M\"uger 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 M\"uger centralizer is given in the terms of the conjugacy cla...

In this paper it is shown that any irreducible representation of a Drinfeld double D(A) of a semisimple Hopf algebra A can be obtained as an induced representation from a certain Hopf subalgebra of D(A). This generalizes a well known result concerning the irreducible representations of Drinfeld doubles of finite groups [11]. Using this description...

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...

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.

In this paper, we describe a Mackey type decomposition for group actions on abelian categories. This allows us to define new Mackey functors which associates to any subgroup the K-theory of the corresponding equivariantized abelian category. In the case of an action by tensor autoequivalences, the Mackey functor at the level of Grothendieck rings h...

It is shown that any coideal subalgebra of a finite-dimensional Hopf algebra is a cyclic module over the dual Hopf algebra. Using this we describe all coideal subalgebras of a cocentral abelian extension of Hopf algebras extending some results from [R. Guralnick and F. Xu, On a subfactor generalization of Wall's conjecture, J. Algebra 332 (2011) 45...

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...

We provide a general formula for M\"{u}ger's centralizer of any fusion subcategory of a braided fusion category containing a tannakian subcategory. This entails a description for M\"{u}ger's centralizer of all fusion subcategories of a group theoretical braided fusion category.

In this paper we prove a formula that relates M\"uger'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 M\"uger's centralizer of some fusion subcategories of the fusion category of finite...

This paper gives necessary and sufficient conditions for subgroups with trivial core to be of odd depth. We show that a subgroup with trivial core is an odd depth subgroup if and only if certain induced modules from it are faithful. Algebraically this gives a combinatorial condition that has to be satisfied by the subgroups with trivial core in ord...

We define a notion of depth for an inclusion of multima- trix algebras B ⊆ A based on a comparison of powers of the induction- restriction table M (and its transpose matrix). The depth of the semisim- ple subalgebra B in the semisimple algebra A is the least positive integer n ≥ 2 for which M n+1 ≤ qM n−1 for some q ∈ Z +. We prove that a depth two...

A description of the commutator of a normal subcategory of the fusion category of representation Rep A of a semisimple Hopf algebra A is given. Formulae for the kernels of representations of Drinfeld doubles D(G) of finite groups G are presented. It is shown that all these kernels are normal Hopf subalgebras.

A Mackey type decomposition for group actions on abelian categories is described. This allows us to define new Mackey functors which associates to any subgroup the $K$-theory of the corresponding equivariantized abelian category. In the case of an action by tensor autoequivalences the Mackey functor at the level of Grothendieck rings has a Green fu...

The quantum doubles of a certain class of rank two pointed Hopf algebras are considered. The socle of the tensor product of two such modules is computed, and formulas similar to the ones in [4] are obtained. Cases when such a tensor product is completely irreducible are also given in the last section.

We introduce the notion of double cosets relative to two fusion subcategories of a fusion category. Given a tensor functor \(F : {\mathcal {C}} \to {\mathcal {D}}\) between fusion categories, we introduce an equivalence relation ≈ F on the set \(\Lambda _{\mathcal {C}}\) of isomorphism classes of simple objects of \({\mathcal {C}}\), and when F is...

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...

We investigate a decomposition of Mackey type for any induced module from a Hopf subalgebra of a semisimple Hopf algebra. This generalizes the well known decomposition of an induced representation from a subgroup of a finite group. Some general examples arising from the universal gradings of the fusion categories of corepresentations are provided.

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...

It shown that any coideal subalgebra of a finite dimensional Hopf algebra is a cyclic module over the dual Hopf algebra. Using this we describe all coideal subalgebras of a cocentral abelian extension of Hopf algebras extending some results from [4].

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...

The restriction functor from the category of representations of a semisimple Hopf algebra to the category of representations of a normal coideal subalgebra is studied. It is shown that this functor has a similar behavior to the restriction functor to the category of representations of a normal Hopf subalgebra. Commutator subalgebras as normal left...

We survey the results obtained so far in the study of normal Hopf subalgebras of semisimple Hopf algebras. The connection between normal Hopf subalgebras and kernels of representations is described. The representation theory of normal Hopf subalgebras and their associated fusion rings are also discussed. KeywordsSemisimple Hopf algebras–Normal Hop...

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.

An extension of $k$-algebras $B \subset A$ is said to have depth one if there exists a positive integer $n$ such that $ A$ is a direct summand of $ B^n$ in $_B\mtr{Mod}_B$. Depth one extensions of semisimple algebras are completely characterized in terms of their centers. For extensions of semisimple Hopf algebras our results are similar to those o...

The classical Clifford correspondence for normal subgroups is considered in the setting of semisimple Hopf algebras. We prove that this correspondence still holds if the extension determined by the normal Hopf subalgebra is cocentral.

We define a notion of depth for an inclusion of complex semisimple algebras, based on a comparison of powers of the induction-restriction table (and its transpose matrix) and a previous notion of depth in an earlier paper of the second author. We prove that a depth two extension of complex semisimple algebras is normal in the sense of Rieffel, and...

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...

It is shown that in the category of semisimple Hopf algebras the categorical Hopf kernels introduced by N. Andruskiewitsch and J. Devoto [in St. Petersbg. Math. J. 7, No. 1, 17-52 (1996); translation from Algebra Anal. 7, No. 1, 22-61 (1995; Zbl 0857.16032)] coincide with the kernels of representations introduced by the present author in 2009 [Proc...

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.

We survey the results obtained so far in the study of semisimple Hopf algebras. The role of Kaplansky's sixth conjecture to the development of this study is presented in detail. The connection between fusion categories and this study is also emphasized. In the final Section we present a survey on the relation between Hopf algebra extensions and fin...

A depth two Hopf subalgebra K of a semisimple Hopf algebra H is shown to be a normal Hopf subalgebra. On the one hand, we prove this using Galois theory of quantum groupoids. On the other hand, we give a second proof using character theory of semisimple Hopf algebras.

We define a notion of depth for an inclusion of multimatrix algebras B < A based on a comparison of powers of the induction-restriction table M (and its transpose matrix). This notion of depth coincides with the depth from [Kadison, 2008]. In particular depth 2 extensions coincides with normal extensions as introduced by Rieffel in 1979. For a grou...

For any normal commutative Hopf subalgebra K = k G of a semisimple Hopf algebra we describe the ring inside kG obtained by the restriction of H-modules. If G = \(G={\mathbb{Z}}\) p this ring determines a fusion ring and we give a complete description for it. The case \(G={\mathbb{Z}}_{p^n}\) and some other applications are presented.

A subalgebra pair of semisimple complex algebras B < A with inclusion matrix M is depth two if MM^t M < nM for some positive integer n and all corresponding entries. If A and B are the group algebras of finite group-subgroup pair H < G, the induction-restriction table equals M and S = MM^t satisfies S^2 < nS iff the subgroup H is depth three in G;...

A criterion for subcoalgebras to be invariant under the adjoint action is given generalizing Masuoka's criterion for normal Hopf subalgebras. At the level of characters, the image of the induction functor from a normal Hopf subalgebra is isomorphic to the image of the restriction functor.

Andruskiewitsch and Schneider classify a large class of pointed Hopf algebras with abelian coradical. The Drinfeld double of each such Hopf algebra is investigated. The Drinfeld doubles of a family of Hopf algebras from the above classification are ribbon Hopf algebras.

The notion of double coset for semisimple finite dimensional Hopf algebras is introduced. This is done by considering an equivalence relation on the set of irreducible characters of the dual Hopf algebra. As an application formulae for the restriction of the irreducible characters to normal Hopf subalgebras are given.

We study the induction and restriction functor from a Hopf subalgebra of a semisimple Hopf algebra. The image of the induction functor is described when the Hopf subalgebra is normal. In this situation, at the level of characters this image is isomorphic to the image of the restriction functor. A criterion for subcoalgebras to be invariant under th...

Andruskiewitsch and Schneider classify a large class of pointed Hopf algebras with abelian coradical. The quantum double of each such Hopf algebra is investigated. The quantum doubles of a family of Hopf algebras from the above classification are ribbon Hopf algebras.

A certain class of rank two pointed Hopf algebras is considered. The simple modules of their Drinfel'd double is described using Radford's method \cite{rad}. The socle of the tensor product of two such modules is computed and a formula similar to the one in \cite{one} is obtained in some conditions. Cases when such a tensor product is completely ir...

In this note the notion of kernel of a representation of a semisimple Hopf algebra is introduced. Similar properties to the kernel of a group representation are proved in some special cases. In particular, every normal Hopf subalgebra of a semisimple Hopf algebra H is the kernel of a representation of H.

We give some general results on the ring structure of Hochschild cohomology of smash products of algebras with Hopf algebras. We compute this ring structure explicitly for a large class of finite dimensional Hopf algebras of rank one.

For H a finite-dimensional semisimple Hopf algebra over an algebraically closed field of characteristic zero the induced representations from H and H∗ to the Drinfel'd double D(H) are studied. The product of two such representations is a sum of copies of the regular representation of D(H). The action of certain irreducible central characters of H∗...

Let H be a cosemisimple Hopf algebra over an algebraically closed field. In the first chapter of the thesis, it is shown that if H has a simple subcoalgebra of dimension 9 and has no simple subcoalgebras of even dimension, then H contains either a grouplike element of order 2 or 3, or a family of simple subcoalgebras whose dimensions are the square...

Let H be a cosemisimple Hopf algebra over an algebraically closed field. It is shown that if H has a simple subcoalgebra of dimension 9 and has no simple subcoalgebras of even dimension, then H contains either a grouplike element of order 2 or 3, or a family of simple subcoalgebras whose dimensions are the squares of each positive odd integer. In p...

Let H be a cosemisimple Hopf algebra over an algebraically closed field k which contains a simple subcoalgebra of dimension 9. We show that if H has no simple subcoalgebras of even dimension then H contains either a grouplike element with order 2 or 3, a Hopf subalgebra of dimension 75, or a family of simple subcoalgebras whose dimensions are the s...

In this note, we introduce the notion of integral on Hopf Galois extensions. A natural question arises: is there a bialgebra structure on these extensions such that this notion corresponds with the integral on a bialgebra? We will solve this problem. In the last section we will give some applications for example Radford's formula and the fact that...

A description for kernels of representations of semisimple Drinfeld doubles $D(A)$ is given. Using this we also obtain a description of all normal Hopf subalgebras of $D(A)$. As an application, kernels of irreducible representations of $D(G)$ are computed and shown that they are all normal Hopf subalgebras of $D(G)$.