Jingcheng Dong

Nanjing Agricultural University | NAU · College of Engineering

Let $q>2$ be a prime number, $d$ be an odd square-free natural number, and $n$ be a non-negative integer. We prove that a semisimple quasitriangular Hopf algebra of dimension $dq^n$ is solvable in the sense of Etingof, Nikshych and Ostrik. In particular, if $n\leq 3$ then it is either isomorphic to $k^G$ for some abelian group $G$, or twist equival...

Let C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {C}$\end{document} be a modular category of Frobenius-Perron dimension dqⁿ, where q > 2 is a prime number a...

We study several classes of braided fusion categories, and prove that they all contain nontrivial Tannakian subcategories. As applications, we classify some fusion categories in terms of solvability and group-theoreticality.

Let C be a self-dual spherical fusion categories of rank 4 with non-trivial grading. We complete the classification of Grothendieck ring K(C) of C; that is, we prove that K(C) ≅= Fib⊗Z[Z2], where Fib is the Fibonacci fusion ring and Z[Z2] is the group ring on Z2. In particular, if C is braided, then it is equivalent to Fib⊠(Formula presented.) as f...

We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius-Perron dimension $p^nm$, where $p$ is a prime number, $m$ is a square-free natural number and ${\rm gcd}(p,m)=1$. We prove that if $n\leq 5$ then they are group-theoretical, and if in addition $m$ is prime with $m<p$ then they are also group-theo...

We classify braided $\mathbb{Z}_q$-extensions of pointed fusion categories, where $q$ is a prime number. As an application, we classify modular categories of Frobenius-Perron dimension $q^3$.

Let q be a prime number, k an algebraically closed field of characteristic 0, and H a non-trivial semisimple Hopf algebra of dimension 2q

About $39$ years ago, Kaplansky conjectured that the dimension of a semisimple Hopf algebra over an algebraically closed field of characteristic zero is divisible by the dimensions of its simple modules. Although it still remains open, some partial answers to this conjecture play an important role in classifying semisimple Hopf algebras. This paper...

We prove that integral modular categories of Frobenius-Perron dimension $pq^5$ are group-theoretical, where $p,q$ are distinct prime numbers. Combining this with previous results in the literature, integral modular categories of Frobenius-Perron dimension $pq^i$, $0\leq i\leq 5$, are group-theoretical. We also prove a sufficient and necessary condi...

Let k be a field of characteristic p > 0, and G be a finite group of order divisible by p. We prove that the almost split sequences of the quantum double D(kG) can be constructed from those of group algebras, where the groups run over all centralizer subgroups of representatives of conjugate classes of G. As a special case, we give an application t...

Let $k$ be an algebraically closedfield of characteristic zero. In this paper we consider an integral fusion category over $k$ in which the Frobenius-Perron dimensions of its simple objects are at most 3. We prove that such fusion category is of Frobenius type. In addition, we also prove that such fusion category is not simple.

Let k be an algebraically closed field of characteristic zero. In this paper we prove that fusion categories of Frobenius-Perron dimensions 84 and 90 are of Frobenius type. Combining this with previous results in the literature, we obtain that every weakly integral fusion category of Frobenius-Perron dimension less than 120 is of Frobenius type.

Let p, q be prime numbers with p(4) < q, and k an algebraically closed field of characteristic 0. We show that semisimple Hopf algebras of dimension p(2)q(2) can be constructed either from group algebras and their duals by means of extensions, or from Radford biproduct R#kG, where kG is the group algebra of group G of order p(2), R is a semisimple...

Let p,q be distinct prime numbers, and k an algebraically closed field of characteristic 0. Under certain restrictions on p,q, we discuss the structure of semisimple Hopf algebras of dimension p 2 q 2 . As an application, we obtain the structure theorems for semisimple Hopf algebras of dimension 9q 2 over k. As a byproduct, we also prove that odd-d...

Let p, q be distinct prime numbers, and k an algebraically closed field of characteristic 0. Under certain restrictions on p, q, we discuss the structure of semisimple Hopf algebras of dimension p 2q 2. As an application, we obtain the structure theorems for semisimple Hopf algebras of dimension 9q 2 over k. As a byproduct, we also prove that odd-d...

Let $q$ be a prime number, $k$ an algebraically closed field of characteristic 0, and $H$ a semisimple Hopf algebra of dimension $2q^3$. This paper proves that $H$ is always semisolvable. That is, such Hopf algebras can be obtained by (a number of) extensions from group algebras or duals of group algebras.

Let $p$ , $q$ be prime numbers with ${{p}^{2}}\,<\,q,\,n\,\in \,\mathbb{N}$ , and $H$ a semisimple Hopf algebra of dimension $p{{q}^{n}}$ over an algebraically closed field of characteristic 0. This paper proves that $H$ must possess one of the following two structures: (1) $H$ is semisolvable; (2) $H$ is a Radford biproduct $R\#kG$ , where $kG$ is...

Let $p,q$ be prime numbers with $p>q^3$, and $k$ an algebraically closed field of characteristic 0. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $pq^3$.

Let $k$ be an algebraically closed field of odd characteristic $p$, and let $D_n$ be the dihedral group of order $2n$ such that $p\mid 2n$. Let $D(kD_n)$ denote the quantum double of the group algebra $kD_n$. In this paper, we describe the structures of all finite dimensional indecomposable left $D(kD_n)$-modules, equivalently, of all finite dimens...

We consider the categories of modules over rings and categories of comodules over corings. By properties of split forks and coseparable corings, we get some sufficient conditions for the equivalence between the above two categories.

Let kG be a group algebra, and D(kG) its quantum double. We first prove that the structure of the Grothendieck ring of D(kG) can be induced from the Grothendieck ring of centralizers of representatives of conjugate classes of G. As a special case, we then give an application to the group algebra kD n , where k is a field of characteristic 2 and D...

Let H be a cosemisimple Hopf algebra over an algebraically closed field k of characteristic zero. We show that if H is of type l:1+m:p+1:q with p 2 <q, or of type 1:1+1:m+1:n in the sense of Larson and Radford, then H has the Frobenius property, that is, Kaplansky’s conjecture is true for these Hopf algebras.

Let H be a cosemisimple Hopf algebra over an algebraically closed field k of characteristic zero. This paper shows that if H is of type l:1+m:2+1:p, where p≠4, then H has the Frobenius property, that is, Kaplansky’s conjecture is true for these Hopf algebras.