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Jingcheng Dong

Publications and source records attributed to Jingcheng Dong.

32 records · Page 2Linked to original sources
arXiv2016

Integral almost square-free modular categories

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 $n\leq 5$ or $m$ is prime with $m<p$ then they are group-theoretical. This generalizes several results in the literature and gives a partial answer to the question posed by the first author and H. Tucker. As an application, we prove that an integral modular category whose Frobenius-Perron dimensions is odd and less than $1125$ is group-theoretical.

math.CT↗
arXiv2015

Integral modular categories of Frobenius-Perron dimension $pq^n$

Integral modular categories of Frobenius-Perron dimension $pq^n$, where $p$ and $q$ are primes, are considered. It is already known that such categories are group-theoretical in the cases of $0 \leq n \leq 4$. In the general case we determine that these categories are either group theoretical or contain a Tannakian subcategory of dimension $q^i$ for $i>1$. We then show that all integral modular categories $\mathcal{C}$ with $\mathrm{FPdim}(\mathcal{C})=pq^5$ are group-theoretical, and, if in addition $p<q$, all with $\mathrm{FPdim}(\mathcal{C})=pq^6$ or $pq^7$ are group-theoretical. In the process we generalize an existing criterion for an integral modular category to be group-theoretical.

math.QA↗
arXiv2014

Frobenius property for fusion categories of small integral dimension

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.

math.QA↗
arXiv2014

On Kaplansky's sixth conjecture

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 focuses on the recent development of Kaplansky's sixth conjecture and its applications in classifying semisimple Hopf algebras.

math.RA↗
arXiv2014

Existence of Tannakian subcategories and its applications

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.

math.CT↗
arXiv2013

On integral fusion categories with low-dimensional simple objects

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.

math.QA↗
arXiv2013

On semisimple Hopf algebras of dimension $2q^3$

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.

math.RA↗
arXiv2012

Semisimple Hopf algebras of dimension $2q^3$

Let $q$ be a prime number, $k$ an algebraically closed field of characteristic 0, and $H$ a non-trivial semisimple Hopf algebra of dimension $2q^3$. This paper proves that $H$ can be constructed either from group algebras and their duals by means of extensions, or from Radford's biproduct $H\cong R#kG$, where $kG$ is the group algebra of $G$ of order 2, $R$ is a semisimple Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^3$.

math.RA↗
arXiv2011

The Representations of Quantum Double of Dihedral Groups

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 dimensional indecomposable Yetter-Drinfeld $kD_n$-modules, and classify them.

math.QA↗
arXiv2011

Structure of semisimple Hopf algebras of dimension $p^2q^2$, II

Let $k$ be an algebraically closed field of characteristic $0$. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $p^2q^2$ over $k$, where $p,q$ are prime numbers with $p^2<q$. As an application, we also obtain the structure theorems for semisimple Hopf algebras of dimension $9p^2$ and $25q^2$ for all primes $3\neq p$ and $5\neq q$.

math.RA↗
arXiv2010

Structure of semisimple Hopf algebras of dimension $p^2q^2$

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 $p^2q^2$ 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 Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^2$. As an application, the special case that the structure of semisimple Hopf algebras of dimension $4q^2$ is given.

math.RA↗