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Gongxiang Liu

Publications and source records attributed to Gongxiang Liu.

At least 37 records · Page 2Linked to original sources
arXiv2022

The Grothendieck algebras of certain smash product semisimple Hopf algebras

Let $H$ be a semisimple Hopf algebra over an algebraically closed field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$ and $p\nmid 2\dim_{\mathbbm{k}}(H)$. In this paper, we consider the smash product semisimple Hopf algebra $H\#\mathbbm{k}G$, where $G$ is a cyclic group of order $n:=2\dim_{\mathbbm{k}}(H)$. Using irreducible representations of $H$ and those of $\mathbbm{k}G$, we determine all non-isomorphic irreducible representations of $H\#\mathbbm{k}G$. There is a close relationship between the Grothendieck algebra $(G_0(H\#\mathbbm{k}G)\otimes_{\mathbb{Z}}\mathbbm{k},*)$ of $H\#\mathbbm{k}G$ and the Grothendieck algebra $(G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k},*)$ of $H$. To establish this connection, we endow with a new multiplication operator $\star$ on $G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k}$ and show that the Grothendieck algebra $(G_0(H\#\mathbbm{k}G)\otimes_{\mathbb{Z}}\mathbbm{k},\ast)$ is isomorphic to the direct sum of $(G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k},*)^{\oplus\frac{n}{2}}$ and $(G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k},\star)^{\oplus\frac{n}{2}}$.

math.RT↗
arXiv2022

Higher Frobenius-Schur indicators for semisimple Hopf algebras in positive characteristic

Let $H$ be a semisimple Hopf algebra over an algebraically closed field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. We show that the antipode $S$ of $H$ satisfies the equality $S^2(h)=\mathbf{u}h\mathbf{u}^{-1}$, where $h\in H$, $\mathbf{u}=S(Λ_{(2)})Λ_{(1)}$ and $Λ$ is a nonzero integral of $H$. The formula of $S^2$ enables us to define higher Frobenius-Schur indicators for the Hopf algebra $H$. This generalizes the notions of higher Frobenius-Schur indicators from the case of characteristic 0 to the case of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. These indicators defined here share some properties with the ones defined over a field of characteristic 0. Especially, all these indicators are gauge invariants for the tensor category Rep$(H)$ of finite dimensional representations of $H$.

math.RT↗
arXiv2022

Invariants from the Sweedler power maps on integrals

For a finite-dimensional Hopf algebra $A$ with a nonzero left integral $Λ$, we investigate a relationship between $P_n(Λ)$ and $P_n^J(Λ)$, where $P_n$ and $P_n^J$ are respectively the $n$-th Sweedler power maps of $A$ and the twisted Hopf algebra $A^J$. We use this relation to give several invariants of the representation category Rep$(A)$ considered as a tensor category. As applications, we distinguish the representation categories of 12-dimensional pointed nonsemisimple Hopf algebras. Also, these invariants are sufficient to distinguish the representation categories Rep$(K_8)$, Rep$(\kk Q_8)$ and Rep$(\kk D_4)$, although they have been completely distinguished by their Frobenius-Schur indicators. We further reveal a relationship between the right integrals $λ$ in $A^*$ and $λ^J$ in $(A^J)^*$. This can be used to give a uniform proof of the remarkable result which says that the $n$-th indicator $ν_n(A)$ is a gauge invariant of $A$ for any $n\in \mathbb{Z}$. We also use the expression for $λ^J$ to give an alternative proof of the known result that the Killing form of the Hopf algebra $A$ is invariant under twisting. As a result, the dimension of the Killing radical of $A$ is a gauge invariant of $A$.

math.QA↗
arXiv2021

Quasitriangular Hopf algebras of dimension pq^2

Let p and q be distinct odd primes and assume k is an algebraically closed field of characteristic zero. We classify all quasitriangular Hopf algebras of dimension pq^2 over k, which are not simple as Hopf algebras. Moreover, we obtained all quasitriangular structures on these Hopf algebras.

math.QA↗
arXiv2020

On the Antipode of Hopf Algebras with the Dual Chevalley Property

In this paper, we study the antipode of a finite-dimensional Hopf algebra $H$ with the dual Chevalley property and obtain an annihilation polynomial for its antipode $S$. The annihilation polynomial is determined by the exponent $N$ of the coradical and the Loewy length. In particular, the order of $S^2$ divides $N$ in characteristic $0$. Moreover, we get two characterizations of the quasi-exponent.

math.RA↗
arXiv2020

On the quasitriangular structures of abelian extensions of $Z_2$

The aim of this paper is to study quasitriangular structures on a class of semisimple Hopf algebras constructed through abelian extensions of $Z_2$ for an abelian group $G$. We prove that there are only two forms of them. Using such description together with some other techniques, we get a complete list of all universal $\mathcal{R}$-matrices on some Hopf algebras.

math.QA↗
arXiv2018

A classification result on prime Hopf algebras of GK-dimension one

In this paper, we classify all prime Hopf algebras $H$ of GK-dimension one satisfying the following two conditions: 1) $H$ has a 1-dimensional representation of order PI.deg$(H)$ and 2) the invariant components of $H$ with respect to this 1-dimensional representation are domains (see Section 2 for related definitions). As consequences, 1) a number of new Hopf algebras of GK-dimension one are found and some of them are not pointed, 2) we give a partial answer to a question posed by Brown and Zhang and 3) some new kinds of finite-dimensional Hopf algebras are found.

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arXiv2016

Finite quasi-quantum groups of diagonal type

The goal of the present paper is to classify an interesting class of elementary quasi-Hopf algebras, or equivalently, finite-dimensional pointed Majid algebras. By a Tannaka-Krein type duality, this determines a big class of pointed finite tensor categories. Based on some interesting observations of normalized 3-cocycles on finite abelian groups, we elucidate an explicit connection between our objective pointed Majid algebras and finite-dimensional pointed Hopf algebras over finite abelian groups. With a help of this connection and the successful theory of diagonal Nichols algebras over abelian groups, we provide a conceptual classification of finite-dimensional graded pointed Majid algebras of diagonal type. Some efficient methods of construction are also given.

math.QA↗
arXiv2015

Finite quasi-quantum groups of rank two

This is a contribution to the structure theory of finite pointed quasi-quantum groups. We classify all finite-dimensional connected graded pointed Majid algebras of rank two which are not twist equivalent to ordinary pointed Hopf algebras.

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arXiv2014

The braided monoidal structures on a class of linear Gr-categories

A linear Gr-category is a category of finite-dimensional vector spaces graded by a finite group together with natural tensor product. We classify the braided monoidal structures of a class of linear Gr-categories via explicit computations of the 3-cocycles and quasi-bicharacters of finite abelian groups which are direct product of two cyclic groups.

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arXiv2014

On Braided Linear Gr-categories

We provide explicit and unified formulae for the normalized 3-cocycles on arbitrary finite abelian groups. As an application, we compute all the braided monoidal structures on linear Gr-categories.

math.CT↗
arXiv2014

Graded elementary quasi-Hopf algebras of tame representation type

The class of graded elementary quasi-Hopf algebras of tame type is classified. Combining with our previous work [19], this completes the trichotomy for such class of algebras according to their representation types. In addition, new examples of genuine elementary quasi-Hopf algebras, and accordingly finite pointed tensor categories, are provided.

math.QA↗
arXiv2014

Representations of the small quasi-quantum group ${Q}\mathbf{u}_{q}(\mathfrak{sl}_{2})$

The quasi-Frobenius-Lusztig kernel ${Q}\mathbf{u}_{q}(\mathfrak{sl}_{2})$ associated with $\mathfrak{sl}_{2}$ has been constructed in \cite{Liu}. In this paper we study the representations of this small quasi-quantum group. We give a complete list of non-isomorphic indecomposables and the tensor product decomposition rules for simples and projectives. A description of the Grothendieck ring is provided.

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arXiv2012

On Total Frobenius-Schur Indicators

We define total Frobenius-Schur indicator for each object in a spherical fusion category $C$ as a certain canonical sum of its higher indicators. The total indicators are invariants of spherical fusion categories. If $C$ is the representation category of a semisimple quasi-Hopf algebra $H$, we prove that the total indicators are non-negative integers which satisfy a certain divisibility condition. In addition, if $H$ is a Hopf algebra, then all the total indicators are positive. Consequently, the positivity of total indicators is a necessary condition for a quasi-Hopf algebra being gauge equivalent to a Hopf algebra. Certain twisted quantum doubles of finite groups and some examples of Tambara-Yamagami categories are discussed for the sufficiency of this positivity condition.

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