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Hui-Xiang Chen

Publications and source records attributed to Hui-Xiang Chen.

At least 19 recordsLinked to original sources
arXiv2025

The Projective Class Rings of Drinfeld doubles of pointed rank one Hopf algebras

Let $\Bbbk$ be an algebraically closed field of characteristic $0$. In this paper, we study the Grothendieck ring $G_0(D(H_\mathcal{D}))$ and the projective class ring $r_p(D(H_\mathcal{D}))$ of the Drinfeld double $D(H_{\mathcal{D}})$ of the rank one pointed Hopf algebra $H_{\mathcal{D}}$. We analyze the tensor products of simple modules with simple modules, simple modules with indecomposable projective modules, and indecomposable projective modules with indecomposable projective modules, providing explicit decomposition rules in each case. Finally, we compute both the Grothendieck ring $G_0(D(H_\mathcal{D}))$ and the projective class ring $r_p(D(H_\mathcal{D}))$, and present these two rings in terms of generators and defining relations.

math.QA↗
arXiv2023

Representations of the small quasi-quantum group

In this paper, we study the representation theory of the small quantum group $\overline{U}_q$ and the small quasi-quantum group $\widetilde{U}_q$, where $q$ is a primitive $n$-th root of unity and $n>2$ is odd. All finite dimensional indecomposable $\widetilde{U}_q$-modules are described and classified. Moreover, the decomposition rules for the tensor products of $\widetilde{U}_q$-modules are given. Finally, we describe the structures of the projective class ring $r_p(\widetilde{U}_q)$ and the Green ring $r(\widetilde{U}_q)$. We show that $r(\overline{U}_q)$ is isomorphic to a subring of $r(\widetilde{U}_q)$, and the stable Green rings $r_{st}(\widetilde{U}_q)$ and $r_{st}(\overline{U}_q)$ are isomorphic.

math.QA↗
arXiv2023

Representations of Drinfeld Doubles of Radford Hopf algebras

In this article, we investigate the representations of the Drinfeld doubles $D(R_{mn}(q))$ of the Radford Hopf algebras $R_{mn}(q)$ over an algebraically closed field $\Bbbk$, where $m>1$ and $n>1$ are integers and $q\in\Bbbk$ is a root of unity of order $n$. Under the assumption ${\rm char}(\Bbbk)\nmid mn$, all the finite dimensional indecomposable modules over $D(R_{mn}(q))$ are displayed and classified up to isomorphism. The Auslander-Reiten sequences in the category of finite dimensional $D(R_{mn}(q))$-modules are also all displayed. It is shown that $D(R_{mn}(q))$ is of tame representation type.

math.QA↗
arXiv2021

Flow Instability Transferability Characteristics within a Reversible Pump Turbine (RPT) under Large Guide Vane Opening (GVO)

Reversible pump turbines are praised for their operational flexibility leading to their recent wide adoption within pumped storage hydropower plants. However, frequently imposed off-design operating conditions in these plants give rise to large flow instability within RPT flow zones, where the vaneless space (VS) between the runner and guide vanes is claimed to be the base. Recent studies have pointed out the possibility of these instabilities stretching to other flow zones causing more losses and subsequent machine operational performance degradation. This study therefore intends to investigate the VS flow instability, its propagation characteristics, and the effect of machine influx and runner blade number on the same. CFD-backed simulations are conducted on ten flow conditions spanning from turbine zone through runaway vicinities to turbine brake (OC1 to OC15), using three runner models with different blades (7BL, 8BL, and 9BL). While VS pressure pulsation amplitudes increased with runner blades number decrease, the continuously decreasing flow led to gradual VS pressure pulsation level drop within the Turbine zone before increasing to Runaway and dropping back to deep turbine brake zone. The effect of the same parameters on the transmission mode to VS upstream flow zones is more remarkable than the downstream flow zones.

physics.flu-dyn↗
arXiv2021

Representations of Hopf-Ore extensions of group algebras

In this paper, we study the representations of the Hopf-Ore extensions $kG(χ^{-1}, a, 0)$ of group algebra $kG$, where $k$ is an algebraically closed field. We classify all finite dimensional simple $kG(χ^{-1}, a, 0)$-modules under the assumption $|χ|=\infty$ and $|χ|=|χ(a)|<\infty$ respectively, and all finite dimensional indecomposable $kG(χ^{-1}, a, 0)$-modules under the assumption that $kG$ is finite dimensional and semisimple, and $|χ|=|χ(a)|$. Moreover, we investigate the decomposition rules for the tensor product modules over $kG(χ^{-1}, a, 0)$ when char$(k)$=0. Finally, we consider the representations of some Hopf-Ore extension of the dihedral group algebra $kD_n$, where $n=2m$, $m>1$ odd, and char$(k)$=0. The Grothendieck ring and the Green ring of the Hopf-Ore extension are described respectively in terms of generators and relations.

math.RA↗
arXiv2020

Dorroh extensions of algebras and coalgebras, II

In this paper, we study Dorroh extensions of bialgebras and Hopf algebras. Let $(H,I)$ be both a Dorroh pair of algebras and a Dorroh pair of coalgebras. We give necessary and sufficient conditions for $H\ltimes_dI$ to be a bialgebra and a Hopf algebra, respectively. We also describe all ideals of Dorroh extensions of algebras and subcoalgebras of Dorroh extensions of coalgebras and compute these ideals and subcoalgebras for some concrete examples.

math.RA↗
arXiv2020

Dorroh extensions of algebras and coalgebras, I

In this article, we study Dorroh extensions of algebras and Dorroh extensions of coalgebras. Their structures are described. Some properties of these extensions are presented. We also introduce the finite duals of algebras and modules which are not necessarily unital. Using these finite duals, we determine the dual relations between the two kinds of extensions.

math.RA↗
arXiv2019

The structures of Hopf $\ast$-algebra on Radford algebras

We investigate the structures of Hopf $\ast$-algebra on the Radford algebras over $\mathbb {C}$. All the $*$-structures on $H$ are explicitly given. Moreover, these Hopf $*$-algebra structures are classified up to equivalence.

math.RA↗
arXiv2018

Green ring of the category of weight modules over the Hopf-Ore extensions of group algebras

In this paper, we continue our study of the tensor product structure of category $\mathcal W$ of weight modules over the Hopf-Ore extensions $kG(χ^{-1}, a, 0)$ of group algebras $kG$, where $k$ is an algebraically closed field of characteristic zero. We first describe the tensor product decomposition rules for all indecomposable weight modules under the assumption that the orders of $χ$ and $χ(a)$ are different. Then we describe the Green ring $r(\mathcal W)$ of the tensor category $\mathcal W$. It is shown that $r(\mathcal W)$ is isomorphic to the polynomial algebra over the group ring $\mathbb{Z}\hat{G}$ in one variable when $|χ(a)|=|χ|=\infty$, and that $r(\mathcal W)$ is isomorphic to the quotient ring of the polynomial algebra over the group ring $\mathbb{Z}\hat{G}$ in two variables modulo a principle ideal when $|χ(a)|<|χ|=\infty$. When $|χ(a)|\le|χ|<\infty$, $r(\mathcal W)$ is isomorphic to the quotient ring of a skew group ring $\mathbb{Z}[X]\sharp\hat{G}$ modulo some ideal, where $\mathbb{Z}[X]$ is a polynomial algebra over $\mathbb{Z}$ in infinitely many variables.

math.RA↗
arXiv2018

Tensor product decomposition rules for weight modules over the Hopf-Ore extensions of group algebras

In this paper, we investigate the tensor structure of the category of finite dimensional weight modules over the Hopf-Ore extensions $kG(χ^{-1}, a, 0)$ of group algebras $kG$. The tensor product decomposition rules for all indecomposable weight modules are explicitly given under the assumptions that $k$ is an algebraically closed field of characteristic zero, and the orders of $χ$ and $χ(a)$ are the same.

math.RT↗
arXiv2017

Green rings of Drinfeld Doubles of Taft algebras

In this article, we investigate the representation ring (or Green ring) of the Drinfeld double $D(H_n(q))$ of the Taft algebra $H_n(q)$, where $n$ is an integer with $n>2$ and $q$ is a root of unity of order $n$. It is shown that the Green ring $r(D(H_n(q)))$ is a commutative ring generated by infinitely many elements subject to certain relations.

math.RA↗
arXiv2017

The Projective Class Rings of a family of pointed Hopf algebras of Rank two

In this paper, we compute the projective class rings of the tensor product $\mathcal{H}_n(q)=A_n(q)\otimes A_n(q^{-1})$ of Taft algebras $A_n(q)$ and $A_n(q^{-1})$, and its cocycle deformations $H_n(0,q)$ and $H_n(1,q)$, where $n>2$ is a positive integer and $q$ is a primitive $n$-th root of unity. It is shown that the projective class rings $r_p(\mathcal{H}_n(q))$, $r_p(H_n(0,q))$ and $r_p(H_n(1,q))$ are commutative rings generated by three elements, three elements and two elements subject to some relations, respectively. It turns out that even $\mathcal{H}_n(q)$, $H_n(0,q)$ and $H_n(1,q)$ are cocycle twist-equivalent to each other, they are of different representation types: wild, wild and tame, respectively.

math.RT↗
arXiv2015

Hopf-Ore Extensions and Hopf Algebras of Rank One

In this paper, we study pointed rank one Hopf algebras and Hopf-Ore extensions of group algebras, over an arbitrary field $k$. It is proved that the rank of a Hopf-Ore extension of a group algebra is one or two or infinite. It is also shown that an arbitrary (finite or infinite dimensional) pointed Hopf algebra of rank one is isomorphic to a quotient of a Hopf-Ore extension of its coradical, a group algebra. We classify the finite dimensional simple modules and describe a family of indecomposable modules over a Hopf-Ore extension $H=kG(χ, a,δ)$ and its quotient $H'$ of rank one, where $χ(a)\neq 1$, $G$ is an abelian group and $k$ is an algebraically closed field. The decomposition of the tensor products of two finite dimensional simple modules into a direct sum of indecomposable modules is given too. We also determine all simple objects and a family of indecomposable projective objects in the categories of all weight modules over $H$ and $H'$.

math.RA↗
arXiv2015

Representations of Hopf Ore extensions of group algebras and pointed Hopf algebras of rank one

In this paper, we study the representation theory of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one over an arbitrary field $k$. Let $H=kG(χ, a,\d)$ be a Hopf-Ore extension of $kG$ and $H'$ a rank one quotient Hopf algebra of $H$, where $k$ is a field, $G$ is a group, $a$ is a central element of $G$ and $χ$ is a $k$-valued character for $G$ with $χ(a)\neq 1$. We first show that the simple weight modules over $H$ and $H'$ are finite dimensional. Then we describe the structures of all simple weight modules over $H$ and $H'$, and classify them. We also consider the decomposition of the tensor product of two simple weight modules over $H'$ into the direct sum of indecomposable modules. Furthermore, we describe the structures of finite dimensional indecomposable weight modules over $H$ and $H'$, and classify them. Finally, when $χ(a)$ is a primitive $n$-th root of unity for some $n>2$, we determine all finite dimensional indecomposable projective objects in the category of weight modules over $H'$.

math.RT↗
arXiv2014

The Green Ring of Drinfeld Double $D(H_4)$

In this paper, we study the Green ring (or the representation ring) of Drinfeld quantum double $D(H_4)$ of Sweedler's 4-dimensional Hopf algebra $H_4$. We first give the decompositions of the tensor products of finite dimensional indecomposable modules into the direct sum of indecomposable modules over $D(H_4)$. Then we describe the structure of the Green ring $r(D(H_4))$ of $D(H_4)$ and show that $r(D(H_4))$ is generated, as a ring, by infinitely many elements subject to a family of relations.

math.RT↗