arXiv subjects

Yuxing Shi

Publications and source records attributed to Yuxing Shi.

10 recordsLinked to original sources

arXiv2023

On Nichols algebras associated to Near-rack solutions of the Yang-Baxter equation

Let $(X, r)$ be any set-theoretical non-degenerate solution of the Yang-Baxter equation and $(X, \tilde r)$ be the derived solution of $(X, r)$. As for any braided vector space $(W_{X, r}, c)$ associated to $(X, r)$, is it possible to find some braided vector space $(W_{X, \tilde r}, \tilde c)$ which is t-equivalent to $(W_{X, r}, c)$? In case that $(X, r)$ is a near-rack solution, we give a sufficient condition to make an affirmative answer to the question. Examples of t-equivalence are constructed, hence finite dimensional Nichols algebras are obtained. In particular, all finite dimensional Nichols algebras associated to involutive near-rack solutions are classified.

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arXiv2023

Finite-dimensional Nichols algebras over the Suzuki algebras III: simple Yetter-Drinfeld modules

In this paper, we continue to investigate finite-dimensional Nichols algebras over simple Yetter-Drinfeld modules of the Suzuki algebras $A_{N\, n}^{μλ}$. It is finished for the case $A_{N\, 2n}^{μλ}$. As for the case $A_{N\, 2n+1}^{μλ}$, it boils down to the long-standing open problem: calculate dimensions of Nichols algebras of dihedral rack type $\Bbb D_{2n+1}$. It is interesting to see that the Suzuki algebras are of set-theoretical type. We pose some question or problems for our future research. In particular, we are curious about how to generalize the correspondence between braidings of rack type and group algebras to braidings and Hopf algebras of set-theoretical type.

math.QA↗

arXiv2023

Automorphism group of Suzuki's Hopf algebra

In this paper, we calculate explicitly automorphism group of the Suzuki's Hopf algebra $A_{Nn}^{\mu\lambda}$ by viewing Yetter-Drinfeld modules as invariants of Hopf algebra automorphisms.

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arXiv2022

On some classification of finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}^*$ of Kashina

Let $H$ be the dual of $16$-dimensional nontrivial semisimple Hopf algebra $H_{b:1}$ in the classification work of Kashina \cite{K00}. We completely determine all finite-dimensional Nichols algebras satisfying $\mathcal{B}(N)\cong \bigotimes_{i\in I}\mathcal{B}(N_i)$, where $N=\bigoplus_{i\in I}N_i$, each $N_i$ is a simple object in $_H^H\mathcal{YD}$. Under this assumption, we classify all those Hopf algebras of finite-dimensional growth from the semisimple Hopf algebra $H$ via the relevant Nichols algebras $\mathcal B(N)$.

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arXiv2022

Multinomial expansion and Nichols algebras associated to non-degenerate involutive solutions of the Yang-Baxter equation

In this paper, we investigate the Nichols algebra $\mathfrak{B}(W_{X,r})$ associated to any non-degenerate involutive solution $(X, r)$ of the Yang-Baxter equation. Infinite examples of finite dimensional Nichols algebras are obtained, including those of dimension $n^m$ with $m$, $n\in\Bbb Z^{\geq 2}$. It turns out that the Nichols algebra $\mathfrak{B}(W_{X, r})$ has interesting relations with multinomial expansion. This is a generalization of the work in arXiv:2103.06489, which built a connection between the Nichols algebras of squared dimension and Pascal's triangle.

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arXiv2021

Finite dimensional Nichols algebras over Suzuki algebra I: simple Yetter-Drinfeld modules of $A_{N\,2n}^{μλ}$

The Suzuki algebra $A_{Nn}^{μλ}$ was introduced by Suzuki Satoshi in 1998, which is a class of cosemisimple Hopf algebras. It is not categorically Morita-equivalent to a group algebra in general. In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over the Suzuki algebra $A_{N\,2n}^{μλ}$ and investigates the Nichols algebras over those simple Yetter-Drinfeld modules. The involved finite dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1\times A_1$, $A_2$, $A_2\times A_2$, Super type ${\bf A}_{2}(q;I_2)$ and the Nichols algebra ufo(8). There are $64$, $4m$ and $m^2$-dimensional Nichols algebras of non-diagonal type over $A_{N\,2n}^{μλ}$. The $64$-dimensional Nichols algebras are of dihedral rack type $\Bbb{D}_4$. The $4m$ and $m^2$-dimensional Nichols algebras $\mathfrak{B}(V_{abe})$ discovered first by Andruskiewitsch and Giraldi can be realized in the category of Yetter-Drinfeld modules over $A_{Nn}^{μλ}$. By using a result of Masuoka, we prove that $\dim\mathfrak{B}(V_{abe})=\infty$ under the condition $b^2=(ae)^{-1}$, $b\in\Bbb{G}_{m}$ for $m\geq 5$.

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arXiv2021

Finite dimensional Nichols algebras over Suzuki algebra II: over simple Yetter-Drinfeld modules of $A_{N\,2n+1}^{μλ}$

In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over Suzuki algebra $A_{N\,2n+1}^{μλ}$ and investigates the Nichols algebras over those irreducible Yetter-Drinfeld modules. The finite dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1\times A_1$, $A_2$, Super type ${A}_{2}(q;I_2)$ and the Nichols algebra $\mathfrak{ufo}(8)$. And the involved finite dimensional Nichols algebras of non-diagonal type are $12$, $4m$ and $m^2$ dimensional. The left three unsolved cases are set as open problems.

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arXiv2021

The Nichols algebra $\mathfrak{B}(V_{abe})$ and a class of combinatorial numbers

We investigate the Nichols algebra $\mathfrak{B}(V_{abe})$ which are from the Yetter-Drinfeld category of Suzuki algebras. The $4n$ and $n^2$ dimensional Nichols algebras, first appeared in \cite{Andruskiewitsch2018}, are obtained again via a different method. And the connection between the Nichols algebra $\mathfrak{B}(V_{abe})$ and a class of combinatorial numbers on the subgroups of symmetric groups is established.

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arXiv2017

Finite dimensional Hopf algebras over Kac-Paljutkin algebra $H_8$

Let $H_8$ be the neither commutative nor cocommutative semisimple eight dimensional Hopf algebra, which is also called Kac-Paljutkin algebra \cite{MR0208401}. All simple Yetter-Drinfel'd modules over $H_8$ are given. As for simple objects and direct sums of two simple objects in ${}_{H_8}^{H_8}\mathcal{YD}$, we calculated dimensions for the corresponding Nichols algebras, except four semisimple cases which are generally difficult. Under the assumption that the four undetermined Nichols algebras are all infinite dimensional, we determine all the finite dimensional Nichols algebras over $H_8$. It turns out that the already known finite dimensional Nichols algebras are all diagonal type. In fact, they are Cartan types $A_1$, $A_2$, $A_2\times A_2$, $A_1\times \cdots \times A_1$, and $A_1\times \cdots \times A_1\times A_2$. By the way, we calculate Gelfand-Kirillov dimensions for some Nichols algebras. As an application, we obtain five families of new finite dimensional Hopf algebras over $H_8$ according to the lifting method.

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arXiv2014

On the Center of Two-parameter Quantum Groups $U_{r,s}(\mathfrak{so}_{2n+1})$

The paper mainly considers the center of two-parameter quantum groups $U_{r,s}(\mathfrak{so}_{2n+1})$ via an analogue of the Harish-Chandra homomorphism. In the case when $n$ is odd, the Harish-Chandra homomorphism is not injective in general. When $n$ is even, the Harish-Chandra homomorphism is injective and the center of two-parameter quantum groups $U_{r,s}{(\mathfrak{so}_{2n+1})}$ is described, up to isomorphism.

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