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Shilin Yang

Publications and source records attributed to Shilin Yang.

10 recordsLinked to original sources
arXiv2024

PBW-deformations of smash products involving Hopf algebra of Kac-Paljutkin type

Let $H_{2n^2}$ be the Kac-Paljutkin type Hopf algebra of dimension $2n^2$, $A$ its graded Koszul Artin-Schelter regular $H_{2n^2}$-module algebra of dimension $2$, $A^!$ the Koszul dual of $A$, and $A^{\mathrm{op}}_c$ the braided-opposite algebra of $A$. This paper describes $(0, 1)$-degree PBW-deformations of the smash product $A \sharp H_{2n^2}$ and those of $A^! \sharp\, H_{2n^2}$ under the condition that the Koszul dual $A^!$ of $A$ is also an $H_{2n^2}$-module algebra. Also, $0$-degree PBW-deformations of $(A \otimes^c A^{\mathrm{op}}_c) \sharp\, H_{2n^2}$ are explored, where $A \otimes^c A^{\mathrm{op}}_c$ is the associated braided tensor product algebra.

math.RA↗
arXiv2024

On the Higher-Rank Askey-Wilson Algebras

In the paper, the algebra $\mathscr{A}(n)$, which is generated by an upper triangular generating matrix with triple relations, is introduced. It is shown that there exists an isomorphism between the algebra $\mathscr{A}(n)$ and the higher-rank Askey-Wilson algebra $\mathfrak{aw}(n)$ introduced by Cramp\'e et al. Furthermore, we establish a series of automorphisms of $\mathscr{A}(n)$, which satisfy braid group relations and coincide with those in $\mathfrak{aw}(n)$.

math.QA↗
arXiv2018

Grothendieck Rings of 2$n^2$ dimensional Hopf Algebras $H_{2n^2}$

In this paper, we construct the Grothendieck ring of a class of 2$n^2$-dimension semisimple Hopf Algebras $H_{2n^2}$, which can be viewed as a generalization of the 8-dimension Kac-Paljutkin Hopf algebra $K_8$. All irreducible $H_{2n^2}$-modules are classified. Furthermore, we describe the Grothendieck ring $r(H_{2n^2})$ by generators and relations explicitly.

math.RA↗
arXiv2018

On $4n$-dimensional neither pointed nor semisimple Hopf algebras and the associated weak Hopf algebras

For a class of neither pointed nor semisimple Hopf algebras $H_{4n}$ of dimension $4n$, it is shown that they are quasi-triangular, which universal $R$-matrices are described. The corresponding weak Hopf algebras $\mathfrak{w}H_{4n}$ and their representations are constructed. Finally, their duality and their Green rings are established by generators and relations explicitly. It turns out that the Green rings of the associated weak Hopf algebras are not commutative even if the Green rings of $H_{4n}$ are commutative.

math.RA↗
arXiv2011

Generalized McKay Quivers, Root System and Kac-Moody Algebras

Let $Q$ be a finite quiver and $G\subseteq\Aut(\mathbbm{k}Q)$ a finite abelian group. Assume that $\hat{Q}$ and $Γ$ is the generalized Mckay quiver and the valued graph corresponding to $(Q, G)$ respectively. In this paper we discuss the relationship between indecomposable $\hat{Q}$-representations and the root system of Kac-Moody algebra $\mathfrak{g}(Γ)$. Moreover, we may lift $G$ to $\bar{G}\subseteq\Aut(\mathfrak{g}(\hat{Q}))$ such that $\mathfrak{g}(Γ)$ embeds into the fixed point algebra $\mathfrak{g}(\hat{Q})^{\bar{G}}$ and $\mathfrak{g}(\hat{Q})^{\bar{G}}$ as $\mathfrak{g}(Γ)$-module is integrable.

math.RT↗
arXiv2010

Skew group algebras of deformed preprojective algebras

Suppose that $Q$ is a finite quiver and $G\subseteq \Aut(Q)$ is a finite group, $k$ is an algebraic closed field whose characteristic does not divide the order of $G$. For any algebra $Λ=kQ/{\mathcal {I}}$, $\mathcal {I}$ is an arbitrary ideal of path algebra $kQ$, we give all the indecomposable $ΛG$-modules from indecomposable $Λ$-modules when $G$ is abelian. In particular, we apply this result to the deformed preprojective algebra $Π_{Q}^λ$, and get a reflection functor for the module category of $Π_{Q}^λG$. Furthermore, we construct a new quiver $Q_{G}$ and prove that $Π_{Q}^λG$ is Morita equivalent to $Π_{Q_{G}}^η$ for some $η$.

math.RT↗
arXiv2005

Quantum groups and double quiver algebras

For a finite dimensional semisimple Lie algebra ${\frak{g}}$ and a root $q$ of unity in a field $k,$ we associate to these data a double quiver $\bar{\cal{Q}}.$ It is shown that a restricted version of the quantized enveloping algebras $U_q(\frak g)$ is a quotient of the double quiver algebra $k\bar{\cal Q}.$

math.QA↗
arXiv2005

Weak Hopf algebras corresponding to Cartan matrices

We replace the group of group-like elements of the quantized enveloping algebra $U_q({\frak{g}})$ of a finite dimensional semisimple Lie algebra ${\frak g}$ by some regular monoid and get the weak Hopf algebra ${\frak{w}}_q^{\sf d}({\frak g})$. It is a new subclass of weak Hopf algebras but not Hopf algebras. Then we devote to constructing a basis of ${\frak{w}}_q^{\sf d}({\frak g})$ and determine the group of weak Hopf algebra automorphisms of ${\frak{w}}_q^{\sf d}({\frak g})$ when $q$ is not a root of unity.

math.QA↗