A duality between non-compact semisimple symmetric pairs and commutative compact semisimple symmetric triads and its general theory

基于组合零点定理的交换半单群代数Wedderburn分解

Wedderburn decomposition of commutative semisimple group algebras using the Combinatorial Nullstellensatz

该研究从几何角度出发，利用Combinatorial Nullstellensatz得到有限阿贝尔群及有限域上半单交换群代数的Wedderburn分解的单分量，完善了相关代数结构的刻画。

On maximal commutative subalgebras of Poisson algebras associated with involutions of semisimple Lie algebras

For any involution $σ$ of a semisimple Lie algebra $\mathfrak g$, one constructs a non-reductive Lie algebra $\mathfrak k$, which is called a $\mathbb Z_2$-contraction of $\mathfrak g$. In this paper, we attack the problem of describing maximal commutative subalgebras of the Poisson algebra $S(\mathfrak k)$. This is closely related to the study of the coadjoint representation of $\mathfrak k$ and the set, $\mathfrak k^*_{reg}$, of the regular elements of $\mathfrak k^*$. By our previous results, in the context of $\mathbb Z_2$-contractions, the argument shift method provides maximal commutativ…

半单李群表示的交换化身

Commutative avatars of representations of semisimple Lie groups

本研究构造了附属于复半单李群有限维不可约表示的基里洛夫代数大交换子代数，证明其为群分类空间上同调上的交换有限平坦代数，与仿射舒伯特簇等变相交上同调同构并赋予其新环结构，还可实现Lusztig q权重数多项式的环化。

关于 $4n$ 维非指针非半单霍普夫代数及其相关的弱霍普夫代数

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

本文研究了 $4n$ 维非指针非半单霍普夫代数及其相关弱霍普夫代数的结构，揭示了其准三角性、 $R$-矩阵描述及绿色环的非可交换性。

Cocycle deformations and Galois objects of semisimple Hopf algebras of dimension $16$

In this article, we determine cocycle deformations and Galois objects of non-commutative and non-cocommutative semisimple Hopf algebras of dimension $16$. We show that these Hopf algebras are pairwise twist inequivalent mainly by calculating their higher Frobenius-Schur indicators, and that except three Hopf algebras which are cocycle deformations of dual group algebras, none of them admit non-trivial cocycle deformations.

Reductive subalgebras of semisimple Lie algebras and Poisson commutativity

Let $\mathfrak g$ be a semisimple Lie algebra, $\mathfrak h\subset\mathfrak g$ a reductive subalgebra such that $\mathfrak h^\perp$ is a complementary $\mathfrak h$-submodule of $\mathfrak g$. In 1983, Bogoyavlenski claimed that one obtains a Poisson commutative subalgebra of the symmetric algebra ${\mathcal S}(\mathfrak g)$ by taking the subalgebra ${\mathcal Z}$ generated by the bi-homogeneous components of all $H\in{\mathcal S}(\mathfrak g)^{\mathfrak g}$. But this is false, and we present a counterexample. We also provide a criterion for the Poisson commutativity of such subalgebras ${\mat…