具有Levi因子\frak{so}（3），\frak{sl}（2，R）的非半单李代数及其不变量

The subalgebras of the rank two symplectic Lie algebra

The semisimple subalgebras of the rank $2$ symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{C})$ are well-known, and we recently classified its Levi decomposable subalgebras. In this article, we classify the solvable subalgebras of $\mathfrak{sp}(4,\mathbb{C})$, up to inner automorphism. This completes the classification of the subalgebras of $\mathfrak{sp}(4,\mathbb{C})$. More broadly speaking, in completing the classification of the subalgebras of $\mathfrak{sp}(4,\mathbb{C})$ we have completed the classification of the subalgebras of the rank $2$ semisimple Lie algebras.

关于维数小于25的sympathetic李代数的不存在性

On the non-existence of sympathetic Lie algebras with dimension less than 25

本文研究 Levi 子代数单一时 sympathetic 李代数的最低维数，通过分析幂零根的模分解与半单导子，证明维数低于15不存在，且含4个单模时维数至少为25。

度量$n$ -李代数

Metric $n$-Lie Algebras

研究了复域$\mathbb C$上度量$n$ -李代数$\mathcal {G}$的结构.设$\mathcal {G}= \mathcal S\oplus {\mathcal R}$是Levi分解，其中$\mathcal R$是$\mathcal {G}$的根，$\mathcal S$是$\mathcal {G}$的强半单子代数.用$m(\mathcal {G})$表示不可分解度量$n$ -李代数的所有极小理想的个数，用$\mathcal R^\bot$表示$R$的正交补.我们得到以下结果。作为$\mathcal S$ -模，$\mathcal R^{\bot}$同构于$\mathcal {G} / \mathcal R.$的对偶模。$\mathcal {G}$上所有非退化不变对称双线性形式所张成的向量空间的维数等于$\mathcal {G}$上某些线性变换的向量空间的维数;这个维数大于或等于$m(\mathcal {G}) + 1$。$\mathcal G$中$\mathcal R$的中心化子等于所有极小理想之和;它是$\mathcal R^\bot$与$\mathcal {G}$的中心的直和。$\mathcal {G}$没有强半单理想的充要条件是$\mathcal R^\bot \subseteq \mathcal R$ .

Structure of H-(co)module Lie algebras

Let $L$ be a finite dimensional Lie algebra over a field of characteristic $0$. Then by the original Levi theorem, $L = B \oplus R$ where $R$ is the solvable radical and $B$ is some maximal semisimple subalgebra. We prove that if $L$ is an $H$-(co)module algebra for a finite dimensional (co)semisimple Hopf algebra $H$, then $R$ is $H$-(co)invariant and $B$ can be chosen to be $H$-(co)invariant too. Moreover, the nilpotent radical $N$ of $L$ is $H$-(co)invariant and there exists an $H$-sub(co)module $S\subseteq R$ such that $R=S\oplus N$ and $[B,S]=0$. In addition, the $H$-(co)invariant analog …

Nilpotent Decomposition of Solvable Lie Algebras

Semisimple Lie algebras have been completely classified by Cartan and Killing. The Levi theorem states that every finite dimensional Lie algebra is isomorphic to a semidirect sum of its largest solvable ideal and a semisimple Lie algebra. These focus the classification of solvable Lie algebras as one of the main challenges of Lie algebra research. One approach towards this task is to take a class of nilpotent Lie algebras and construct all extensions of these algebras to solvable ones. In this paper, we propose another approach, i.e., to decompose a solvable nonnilpotent Lie algebra to two nil…

通过派生括号从 $\mathfrak{gl}_{m|n}$ 构造的李代数的分类

Classification of Lie algebras constructed from $\mathfrak{gl}_{m|n}$ via Derived Bracket

本文通过奇元素 $B$ 生成的派生括号，对一般线性李超代数 $\mathfrak{gl}_{m|n}$ 构造的一族李代数进行了完全的结构和同构分类，证明了同构类型完全由秩 $r$ 和维数对决定，并给出了 Levi-Malcev 分解。

Derivations, gradings, actions of algebraic groups, and codimension growth of polynomial identities

Suppose a finite dimensional semisimple Lie algebra $\mathfrak g$ acts by derivations on a finite dimensional associative or Lie algebra $A$ over a field of characteristic $0$. We prove the $\mathfrak g$-invariant analogs of Wedderburn - Mal'cev and Levi theorems, and the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial identities with derivations of $A$. It turns out that for associative algebras the differential PI-exponent coincides with the ordinary one. Also we prove the analog of Amitsur's conjecture for finite dimensional associative algebras with an …

截断流李代数的范畴 $\mathcal{O}$

Category $\mathcal{O}$ for truncated current Lie algebras

本文研究截断流李代数的范畴 $\mathcal{O}$，给出通过 Levi 子代数低一阶情形的合成重数归纳计算 Verma 模中单模合成重数的方法，并将其表示为 Kazhdan–Lusztig 多项式在 1 处取值的整系数线性组合。

截断电流李代数的模表示

Modular Representations of Truncated current Lie algebras

本文研究正特征域上截断电流代数的结构与表示，分类半单与幂零元，证明 Morita 等价，给出单模维数界，发展 baby Verma 模理论，并在标准 Levi 型下完全分类单模，最后计算 Cartan 不变量。

$G$-表与海森堡李群余切丛偶上同调的泊松结构

$G$-tables and the Poisson structure of the even cohomology of cotangent bundle of the Heisenberg Lie group

本文定义了 $G$-(余)代数的 $G$-表及相关代数 $P(A)$，并利用该工具计算了 3 维海森堡李群余切丛偶上同调泊松结构的 $SL(2)$-表，揭示了其底层李代数结构及 Levi 因子的非平凡性。