| 摘要: |
| 特殊线性李超代数sl(0,n) 是一类具有特定标准基的特殊线性李超代数子代数. 素特征域上的有限维 Cartan 型模李超代数 S(m,n,t) 是广义 Witt 型李超代数 W(m,n,t) 的一个子代数.本文采用上同调理论与模分解相结合的方法,研究 sl(0,n) 在 S(m,n,t) 上的中心化子结构. 我们建立了 S(m,n,t) 作为 sl(0,n)-模的典范直和分解, 并以此为基础构造了一个基于模分解与上同调的系统分析框架. 基于该框架,对各子模的零维上同调群进行了计算,从而完整刻画了中心化子 C_S(sl(0,n)) 的结构,并证明其具有与多项式代数 O(m) 张量积的代数形式. 所得的结构定理不仅给出了中心化子的具体表达式, 而且从表示论的角度揭示了其内在本质,为进一步研究 Cartan 型李超代数的中心化子提供了一种可推广的理论范式. |
| 关键词: 模李超代数 中心化子 特殊线性李超代数sl(0,n) 模分解 上同调 |
| DOI： |
| 分类号:O152.5 |
| 基金项目:黑龙江省博士后资助经费(LBH-Z23068) |
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| The Decomposition of S(m,n,t) as a sl(0,n)-Module and Its Centralizer Structure over Fields of Prime Characteristic |
|
Zheng Keli
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| Abstract: |
| The special linear Lie superalgebra sl(0,n) is a subclass of special linear Lie superalgebras characterized by a specific set of standard bases. The finite-dimensional modular Lie superalgebra S(m,n,t) of Cartan type over fields of prime characteristic is a subalgebra of the generalized Witt-type Lie superalgebra W(m,n,t) . Employing methods from cohomology theory and module decomposition, this work investigates the centralizer structure of sl(0,n) on S(m,n,t) . A canonical decomposition of S(m,n,t) as a sl(0,n) -module is established, forming the foundation of an analytical framework that integrates module decomposition with cohomological techniques. Within this framework, the zero-dimensional cohomology groups of the constituent submodules are computed, leading to a complete description of the centralizer C_S(sl(0,n)) and a proof of its tensor product structure with the polynomial algebra O(m). This structural theorem not only furnishes an explicit expression for the centralizer but also elucidates its representation-theoretic nature, thereby providing a transferable theoretical paradigm for studying centralizers of Cartan-type Lie superalgebras. |
| Key words: Modular Lie superalgebra Centralizer Special linear Lie superalgebra $\mathfrak{sl}(0,n)$ Module decomposition Cohomology |
