非分次Witt李代数上的单尖顶模分类
Classification of Simple Cuspidal Modules over Nongraded Witt Lie Algebras
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中文总结 AI 辅助
针对非分次Witt李代数，先确定Weyl型代数上单模P与单gl_n-模V的张量模T(P,V)为单模的条件，再通过代数同构证明所有单尖顶模均为特定张量模的单商，完成其分类。
中文摘要 AI 辅助
对于正整数$n$，令$A_n=\boldsymbol{\textup{C}}[t_1^{\boldsymbol{\textup{±1}}},\boldsymbol{\textup{…}},t_n^{\boldsymbol{\textup{±1}}},x_1,\boldsymbol{\textup{…}},x_n]$，$\boldsymbol{\textup{g}}_n=\bigoplus_{i=1}^n A_nd_i$，其中$d_i=t_i\frac{\boldsymbol{\textup{∂}}}{\boldsymbol{\textup{∂}}t_i}+\frac{\boldsymbol{\textup{∂}}}{\boldsymbol{\textup{∂}}x_i}$。首先确定张量模$T(P,V)=P\boldsymbol{\textup{⊗}}V$为单模的条件，其中$P$是Weyl型代数$D_n$上的单模，$V$是单$\boldsymbol{\textup{gl}}_n$-模；随后证明典范代数同构$A_n\boldsymbol{\textup{#}}U(\boldsymbol{\textup{g}}_n)\boldsymbol{\textup{≅}}D_n\boldsymbol{\textup{⊗}}U(\boldsymbol{\textup{m}}_{\boldsymbol{\textup{1}},\boldsymbol{\textup{0}}}\boldsymbol{\textup{Δ}})$，并利用该同构证明所有单尖顶$\boldsymbol{\textup{g}}_n$-模都同构于某$T(A_n(\boldsymbol{\textup{λ}}),V)$的单商，其中$V$是有限维单$\boldsymbol{\textup{gl}}_n$-模。
英文摘要
For a positive integer $n$, let $A_n=\mathbb{C}[t_1^{\pm1},\ldots,t_n^{\pm1},x_1,\ldots,x_n]$ and $\mathfrak{g}_n=\bigoplus_{i=1}^n A_nd_i$, where $d_i=t_i\frac{\partial}{\partial t_i} +\frac{\partial}{\partial x_i}$. We first determine when the tensor module $T(P,V)=P\otimes V$ is simple, where $P$ is a simple module over the Weyl type algebra $D_n$ and $V$ is a simple $\mathfrak{gl}_n$-module. We then prove a canonical algebra isomorphism $A_n\#U(\mathfrak{g}_n)\cong D_n\otimes U(\mathfrak{m}_{\mathbf{1},\mathbf{0}}Δ)$, and use it to show that every simple cuspidal $\mathfrak{g}_n$-module is isomorphic to a simple quotient of some $T(A_n(λ),V)$, where $V$ is a finite-dimensional simple $\mathfrak{gl}_n$-module.
