论文标题
$ \ mathfrak {sl} _ {2} $上的takiff lie代数上的简单模块
Simple modules over the Takiff Lie algebra for $\mathfrak{sl}_{2}$
论文作者
论文摘要
在本文中，我们在某些情况下构建，调查并在Takiff $ \ Mathfrak {Sl} _ {2} $上对几个新类（简单）模块进行了分类。更确切地说，我们首先明确构建和分类，直到同构为同构，所有模块在Takiff $ \ Mathfrak {sl} _ {2} $是$ u \ left（\ overline {\ mathfrak {h h}}}} \ right（\ mathfrak {\ mathfrak {h h}} \ right）$。这些分为三个通用模块系列。提出了这些模块的充分和必要条件，并确定它们的同构类别。使用矢量空间双重性和Mathieu的扭曲函数，这三类模块用于在Takiff $ \ Mathfrak {SL} _ {2} $上构建重量模块的新家族。我们提供了这些重量模块简单的必要条件，并且在某些情况下完全确定了它们的子模块结构。
In this paper, we construct, investigate and, in some cases, classify several new classes of (simple) modules over the Takiff $\mathfrak{sl}_{2}$. More precisely, we first explicitly construct and classify, up to isomorphism, all modules over the Takiff $\mathfrak{sl}_{2}$ that are $U\left(\overline{\mathfrak{h}}\right)$-free of rank one. These split into three general families of modules. The sufficient and necessary conditions for simplicity of these modules are presented, and their isomorphism classes are determined. Using the vector space duality and Mathieu's twisting functors, these three classes of modules are used to construct new families of weight modules over the Takiff $\mathfrak{sl}_{2}$. We give necessary and sufficient conditions for these weight modules to be simple and, in some cases, completely determine their submodule structure.
