Simple modules over the Takiff Lie algebra for $\mathfrak{sl}_{2}$

TL;DR

This paper constructs and classifies new simple modules over the Takiff f1f1f1sl_2 algebra, expanding understanding of its module categories and providing explicit criteria for simplicity and structure.

Contribution

It explicitly classifies all rank-one free modules over the Takiff f1f1f1sl_2 and constructs new weight modules using duality and twisting functors.

Findings

Classified all rank-one free modules over the Takiff f1f1f1sl_2.

Established necessary and sufficient conditions for module simplicity.

Constructed new families of weight modules and analyzed their submodule structures.

Abstract

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.