Irreducible tensor product modules over the Takiff Lie algebra for $\mathfrak{sl}_{2}$

TL;DR

This paper constructs and analyzes a new class of irreducible non-weight modules over the Takiff $ ext{sl}_2$ algebra, characterizing their irreducibility, isomorphisms, and relation to induced modules.

Contribution

It introduces a novel class of irreducible non-weight modules over the Takiff $ ext{sl}_2$, expanding the understanding of module structures and their reducibility conditions.

Findings

Constructed non-weight modules via tensor products of irreducible modules.

Characterized irreducibility and isomorphism conditions for these modules.

Reformulated some modules as induced modules and determined reducibility criteria.

Abstract

In this paper, we construct a class of non-weight modules over the Takiff $\mathfrak{sl}_{2}$ by taking the tensor products of the irreducible free $U(\overline{\mathfrak{h}})$-modules of rank 1, where $\overline{\mathfrak{h}}$ is a natural Cartan subalgebra of the Takiff $\mathfrak{sl}_{2}$, with the irreducible highest weight modules. We characterize the irreducibility of these tensor product modules and determine the necessary and sufficient conditions for isomorphisms between them. We further prove that these non-weight modules are distinct from the known non-weight modules. Finally, we reformulate some tensor product modules over the Takiff $\mathfrak{sl}_{2}$ as induced modules derived from modules over certain subalgebras, and determine the necessary and sufficient conditions for the reducibility of these induced modules.