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

TL;DR

The paper introduces new non-weight modules for the Takiff algebra $rak g=rak{sl}_2\otimes rak D$ by forming tensor products $V( u,a,b_{eta}) ensor L( ho, au)$ of irreducible free $U(ar{rak h})$-modules of rank 1 with irreducible highest-weight modules. It establishes irreducibility (with the condition $a eq 0$ when $V= Omega$), classifies isomorphism classes, and proves these modules are novel compared with known non-weight $rak g$-modules. Furthermore, it reformulates the tensor products as induced modules from suitable subalgebras and derives precise reducibility criteria for these induced modules, linking their behavior to the irreducibility of the original tensor products. Collectively, the results extend the landscape of non-weight representations for Takiff $rak{sl}_2$ and provide tools for constructing and recognizing such modules via induction and tensor-product techniques.

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.

Theorems & Definitions (20)

• Definition 2.1
• Proposition 2.2
• Theorem 3.1
• proof
• Claim 1
• Claim 2
• Theorem 4.1
• proof
• Lemma 5.1
• proof