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

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.