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Tensor products and intertwining operators for uniserial representations of the Lie algebra $\mathfrak{sl}(2)\ltimes V(m)$

Let $\mathfrak{g}_m=\mathfrak{sl}(2)\ltimes V(m)$, $m\ge 1$, where $V(m)$ is the irreducible $\mathfrak{sl}(2)$-module of dimension $m+1$ viewed as an abelian Lie algebra. It is known that the isomorphism classes of uniserial $\mathfrak{g}_m$-modules consist of a family, say of type $Z$, containing modules of arbitrary composition length, and some exceptional modules with composition length $\le 4$. Let $V$ and $W$ be two uniserial $\mathfrak{g}_m$-modules of type $Z$. In this paper we obtain the $\mathfrak{sl}(2)$-module decomposition of $\text{soc}(V\otimes W)$ by giving explicitly the highest weight vectors. It turns out that $\text{soc}(V\otimes W)$ is multiplicity free. Roughly speaking, $\text{soc}(V\otimes W)=\text{soc}(V)\otimes \text{soc}(W)$ in half of the cases, and in these cases we obtain the full socle series of $V\otimes W$ by proving that $ \text{soc}^{t+1}(V\otimes W)=\sum_{i=0}^{t} \text{soc}^{i+1}(V)\otimes \text{soc}^{t+1-i}(W)$ for all $t\ge0$. As applications of these results, we obtain for which $V$ and $W$, the space of $\mathfrak{g}_m$-module homomorphisms $\text{Hom}_{\mathfrak{g}_m}(V,W)$ is not zero, in which case is 1-dimensional. Finally we prove, for $m\ne 2$, that if $U$ is the tensor product of two uniserial $\mathfrak{g}_m$-modules of type $Z$, then the factors are determined by $U$. We provide a procedure to identify the factors from $U$.