Tensor products and intertwining operators between two uniserial representations of the Galilean Lie algebra $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$

Let $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$, $n\ge 1$, be the Galilean Lie algebra over a field of characteristic zero, here $\mathfrak{h}_{n}$ is the Heisenberg Lie algebra of dimension $2n+1$, and $\mathfrak{sl}(2)$ acts on $\mathfrak{h}_{n}$ so that $\mathfrak{h}_n\simeq V(2n-1)\oplus V(0)$ as $\mathfrak{sl}(2)$-modules (here $V(k)$ denotes the irreducible $\mathfrak{sl}(2)$-module of highest weight $k$). In this paper, we study the tensor product of two uniserial representations of $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$. We obtain the $\mathfrak{sl}(2)$-module structure of the socle of $V\otimes W$ and we describe the space of intertwining operators $\text{Hom}_{\mathfrak{sl}(2)\ltimes \mathfrak{h}_n}(V,W)$, where $V$ and $W$ are uniserial representations of $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$. The structure of the radical of $V\otimes W$ follows from that of the socle of $V^*\otimes W^*$. The result is subtle and shows how difficult is to obtain the whole socle series of arbitrary tensor products of uniserials. In contrast to the serial associative case, our results for $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$ reveal that these tensor products are far from being a direct sum of uniserials; in particular, there are cases in which the tensor product of two uniserial $\big(\mathfrak{sl}(2)\ltimes \mathfrak{h}_n\big)$-modules is indecomposable but not uniserial. Recall that a foundational result of T. Nakayama states that every finitely generated module over a serial associative algebra is a direct sum of uniserial modules. This article extends a previous work in which we obtained the corresponding results for the Lie algebra $\mathfrak{sl}(2)\ltimes \mathfrak{a}_m$ where $\mathfrak{a}_m$ is the abelian Lie algebra of dimension $m+1$ and $\mathfrak{sl}(2)$ acts so that $\mathfrak{a}_m\simeq V(m)$ as $\mathfrak{sl}(2)$-modules.