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

Leandro Cagliero, Iván Gómez Rivera

Published 2023-01-26Version 1

Let $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$, $n\ge 1$, be the Galilean Lie algebra over a field of characteristic zero, where $\mathfrak{h}_{n}$ 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 $k$). The isomorphism classes of uniserial $\big(\mathfrak{sl}(2)\ltimes \mathfrak{h}_n\big)$-modules are known. In this paper we study the tensor product of two uniserial representations of $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$. Among other things, 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 $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$. This article extends a previous work in which we obtained analogous results for the Lie algebra $\mathfrak{sl}(2)\ltimes \mathfrak{a}_m$ where $\mathfrak{a}_m$ is the abelian Lie algebra and $\mathfrak{sl}(2)$ acts so that $\mathfrak{a}_m\simeq V(m-1)$ as $\mathfrak{sl}(2)$-modules.