Tensor products and intertwining operators between two uniserial representations of the Galilean Lie algebra sl(2)\ltimes h_n

LEANDRO CAGLIERO; IVÁN GÓMEZ RIVERA

ANNALI DI MATEMATICA PURA ED APPLICATA

SPRINGER HEIDELBERG

Lugar: HEIDELBERG; Año: 2024

0373-3114

Let $mathfrak{sl}(2)ltimes mathfrak{h}_n$, $nge 1$, 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 $mathfrak{h}_nsimeq V(2n-1)oplus V(0)$ 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 $Votimes W$ and we describe the space of intertwining operators $ext{Hom}_{mathfrak{sl}(2)ltimes mathfrak{h}_n}(V,W)$, where $V$ and $W$ are uniserial representations of $mathfrak{sl}(2)ltimes mathfrak{h}_n$. The structure of the radical of $Votimes W$ 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 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, there are cases in which even the tensor product of two uniserial $ig(mathfrak{sl}(2)ltimes mathfrak{h}_nig)$-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 $mathfrak{a}_msimeq V(m)$ as $mathfrak{sl}(2)$-modules.p, li { white-space: pre-wrap; }