Tensor products and intertwining operators between two uniserial representations of the Galilean Lie algebra sl(2)⋉hn.

Let sl (2) ⋉ h n , n ≥ 1 , be the Galilean Lie algebra over a field of characteristic zero, here h n is the Heisenberg Lie algebra of dimension 2 n + 1 , and sl (2) acts on h n so that, sl (2) -modules, h n ≃ V (2 n - 1) ⊕ V (0) (here V(k) denotes the irreducible sl (2) -module of highest weight k). In this paper, we study the tensor product of two uniserial representations of sl (2) ⋉ h n . We obtain the sl (2) -module structure of the socle of V ⊗ W and we describe the space of intertwining operators Hom sl (2) ⋉ h n (V , W) , where V and W are uniserial representations of sl (2) ⋉ h n . The structure of the radical of V ⊗ W follows from that of the socle of V ∗ ⊗ 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 sl (2) ⋉ 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 (sl (2) ⋉ h n) -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 sl (2) ⋉ a m where a m is the abelian Lie algebra of dimension m + 1 and sl (2) acts so that a m ≃ V (m) as sl (2) -modules. [ABSTRACT FROM AUTHOR]