Tensor products and intertwining operators for uniserial representations of the Lie algebra [formula omitted

Abstract

Let gm=sl(2)⋉V(m), m≥1, where V(m) is the irreducible sl(2)-module of dimension m+1 viewed as an abelian Lie algebra. It is known that the isomorphism classes of uniserial gm-modules consist of a family, say of type Z, containing modules of arbitrary composition length, and some exceptional modules with composition length ≤4.Let V and W be two uniserial gm-modules of type Z. In this paper we obtain the sl(2)-module decomposition of soc(V⊗W) by giving explicitly the highest weight vectors. It turns out that soc(V⊗W) is multiplicity free. Roughly speaking, soc(V⊗W)=soc(V)⊗soc(W) in half of the cases, and in these cases we obtain the full socle series of V⊗W by proving that soct+1(V⊗W)=∑i=0tsoci+1(V)⊗soct+1−i(W) for all t≥0.As applications of these results, we obtain for which V and W, the space of gm-module homomorphisms Homgm(V,W) is not zero, in which case is 1-dimensional. Finally we prove, for m≠2, that if U is the tensor product of two uniserial gm-modules of type Z, then the factors are determined by U. We provide a procedure to identify the factors from U.