2022/01/25 by Leandro Cagliero, Cagliero, Leandro, Iván Gómez Rivera +1
Mathematics · #16G10 #17B10 #17B30 #22E27 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2201.10605
openalex publication_date 2022/01/25 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
Let \mathfrakgm=\mathfraksl(2)\ltimes V(m), m≥ 1, where V(m) is the irreducible \mathfraksl(2)-module of dimension m+1 viewed as an abelian Lie algebra. It is known that the isomorphism classes of uniserial \mathfrakgm-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 \mathfrakgm-modules of type Z. In this paper we obtain the \mathfraksl(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=0t soci+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 \mathfrakgm-module homomorphisms Hom_\mathfrakgm(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 \mathfrakgm-modules of type Z, then the factors are determined by U. We provide a procedure to identify the factors from U.