2012/02/01 by Leandro Cagliero, Cagliero, Leandro, Fernando Szechtman +1
Mathematics · #17B10 #22E70 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:17B10 #msc:22E70
paper · pdf · doi:10.48550/arxiv.1202.0066
33 pages
arxiv created 2012/02/01 · arxiv updated 2012/02/02
All Lie algebras and representations will be assumed to be finite dimensional over the complex numbers. Let V(m) be the irreducible \sl(2)-module with highest weight m≥ 1 and consider the perfect Lie algebra \g=\sl(2)\ltimes V(m). Recall that a \g-module is uniserial when its submodules form a chain. In this paper we classify all uniserial \g-modules. The main family of uniserial \g-modules is actually constructed in greater generality for the perfect Lie algebra \g=\s\ltimes V(μ), where \s is a semisimple Lie algebra and V(μ) is the irreducible \s-module with highest weight μ≠ 0. The fact that the members of this family are, but for a few exceptions of lengths 2, 3 and~4, the only uniserial \sl(2)\ltimes V(m)-modules depends in an essential manner on the determination of certain non-trivial zeros of Racah-Wigner 6j-symbol.