The classification of uniserial sl(2)⋉V(m)-modules and a new interpretation of the Racah–Wigner 6j-symbol

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)⋉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⋉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)⋉V(m)-modules depends in an essential manner on the determination of certain non-trivial zeros of Racah–Wigner 6j-symbol.