THE CLASSIFICATION OF UNISERIAL sl(2)n 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) n 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 = sn 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)nV (m)-modules depends in an essential manner on the determination of certain non-trivial zeros of Racah-Wigner 6j-symbol.