The classification of uniserial sl(2)xV(m)-modules and a new interpretation of the Racah-Wigner 6j-symbols

L. CAGLIERO; F. SZCHETMAN

JOURNAL OF ALGEBRA

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2013 vol. 386 p. 142 - 175

0021-8693

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 $m geq 1$ 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 $g=sltimes V(mu)$, where $s$ is a semisimple Lie algebra and $V(mu)$ is the irreducible $s$-module with highest weight $mu eq 0$. 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.