A new interpretation of the Racah-Wigner $6j$-symbol and the classification of uniserial $sl(2)\ltimes V(m)$-modules

TL;DR

This paper classifies all uniserial modules over a specific class of Lie algebras, revealing a deep connection to the Racah-Wigner 6j-symbols and extending the classification to more general semisimple Lie algebra modules.

Contribution

It provides a complete classification of uniserial modules for the Lie algebra  7 V(m), linking their structure to the zeros of Racah-Wigner 6j-symbols and generalizing to semisimple Lie algebras.

Findings

Most uniserial  7 V(m)-modules are characterized by zeros of 6j-symbols.

The classification extends to  7 V() modules with few exceptions.

The work connects representation theory with special functions in mathematical physics.

Abstract

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 $\g=\s\ltimes V(\mu)$, where $\s$ is a semisimple Lie algebra and $V(\mu)$ is the irreducible $\s$-module with highest weight $\mu\neq 0$. 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.