Wigner coefficient for Lie algebras of series $B,C,D$ and a base of Gelfand-Tsetlin type

TL;DR

This paper develops a simple construction of bases in irreducible representations for certain Lie algebras of series B, C, D using Z-invariants and Wigner coefficients, extending Gelfand-Tsetlin methods.

Contribution

It introduces a new base construction for $g_n$ Lie algebras utilizing Z-invariants and Wigner coefficients, relating matrix elements to those of $rak{gl}_{n+1}$.

Findings

Constructed bases for $g_n$ Lie algebras using Z-invariants.

Established relations between matrix elements and Wigner coefficients.

Extended Gelfand-Tsetlin techniques to series B, C, D Lie algebras.

Abstract

For the Lie algebras $g_n= \mathfrak{o}_{2n+1},\mathfrak{sp}_{2n},\mathfrak{o}_{2n}$ a simple construction of a base in an irreducible representation is given. The construction of this base uses the method of $Z$-invariants of Zhelobenko and the technique of Wigner coefficients, which was applied by Biedenharn and Baird to the construction of a Gelfand-Tsetlin base in the case $\mathfrak{gl}_n$. A relation between matrix elements and Wigner coefficients for $g_n$ and analogous objects for $\mathfrak{gl}_{n+1}$ is established.