Polynomial Algebras and Higher Spins

TL;DR

This paper explores polynomial relations in $su(2)$ Lie algebra generators across various representations, generalizing spin-1/2 relations, and connects these to q-oscillators and quantum mechanics on a sphere.

Contribution

It introduces polynomial relations for $su(2)$ generators in arbitrary representations and links them to q-oscillators with roots of unity, expanding the algebraic framework.

Findings

Polynomial relations for $su(2)$ generators are derived for arbitrary representations.

Modified Holstein-Primakoff transformations are constructed in finite-dimensional spaces.

Connections between $su(2)$ algebra, q-oscillators, and quantum mechanics on a sphere are established.

Abstract

Polynomial relations for generators of $su(2)$ Lie algebra in arbitrary representations are found. They generalize usual relation for Pauli operators in spin 1/2 case and permit to construct modified Holstein-Primakoff transformations in finite dimensional Fock spaces. The connection between $su(2)$ Lie algebra and q-oscillators with a root of unity q-parameter is considered. The meaning of the polynomial relations from the point of view of quantum mechanics on a sphere are discussed.