Polynomial algebras and higher spins

Abstract Polynomial relations for the generators of the su(2) Lie algebra in arbitrary representations are found. They generalize the usual relations for the Pauli operators in the spin 1 2 case and allow one to construct modified Holstein-Primakoff transformations in finite-dimensional Fock spaces. The connection between the 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 and non-commutative geometry is discussed.