A Casimir element inexpressible as a Lie polynomial

TL;DR

This paper proves that the Casimir element of the quantum algebra $U_q'( ext{so}_3)$ cannot be expressed solely through Lie algebra operations on its generators, highlighting a fundamental algebraic independence.

Contribution

It establishes that the Casimir element is inexpressible as a Lie polynomial, demonstrating a structural distinction in the algebra's center and Lie subalgebra.

Findings

The Casimir element cannot be written as a Lie polynomial in the generators.

The center and the Lie subalgebra sum directly, with no overlap.

This reveals a fundamental algebraic independence in $U_q'( ext{so}_3)$.

Abstract

Let $q$ be a scalar that is not a root of unity. We show that any polynomial in the Casimir element of the Fairlie-Odesskii algebra $U_q'(\mathfrak{so}_3)$ cannot be expressed in terms of only Lie algebra operations performed on the generators $I_1,I_2,I_3$ in the usual presentation of $U_q'(\mathfrak{so}_3)$. Hence, the vector space sum of the center of $U_q'(\mathfrak{so}_3)$ and the Lie subalgebra of $U_q'(\mathfrak{so}_3)$ generated by $I_1,I_2,I_3$ is direct.