A Casimir element inexpressible as a Lie polynomial

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 $U_q^{\prime }({\mathfrak{so}}_3)$ 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 $U_q^{\prime }({\mathfrak{so}}_3)$. Hence, the vector space sum of the center of $U_q^{\prime }({\mathfrak{so}}_3)$ and the Lie subalgebra of $U_q^{\prime }({\mathfrak{so}}_3)$ generated by $I_1,I_2,I_3$ is direct.

References in corpus (2)

• The Universal Askey-Wilson Algebra
• Torsion-type $q$-deformed Heisenberg algebra and its Lie polynomials

Cited by in corpus (2)

• Lie polynomials in an algebra defined by a linearly twisted commutation relation
• Lie polynomials in a $q$-deformed universal enveloping algebra of a low-dimensional Lie algebra