A CASIMIR ELEMENT INEXPRESSIBLE AS A LIE POLYNOMIAL

Year 2021, Volume: 30 Issue: 30, 1 - 15, 17.07.2021

Rafael Reno S. Cantuba

https://doi.org/10.24330/ieja.969570

Cited By: 1

Abstract

Let $q$ be a scalar that is not a root of unity. We show that any
nonzero 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.

Keywords

Lie polynomial, Casimir element, quantum group, quantum algebra

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