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 nonzero polynomial in the Casimir element of the Fairlie-Odesskii algebra U q (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 (so 3 ). Hence, the vector space sum of the center of U q (so 3 ) and the Lie subalgebra of U q (so 3 ) generated by I 1 , I 2 , I 3 is direct.