Lie polynomials in q-deformed Heisenberg algebras

Abstract

Let F be a field, and let q∈F. The q-deformed Heisenberg algebra is the unital associative F-algebra H(q) with generators A,B and relation AB−qBA=I, where I is the multiplicative identity in H(q). The set of all Lie polynomials in A,B is the Lie subalgebra L(q) of H(q) generated by A,B. If q≠1 or the characteristic of F is not 2, then the equation AB−qBA=I cannot be expressed in terms of Lie algebra operations only, yet this equation still has consequences on the Lie algebra structure of L(q), which we investigate. We show that if q is not a root of unity, then L(q) is a Lie ideal of H(q), and the resulting quotient Lie algebra is infinite-dimensional and one-step nilpotent.