Lie polynomials in q-deformed Heisenberg algebras

Let F be a field, and let q is an element of 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 not equal 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. (C) 2018 Elsevier Inc. All rights reserved.