Lie polynomials in an algebra defined by a linearly twisted commutation relation

Abstract

We present an elementary approach in characterizing Lie polynomials in the generators $A,B$ of an algebra with a defining relation that is in the form of a deformed or twisted commutation relation $AB=\sigma (BA)$ where the deformation or twisting map $\sigma$ is a linear polynomial with a slope parameter that is not a root of unity. The class of algebras defined as such encompasses $q$-deformed Heisenberg algebras, rotation algebras, and some types of $q$-oscillator algebras whose deformation parameters are not roots of unity, and so we have a general solution for the Lie polynomial characterization problem for these algebras.

References in corpus (2)

• An extension of a $q$-deformed Heisenberg algebra and its Lie polynomials
• A Casimir element inexpressible as a Lie polynomial

Cited by in corpus (1)

• Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra