Two-sided ideals in q-deformed Heisenberg algebras

In this article, the structure of two-sided ideals in the q-deformed Heisenberg algebras defined by the q-deformed Heisenberg canonical commutation relationAB-qBA=Iis investigated. We show that these algebras are simple if and only if q=1. For q≠1,0 we present an infinite descending chain of non-trivial two-sided ideals, thus deducing by explicit construction that the q-deformed Heisenberg algebras are not just non-simple but also non-artinian for q≠1,0. We establish a connection between the quotients of the q-deformed Heisenberg algebras by these ideals and the quotients of the quantum plane. We also present a number of reordering formulae in q-deformed Heisenberg algebras, investigate properties of deformed commutator mappings, show their fundamental importance for investigation of ideals in q-deformed Heisenberg algebras, and demonstrate how to apply these results to the investigation of faithfulness of representations of q-deformed Heisenberg algebras.