Two-sided ideals in [formula omitted]-deformed Heisenberg algebras

Abstract

In this article, the structure of two-sided ideals in the q -deformed Heisenberg algebras defined by the q -deformed Heisenberg canonical commutation relation AB - qBA = I is 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.