A description of pseudo-bosons in terms of nilpotent Lie algebras

We show how the one-mode pseudo-bosonic ladder operators provide concrete examples of nilpotent Lie algebras of dimension five. It is the first time that an algebraic geometric structure of this kind is observed in the context of pseudo-bosonic operators. Indeed we do not find the well known Heisenberg algebras, which are involved in several quantum dynamical systems, but different Lie algebras which may be decomposed into the sum of two abelian Lie algebras in a prescribed way. We introduce the notion of semidirect sum (of Lie algebras) for this scope and find that it describes very well the behavior of pseudobosonic operators in many quantum models. (C) 2017 Elsevier B.V. All rights reserved.