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

TL;DR

This paper explores how pseudo-bosonic ladder operators form specific five-dimensional nilpotent Lie algebras, revealing a new algebraic-geometric structure distinct from the traditional Heisenberg algebra in quantum systems.

Contribution

It introduces the first example of nilpotent Lie algebras arising from pseudo-bosonic operators and defines a semidirect sum concept to describe their structure in quantum models.

Findings

Pseudo-bosonic operators generate five-dimensional nilpotent Lie algebras.

These algebras are decomposable into sums of two abelian Lie algebras.

The semidirect sum framework effectively models pseudo-bosonic behavior in quantum systems.

Abstract

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 don't find the well known Heisenberg algebras, which are involved in several quantum dynamical systems, but different Lie algebras which may be decomposed in 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 behaviour of pseudo-bosonic operators in many quantum models.