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

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.

In press in Journal of Geometry and Physics

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Cited by in corpus (8)

• Abstract ladder operators and their applications
• On the presence of families of pseudo-bosons in nilpotent Lie algebras of arbitrary corank
• Topological decompositions of the Pauli group and their influence on dynamical systems
• On the Pauli group on 2-qubits in dynamical systems with pseudofermions
• Realization of Lie algebras of high dimension via pseudo-bosonic operators
• Coupled Susy, pseudo-bosons and a deformed $\mathfrak{su}(1,1)$ Lie algebra
• Decomposition of Pauli groups via weak central products
• Some Consequences of the Grunewald-O'Halloran Conjecture for Pseudoquonic Operators