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Journal of Lie Theory 30 (2020), No. 4, 925--938
Copyright Heldermann Verlag 2020

Realization of Lie Algebras of High Dimension via Pseudo-Bosonic Operators

Fabio Bagarello
Dip. di Ingegneria, UniversitÓ di Palermo, 90128 Palermo, Italy
and: Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, 80126 Napoli, Italy

Francesco G. Russo
Dept. of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, Cape Town, South Africa

The present paper is the third contribution of a series of works, where we investigate pseudo-bosonic operators and their connections with finite dimensional Lie algebras. We show that all finite dimensional nilpotent Lie algebras (over the complex field) can be realized by central extensions of Lie algebras of pseudo-bosonic operators. This result is interesting, because it provides new examples of dynamical systems for nilpotent Lie algebras of any dimension. One could ask whether these operators are intrinsic with the notion of nilpotence or not, but this is false. In fact we exibit both a simple Lie algebra and a solvable nonnilpotent Lie algebra, which can be realized in terms of pseudo-bosonic operators.

Keywords: Pseudo-bosonic operators, Hilbert space, Schur multiplier, nilpotent Lie algebras, homology.

MSC: 47L60, 17B30; 17B60, 46K10.

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