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

Realization of Lie algebras of high dimension via pseudo-bosonic operators