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We introduce the generalized Heisenberg algebra $\mathcal{H}_n$ and constructrealizations of the orthogonal and Lorentz algebras by power series in asemicompletion of $\mathcal{H}_n$. The obtained realizations are given in termsof the generating functions for the Bernoulli numbers. We also introduce anextension of the orthogonal and Lorentz algebras by quantum angles and studyrealizations of the extended algebras in $\mathcal{H}_n$. Furthermore, we showthat by extending the generalized Heisenberg algebra $\mathcal{H}_n$ one canalso obtain realizations of the Poincare algebra and its extension by quantumangles.

Generalized Heisenberg algebra applied to realizations of the orthogonal, Lorentz and Poincare algebras and their dual extensions