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Generalized Heisenberg algebra applied to realizations of the orthogonal, Lorentz, and Poincaré algebras and their dual extensions. (English) Zbl 1443.81044
Summary: We introduce the generalized Heisenberg algebra \(\mathcal{H}_n\) and construct realizations of the orthogonal and Lorentz algebras by a formal power series in a semicompletion of \(\mathcal{H}_n\). The obtained realizations are given in terms of the generating function for the Bernoulli numbers. We also introduce an extension of the orthogonal and Lorentz algebras by quantum angles and study realizations of the extended algebras in \(\mathcal{H}_n\). Furthermore, we show that by extending the generalized Heisenberg algebra \(\mathcal{H}_n\), one can also obtain realizations of the Poincaré algebra and its extension by quantum angles.
©2020 American Institute of Physics
MSC:
81R60 Noncommutative geometry in quantum theory
33C65 Appell, Horn and Lauricella functions
14D15 Formal methods and deformations in algebraic geometry
11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.)
22E43 Structure and representation of the Lorentz group
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