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

Abstract

We introduce the generalized Heisenberg algebra H n and construct realizations of the orthogonal and Lorentz algebras by formal power series in a semicompletion of 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 H n . Furthermore, we show that by extending the generalized Heisenberg algebra H n one can also obtain realizations of the Poincaré aglebra and its extension by quantum angles.

Generalized Heisenberg algebra, realizations of the $\mathfrak{gl}(n)$ algebra and applications

“…For certain Lie algebras, such as the orthogonal algebra so(n), the Lorentz algebra so(1, n − 1) and the gl(n) algebra, the realizations are not well adapted due to the structure of their commutation relations. This was the motivation for introducing the generalized Heisenberg algebra and constructing an analogue of the Weyl realization of so(n) and so(1, n − 1) by formal power series in a semicompletion of the Heisenberg algebra [14]. This construction was applied to the extended Snyder model [15] and to the unification of the κ-Minkowski and extended Snyder spaces [16,17].…”

Heisenberg doubles for Snyder type models

“…This will be considered in Section IV B. In Section IV C we will also present the realizations for the Lorentz matrices in the so-called Weyl realization [24], [27], [29].…”

“…To discuss the extended phase space associated with this extended Snyder model ( 23) as a result of the Heisenberg double construction, we need to first recall few facts about the generalized Heisenberg algebra and Weyl realization of the Lorentz algebra based on results presented in [27].…”

“…The Weyl realization of the Lorentz algebra in terms of this generalized Heisenberg algebra ( 24)-( 26) (as formal power series) have been discussed in detail in [27].…”

“…III A. For more details we refer the reader to [27] where the Lorentz algebra extension by its dual counterpart has been discussed in detail. The formulas presented below base on the Theorem III.1 from [27].…”