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

Abstract

We introduce the generalized Heisenberg algebra ${\mathcal{H}}_n$ and construct realizations of the orthogonal and Lorentz algebras by power series in a semicompletion of ${\mathcal{H}}_n$. The obtained realizations are given in terms of the generating functions 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 Poincare algebra and its extension by quantum angles.

arXiv comment20 pages

References in corpus (27)

• The quantum structure of spacetime at the Planck scale and quantum fields
• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
• Introduction to Doubly Special Relativity
• Non-commutative space-time of Doubly Special Relativity theories
• Kappa-Minkowski space-time and the star product realizations
• New realizations of Lie algebra kappa-deformed Euclidean space
• New Lie-Algebraic and Quadratic Deformations of Minkowski Space from Twisted Poincare Symmetries
• Twisted Statistics in kappa-Minkowski Spacetime
• Covariant realizations of kappa-deformed space
• Deformed Oscillator Algebras and QFT in $\kappa$-Minkowski Spacetime
• Towards Quantum Noncommutative $\kappa$-deformed Field Theory
• Generalized kappa-deformed spaces, star-products, and their realizations
• Geodesic equation in $k$-Minkowski spacetime
• Noncommutative Spaces and Poincaré Symmetry
• Kappa-deformed Snyder spacetime
• Deformed Covariant Quantum Phase Spaces as Hopf Algebroids
• Lie-deformed quantum Minkowski spaces from twists: Hopf-algebraic versus Hopf-algebroid approach
• Remarks on simple interpolation between Jordanian twists
• Snyder-type spaces, twisted Poincaré algebra and addition of momenta
• Twisted statistics and the structure of Lie-deformed Minkowski spaces
• $\kappa$-deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra and dispersion relations
• Exponential Formulas and Lie Algebra Type Star Products
• Quantum Twist-Deformed D=4 Phase Spaces with Spin Sector and Hopf Algebroid Structures
• kappa-Deformed Covariant Phase Space and Quantum-Gravity Uncertainty Relations
• The Weyl realizations of Lie algebras and left-right duality
• Realization of bicovariant differential calculus on the Lie algebra type noncommutative spaces
• Generalization of Weyl realization to a class of Lie superalgebras

Cited by in corpus (8)

• Associative realizations of the extended Snyder model
• Heisenberg doubles for Snyder type models
• Unification of $\kappa$-Minkowski and extended Snyder spaces
• Fermi equation of state with finite temperature corrections in quantum space-times approach: Snyder model vs GUP case
• Quantum perturbative solutions of extended Snyder and Yang models with spontaneous symmetry breaking
• Symmetric ordering and Weyl realizations for quantum Minkowski spaces
• Generalized Heisenberg algebra, realizations of the $\mathfrak{gl}(n)$ algebra and applications
• Yang model revisited