Twists, realizations and Hopf algebroid structure of kappa-deformed phase space

Abstract

The quantum phase space described by Heisenberg algebra possesses undeformed Hopf algebroid structure. The $\kappa$-deformed phase space with noncommutative coordinates is realized in terms of undeformed quantum phase space. There are infinitely many such realizations related by similarity transformations. For a given realization we construct corresponding coproducts of commutative coordinates and momenta (bialgebroid structure). The $\kappa$-deformed phase space has twisted Hopf algebroid structure. General method for the construction of twist operator (satisfying cocycle and normalization condition) corresponding to deformed coalgebra structure is presented. Specially, twist for natural realization (classical basis) of $\kappa$-Minkowski spacetime is presented. The cocycle condition, $\kappa$-Poincaré algebra and $R$-matrix are discussed. Twist operators in arbitrary realizations are constructed from the twist in the given realization using similarity transformations. Some examples are presented. The important physical applications of twists, realizations, $R$-matrix and Hopf algebroid structure are discussed.

arXiv comment34 pages, revised version, accepted in IJMPA

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• Noncommutative duality and fermionic quasinormal modes of the BTZ black hole
• Toward the classification of differential calculi on $\kappa$-Minkowski space and related field theories
• Deformed Covariant Quantum Phase Spaces as Hopf Algebroids
• Effects of Noncommutativity on the Black Hole Entropy
• Remarks on simple interpolation between Jordanian twists
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• Realizations of $\kappa$-Minkowski space, Drinfeld twists and related symmetry algebras
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• Minimal and maximal lengths from position-dependent noncommutativity
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