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

TL;DR

This paper explores the Hopf algebroid structure of kappa-deformed phase space, presenting a method to construct twist operators for various realizations, with applications to quantum spacetime symmetries.

Contribution

It introduces a general method to construct twist operators for kappa-deformed phase space and analyzes their Hopf algebroid structures across different realizations.

Findings

Constructed twist operators for natural and arbitrary realizations.

Established the relation between realizations via similarity transformations.

Discussed physical implications of the Hopf algebroid structure and twists.

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, $\kappa$-Poincar\'{e} 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.