Realizations of $\kappa$-Minkowski space, Drinfeld twists and related symmetry algebras

TL;DR

This paper classifies all linear realizations of $$-Minkowski space in terms of $(n)$ generators, constructs corresponding Drinfeld twists, and explores their algebraic structures and physical implications.

Contribution

It provides a complete classification of linear realizations and associated Drinfeld twists for $$-Minkowski space, extending understanding of related symmetry algebras.

Findings

Three one-parameter families of realizations for time-like and space-like deformations.

Four realizations for light-like deformations.

Construction of $$-deformed Hopf algebras and derivation of all known Drinfeld twists.

Abstract

Realizations of $\kappa$-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of $\mathfrak{gl}(n)$ generators. There are three one-parameter families of linear realizations for time-like and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between deformed Heisenberg algebra, star product, coproduct of momenta and twist operator is presented. It is proved that for each linear realization there exists Drinfeld twist satisfying normalization and cocycle conditions. $\kappa$-deformed $\mathfrak{igl}(n)$-Hopf algebras are presented for all cases. The $\kappa$-Poincar\'e-Weyl and $\kappa$-Poincar\'e-Hopf algebras are discussed. Left-right dual $\kappa$-Minkowski algebra is constructed from the transposed twists. The corresponding realizations are nonlinear. All known Drinfeld twists related to $\kappa$-Minkowski space are obtained from our construction. Finally, some physical applications are discussed.