Kappa Snyder deformations of Minkowski spacetime, realizations and Hopf algebra

TL;DR

This paper explores Lie-algebraic deformations of Minkowski space that connect Snyder and κ-Minkowski models, providing realizations, algebraic structures, and mappings, with implications for noncommutative geometry and quantum spacetime models.

Contribution

It introduces a unified framework for Snyder and κ-Minkowski deformations, including realizations, Hopf algebra structures, and mappings, extending previous models.

Findings

Realizations of noncommutative coordinates in terms of commutative ones.

Construction of a general mapping between Snyder and κ-deformed spaces.

Perturbative results up to second order in deformation parameters.

Abstract

We present Lie-algebraic deformations of Minkowski space with undeformed Poincar\'{e} algebra. These deformations interpolate between Snyder and $\kappa$-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. By introducing modules, it is shown that although deformed and undeformed structures are not isomorphic at the level of vector spaces, they are however isomorphic at the level of Hopf algebraic action on corresponding modules. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from $\kappa$-deformed Snyder to Snyder space is constructed. Deformed Leibniz rule, the Hopf structure and star product are found. Special cases, particularly Snyder and $\kappa$-Minkowski in Maggiore-type realizations are discussed. The same generalized Hopf algebraic structures are as well considered in the case of an arbitrary allowable kind of realisation and results are given perturbatively up to second order in deformation parameters.