Associative realizations of $\kappa$-deformed extended Snyder model

Abstract

Usually, the realizations of the noncommutative Snyder model lead to a nonassociative star product. However, it has been shown that this problem can be avoided by adding to the spacetime coordinates new tensorial degrees of freedom. The model so obtained, called extended Snyder model, can be subject to a $\kappa$-deformation, giving rise to a unification of the Snyder and the $\kappa$-Poincar\'e algebras in the formalism of extended spacetime. In this paper we review this construction and consider the generic realizations of the $\kappa$-deformed extended Snyder model, calculating the associated star product, coproduct and twist in a perturbative setting. We also introduce a representation of the Lorentz algebra in the extended space and speculate on possible interpretations of the tensorial degrees of freedom.