Towards new relativistic doubly \(\kappa\)-deformed D=4 quantum phase spaces

We propose new noncommutative (NC) models of quantum phase spaces, containing a pair of \(\kappa\)-deformed Poincaré algebras, with two independent \(\kappa\) and \(\tilde{\kappa}\)-deformations in space-time and fourmomenta sectors. The first quantum phase space can be obtained by contractions \(M,R\to \infty\) of recently introduced doubly \(\kappa\)-deformed \((\kappa,\tilde{\kappa})\)-Yang models, with the parameters \(M,R\) describing inverse space-time and fourmomenta curvatures and constant four-vectors \(a_\mu, b_\mu\) determining one of nine types of \((\kappa,\tilde{\kappa})\) deformations. The second model is provided by the non-linear doubly \(\kappa\)-deformed TSR algebra spanned by 14 coset \(\hat{o}(1,5)/\hat {o}(2)\) generators. The basic algebraic difference between the two models is the following: the first one, described by \(\hat{o}(1,5)\) Lie algebra can be supplemented by the Hopf algebra structure, while the second model contains the quantum phase space commutators \([\hat{x}_\mu,\hat{q}_\nu]\) describing the quantum-deformed Heisenberg algebra relations with the standard numerical \(i\hbar\eta_{\mu\nu}\) term.