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

We propose new noncommutative models of quantum phase spaces, containing a pair of $\kappa$-deformed Poincaré algebras, with two independent double ($\kappa ,\tilde{\kappa }$)-deformations in space-time and four-momenta sectors. The first such quantum phase space can be obtained by contractions $M,R\to \mathrm{\infty }$ of recently introduced doubly $\kappa$-deformed $(\kappa ,\tilde{\kappa })$-Yang models, with the parameters $M,R$ describing inverse space-time and four-momenta curvatures and constant four-vectors $a_{\mu },b_{\mu }$ determining nine types of $(\kappa ,\tilde{\kappa })$-deformations. The second considered model is provided by the nonlinear doubly $\kappa$-deformed TSR algebra spanned by 14 coset $\widehat{o}(1,5)\mathrm{/}\widehat{o}(2)$ generators. The basic algebraic difference between the two models is the following: the first one, described by $\widehat{o}(1,5)$ Lie algebra can be supplemented by the Hopf algebra structure, while the second model contains the quantum phase space commutators $[{\widehat{x}}_{\mu },{\widehat{q}}_{\nu }]$, with the standard numerical $i\mathrm{\hslash }{\eta }_{\mu \nu }$ term; therefore it describes the quantum-deformed Heisenberg algebra relations which cannot be equipped with the Hopf algebra.