Realizations and star-product of doubly $\kappa$-deformed Yang models

The Yang algebra was proposed a long time ago as a generalization of the Snyder algebra to the case of curved background spacetime. It includes as subalgebras both the Snyder and the de Sitter algebras and can therefore be viewed as a model of noncommutative curved spacetime. A peculiarity with respect to standard models of noncommutative geometry is that it includes translation and Lorentz generators, so that the definition of a Hopf algebra and the physical interpretation of the variables conjugated to the primary ones is not trivial. In this paper we consider the realizations of the Yang algebra and its $\kappa$-deformed generalization on an extended phase space and in this way we are able to define a Hopf structure and a twist.