Doubly -deformed Yang models, Born-selfdual -deformed quantum phase spaces and two generalizations of Yang models

Recently it was shown that by using two different realizations of o (1, 4) Lie algebra one can describe one-parameter standard Snyder model and two-parameter -deformed Snyder model. In this paper, by using the generalized Born duality and Jacobi identities we obtain from the -deformed Snyder model the doubly -deformed Yang model which provides the new class of quantum relativistic phase spaces. These phase spaces contain as subalgebras the -deformed Minkowski space-time as well as quantum -deformed fourmomenta and are depending on five independent parameters. Such a large class of quantum phase spaces can be described in D=4 by particular realizations of o (1, 5) algebra, what illustrates the property that in noncommutative geometry different D=4 physical models may be described by various realizations of the same algebraic structure. Finally, in the last Section we propose two new ways of generalizing Yang models: by introducing o (1, 3+2N) algebras (N=1, 2) we provide internal symmetries O (N) symmetries in Kaluza-Klein extended Yang model, and by replacing the classical o (1, 5) algebras which describe the algebraic structure of Yang models by o (1, 5) quantum groups with suitably chosen nonprimitive coproducts.