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

TL;DR

This paper introduces doubly ${kappa}$-deformed Yang models and quantum phase spaces, extending previous Snyder models using algebraic dualities, and proposes new generalizations of Yang models with internal symmetries and quantum group structures.

Contribution

It develops a new class of quantum relativistic phase spaces via ${kappa}$-deformations and generalized dualities, and introduces two novel generalizations of Yang models with internal symmetries and quantum groups.

Findings

Constructed doubly ${kappa}$-deformed Yang models from Snyder models.

Described quantum phase spaces with ${o}(1,5)$ algebra realizations.

Proposed two new generalizations of Yang models with internal symmetries and quantum group structures.

Abstract

Recently it was shown that by using two different realizations of $\hat{o}(1,4)$ Lie algebra one can describe one-parameter standard Snyder model and two-parameter $\kappa$-deformed Snyder model. In this paper, by using the generalized Born duality and Jacobi identities we obtain from the $\kappa$-deformed Snyder model the doubly $\kappa$-deformed Yang model which provides the new class of quantum relativistic phase spaces. These phase spaces contain as subalgebras the $\kappa$-deformed Minkowski space-time as well as quantum $\tilde{\kappa}$-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 $\hat{o}(1,5)$ 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 $\hat o(1,3+2N)$ algebras ($N=1,2\ldots$) we provide internal symmetries $O(N)$ symmetries in Kaluza-Klein extended Yang model, and by replacing the classical $\hat{o}(1,5)$ algebras which describe the algebraic structure of Yang models by $\hat o(1,5)$ quantum groups with suitably chosen nonprimitive coproducts.