Quantum groups, non-commutative Lorentzian spacetimes and curved momentum spaces

TL;DR

This paper explores quantum group deformations of classical symmetries in General Relativity with a cosmological constant, describing non-commutative spacetimes and curved momentum spaces in low dimensions, and discusses potential extensions to four dimensions.

Contribution

It provides a detailed semiclassical construction of non-commutative (anti-)de Sitter spacetimes and curved momentum spaces from $$-deformed quantum groups, including the cosmological constant as a parameter.

Findings

Explicit description of non-commutative (anti-)de Sitter spacetimes in 1+1 and 2+1 dimensions.

Construction of curved momentum spaces as orbits from quantum group symmetries.

Discussion on extending results to 3+1 dimensions.

Abstract

The essential features of a quantum group deformation of classical symmetries of General Relativity in the case with non-vanishing cosmological constant $\Lambda$ are presented. We fully describe (anti-)de Sitter non-commutative spacetimes and curved momentum spaces in (1+1) and (2+1) dimensions arising from the $\kappa$-deformed quantum group symmetries. These non-commutative spacetimes are introduced semiclassically by means of a canonical Poisson structure, the Sklyanin bracket, depending on the classical $r$-matrix defining the $\kappa$-deformation, while curved momentum spaces are defined as orbits generated by the $\kappa$-dual of the Hopf algebra of quantum symmetries. Throughout this construction we use kinematical coordinates, in terms of which the physical interpretation becomes more transparent, and the cosmological constant $\Lambda$ is included as an explicit parameter whose $\Lambda \rightarrow 0$ limit provides the Minkowskian case. The generalization of these results to the physically relevant (3+1)-dimensional deformation is also commented.