Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications

TL;DR

This paper applies the Cayley-Klein formalism to classify and construct noncommutative spaces and Lie bialgebras related to various (anti-)de Sitter and Poincaré algebras, with applications to kinematical models and quantum deformations.

Contribution

It introduces a unified framework for noncommutative spaces and Lie bialgebras for a wide class of kinematical algebras using Cayley-Klein contractions and Drinfel'd doubles.

Findings

Classified 63 real Lie bialgebras with noncommutative spaces.

Derived 14 classical r-matrices from Drinfel'd doubles for (anti-)de Sitter algebras.

Established connections between geometric CK generators and physical parameters like cosmological constant and speed of light.

Abstract

The Cayley-Klein (CK) formalism is applied to the real algebra ${so}(5)$ by making use of four graded contraction parameters describing in a unified setting 81 Lie algebras, which cover the (anti-)de Sitter, Poincar\'e, Newtonian and Carrollian algebras. Starting with the Drinfel'd-Jimbo real Lie bialgebra for ${so}(5)$ together with its Drinfel'd double structure, we obtain the corresponding CK bialgebra and the CK $r$-matrix coming from a Drinfel'd double. As a novelty, we construct the (first-order) noncommutative CK spaces of points, lines, 2-planes and 3-hyperplanes, studying their structural properties. By requiring to deal with real structures, it comes out that there exist 63 specific real Lie bialgebras together with their sets of four noncommutative spaces. Furthermore, we find 14 classical $r$-matrices coming from Drinfel'd doubles, obtaining new results for the de Sitter ${so}(4,1)$ and anti-de Sitter ${so}(3,2)$ and for some of their contractions. These geometric results are exhaustively applied onto the (3+1)D kinematical algebras, not only considering the usual (3+1)D spacetime but also the 6D space of lines. We establish different assignations between the geometrical CK generators and the kinematical ones which convey physical identifications for the CK contraction parameters in terms of the cosmological constant/curvature $\Lambda$ and speed of light $c$. We finally obtain four classes of kinematical $r$-matrices together with their noncommutative spacetimes and spaces of lines, comprising all $\kappa$-deformations as particular cases.