CayleyKlein Lie Bialgebras: Noncommutative Spaces, Drinfel'd Doubles and Kinematical Applications
Abstract
The CayleyKlein (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, Poincaré, Newtonian and Carrollian algebras. Starting with the Drinfel'dJimbo 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 (firstorder) noncommutative CK spaces of points, lines, 2planes and 3hyperplanes, 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 antide 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.
 Publication:

Symmetry
 Pub Date:
 July 2021
 DOI:
 10.3390/sym13071249
 arXiv:
 arXiv:2106.03817
 Bibcode:
 2021Symm...13.1249G
 Keywords:

 High Energy Physics  Theory;
 General Relativity and Quantum Cosmology;
 Mathematical Physics;
 17B37;
 17B62;
 14M17;
 81R60
 EPrint:
 66 pages. New paragraphs have been added to the Introduction