Cayley–Klein Lie Bialgebras: Noncommutative Spaces, Drinfel’d Doubles and Kinematical Applications

Symmetry

Francisco Jose Herranz¹  Ivan Gutierrez-Sagredo¹ 

[1] Departamento de Física, Universidad de Burgos, 09001 Burgos, Spain;

关键词: quantum groups;    classical r-matrices;    contractions;    symmetric homogeneous spaces;    Anti-de Sitter;    Carroll;   

DOI  :  10.3390/sym13071249

来源: DOAJ

【 摘 要 】

The Cayley–Klein (CK) formalism is applied to the real algebra $\mathfrak{so}(5)$ 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’d–Jimbo real Lie bialgebra for $\mathfrak{so}(5)$ 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 dealing with real structures, we found that there exist 63 specific real Lie bialgebras together with their sets of four noncommutative spaces. Furthermore, we found 14 classical r-matrices coming from Drinfel’d doubles, obtaining new results for the de Sitter $\mathfrak{so}(4,1)$ and anti-de Sitter $\mathfrak{so}(3,2)$ as well as for some of their contractions. These geometric results were exhaustively applied onto the (3 + 1)D kinematical algebras, considering not only the usual (3 + 1)D spacetime but also the 6D space of lines. We established 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 $\mathsf{\Lambda }$ and the speed of light c. We, finally, obtained four classes of kinematical r-matrices together with their noncommutative spacetimes and spaces of lines, comprising all $\kappa$-deformations as particular cases.

【 授权许可】

Unknown