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, Poincaré, 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 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 so(4,1) and anti-de Sitter 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 Λ 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 κ-deformations as particular cases.
Highlights
The notion of Cayley–Klein (CK) Lie algebras along with their corresponding Lie groups and symmetric homogeneous spaces date back to early studies of projective metrics
We shall focus on quantum deformations of Lie algebras along with their connection to Lie bialgebras, Poisson–Lie groups, Poisson homogeneous spaces, noncommutative spaces and Drinfel’d doubles
This is a consequence of the presence of the term r13 = zJ23 ∧ J12 in r04,13, which does not appear in either the Drinfel’d–Jimbo quantum deformation of so(3) [7] or in so(4) [135]; for these latter deformations, the whole initial Z2⊗2- and Z2⊗3-grading is kept, respectively; for (so(5), δ(r04,13)) there only remains a Z2⊗2-grading spanned by Θ(z2) and Θ(z3)
Summary
The notion of Cayley–Klein (CK) Lie algebras along with their corresponding Lie groups and symmetric homogeneous spaces date back to early studies of projective metrics. In this process, we obtain four classes of kinematical r-matrices and, for some algebras, r-matrices coming from Drinfel’d doubles.
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