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

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.

Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries

“…In the approach here presented, Lie bialgebra structures are used as the defining objects for quantum deformations, and the type of interplay among all the parameters arising in them can be already studied at the Lie bialgebra level (in particular, the theory of quantum group contractions is based on the contraction theory for Lie bialgebras [38]). A detailed description of Lie bialgebras and their role in quantum group theory can be found in [32], and a complete presentation of kinematical Lie bialgebras has been given in [48].…”

“…In general one can perform two kinds of contractions, either working at the level of the rmatrix (this is a "coboundary" contraction), or working directly at the level of the co-commutators (this is the so-called "fundamental" contraction) [38,48]. As it was shown in [30], this distinction is especially relevant in the case of the Galilean limit of κ-Poincaré, where the two procedures are nonequivalent.…”

Interplay between Spacetime Curvature, Speed of Light and Quantum Deformations of Relativistic Symmetries

“…In the approach presented here, Lie bialgebra structures are used as the defining objects for quantum deformations, and the type of interplay among all the parameters arising in them can be already studied at the Lie bialgebra level (in particular, the theory of quantum group contractions is based on the contraction theory for Lie bialgebras [38]). A detailed description of Lie bialgebras and their role in quantum group theory can be found in [32], and a complete presentation of kinematical Lie bialgebras has been given in [48].…”

“…In general, one can perform two kinds of contractions, either working at the level of the r-matrix (this is a "coboundary" contraction), or working directly at the level of the co-commutators (this is the so-called "fundamental" contraction) [38,48]. As was shown in [30], this distinction is especially relevant in the case of the Galilean limit of κ-Poincaré, where the two procedures are nonequivalent.…”

The noncommutative space of light-like worldlines

“…In this way, the latter provides the deformed analogue of the classical relativistic symmetries for the quantum spacetime, and allows the connection with the Deformed Special Relativity (DSR) approach to quantum gravity phenomenology in which the quantum deformation parameter is assumed to be related to the Planck scale [12][13][14][15][16][17][18]. So far, the most studied noncommutative spacetime model arising in this framework is the so-called κ-Minkowski spacetime (see [19][20][21][22][23][24][25][26][27][28][29][30] and references therein), which is covariant under the (time-like) quantum deformation known as the κ-Poincaré algebra [19,[31][32][33][34][35]. Also, its non-vanishing cosmological constant counterpart, the κ-(A)dS noncommutative spacetime, has also been recently constructed [27] from the corresponding κ-(A)dS quantum group introduced in [36].…”