Geodesic
dS--AdS Structures in Noncommutative Minkowski Spaces
Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 3, pp. 513-543

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We consider a family of noncommutative four-dimensional Minkowski spaces with the signature $(1,3)$ and two types of spaces with the signature $(2,2)$. The Minkowski spaces are defined by the common reflection equation and differ in anti-involutions. There exist two Casimir elements, and. xing one of them leads to the noncommutative “homogeneous” spaces $H_3$, $dS_3$, $AdS_3$, and light cones. We present a semiclassical description of the Minkowski spaces. There are three compatible Poisson structures: quadratic, linear, and canonical. Quantizing the first leads to the Minkowski spaces. We introduce horospheric generators of the Minkowski spaces, and they lead to the horospheric description of $H_3$, $dS_3$, and $AdS_3$. We construct irreducible representations of the Minkowski spaces $H_3$ and $dS_3$. We find eigenfunctions of the Klein–Gordon equation in terms of the horospheric generators of the Minkowski spaces, and they lead to eigenfunctions on $H_3$, $dS_3$, $AdS_3$, and light cones.
Keywords: noncommutative geometry, Yang–Baxter equation, reflection equation, harmonic analysis on noncommutative spaces. 
@article{TMF_2005_144_3_a5,
     author = {M. A. Olshanetsky and V.-B. K. Rogov},
     title = {dS--AdS {Structures} in {Noncommutative} {Minkowski} {Spaces}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {513--543},
     publisher = {mathdoc},
     volume = {144},
     number = {3},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2005_144_3_a5/}
}
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M. A. Olshanetsky; V.-B. K. Rogov. dS--AdS Structures in Noncommutative Minkowski Spaces. Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 3, pp. 513-543. http://geodesic.mathdoc.fr/item/TMF_2005_144_3_a5/