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TMF, 2005, Volume 144, Number 3, Pages 513–543 (Mi tmf1874)  

dS–AdS Structures in Noncommutative Minkowski Spaces

M. A. Olshanetskyab, V.-B. K. Rogovc

a Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)
b University of Aarhus
c Moscow State University of Railway Communications

Abstract: 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

DOI: https://doi.org/10.4213/tmf1874

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English version:
Theoretical and Mathematical Physics, 2005, 144:3, 1315–1343

Bibliographic databases:

Received: 17.11.2004

Citation: M. A. Olshanetsky, V.-B. K. Rogov, “dS–AdS Structures in Noncommutative Minkowski Spaces”, TMF, 144:3 (2005), 513–543; Theoret. and Math. Phys., 144:3 (2005), 1315–1343

Citation in format AMSBIB
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\paper dS--AdS Structures in Noncommutative Minkowski Spaces
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\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2005TMP...144.1315O}
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\transl
\jour Theoret. and Math. Phys.
\yr 2005
\vol 144
\issue 3
\pages 1315--1343
\crossref{https://doi.org/10.1007/s11232-005-0162-2}
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