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Journal of High Energy Physics

, 2017:187 | Cite as

Basic quantizations of D = 4 Euclidean, Lorentz, Kleinian and quaternionic \( {\mathfrak{o}}^{\star }(4) \) symmetries

  • A. Borowiec
  • J. Lukierski
  • V.N. Tolstoy
Open Access
Regular Article - Theoretical Physics

Abstract

We construct firstly the complete list of five quantum deformations of D = 4 complex homogeneous orthogonal Lie algebra \( \mathfrak{o}\left(4;\mathbb{C}\right)\cong \mathfrak{o}\left(3;\mathbb{C}\right)\oplus \mathfrak{o}\left(3;\mathbb{C}\right) \), describing quantum rotational symmetries of four-dimensional complex space-time, in particular we provide the corresponding universal quantum R-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of \( \mathfrak{o}\left(4;\mathbb{C}\right) \): Euclidean \( \mathfrak{o}(4) \), Lorentz \( \mathfrak{o}\left(3,\ 1\right) \), Kleinian \( \mathfrak{o}\left(2,\ 2\right) \) and quaternionic \( {\mathfrak{o}}^{\star }(4) \). For \( \mathfrak{o}\left(3,\ 1\right) \) we only recall well-known results obtained previously by the authors, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) as well as for the complex Lie algebra \( \mathfrak{o}\left(4;\mathbb{C}\right) \) we present new results.

Keywords

Quantum Groups Models of Quantum Gravity Non-Commutative Geometry 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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© The Author(s) 2017

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsUniversity of WroclawWroclawPoland
  2. 2.Skobeltsyn Institute of Nuclear PhysicsLomonosov Moscow State UniversityMoscowRussian Federation

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