Quantizations of $$D=3$$ D = 3 Lorentz symmetry

J Lukierski,V N Tolstoy

doi:10.1140/epjc/s10052-017-4786-9

J Lukierski, V N Tolstoy

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https://doi.org/10.1140/epjc/s10052-017-4786-9

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Abstract

Using the isomorphism {mathfrak {o}}(3;{mathbb {C}})simeq {mathfrak {sl}}(2;{mathbb {C}}) we develop a new simple algebraic technique for complete classification of quantum deformations (the classical r-matrices) for real forms {mathfrak {o}}(3) and {mathfrak {o}}(2,1) of the complex Lie algebra {mathfrak {o}}(3;{mathbb {C}}) in terms of real forms of {mathfrak {sl}}(2;{mathbb {C}}): {mathfrak {su}}(2), {mathfrak {su}}(1,1) and {mathfrak {sl}}(2;{mathbb {R}}). We prove that the D=3 Lorentz symmetry {mathfrak {o}}(2,1)simeq {mathfrak {su}}(1,1)simeq {mathfrak {sl}}(2;{mathbb {R}}) has three different Hopf-algebraic quantum deformations, which are expressed in the simplest way by two standard {mathfrak {su}}(1,1) and {mathfrak {sl}}(2;{mathbb {R}})q-analogs and by simple Jordanian {mathfrak {sl}}(2;{mathbb {R}}) twist deformation. These quantizations are presented in terms of the quantum Cartan–Weyl generators for the quantized algebras {mathfrak {su}}(1,1) and {mathfrak {sl}}(2;{mathbb {R}}) as well as in terms of quantum Cartesian generators for the quantized algebra {mathfrak {o}}(2,1). Finally, some applications of the deformed D=3 Lorentz symmetry are mentioned.

Highlights

The search for quantum gravity is linked with studies of noncommutative space-times and quantum deformations of space-time symmetries
We recall that in relativistic theories the basic role is played by Lorentz symmetries and the Lorentz algebra, i.e. all aspects of their quantum deformations should be studied in a very detailed and careful way
The Lie bialgebra (g, δ) is a correct infinitesimalization of the quantum Hopf deformation of U (g) and the operation δ is an infinitesimal part of the difference between a coproduct and an opposite coproductin the Hopf algebra, δ(x) = h−1( − ̃ ) mod h where h is a deformation parameter

Summary

Introduction

The search for quantum gravity is linked with studies of noncommutative space-times and quantum deformations of space-time symmetries. 5 all three Hopf-algebraic quantizations (explicit quantum deformations) of the real D = 3 Lorentz symmetry are presented in detail: quantized bases, coproducts and universal R-matrices are given.

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