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

Abstract Using the isomorphism $${\mathfrak {o}}(3;{\mathbb {C}})\simeq {\mathfrak {sl}}(2;{\mathbb {C}})$$ o ( 3 ; C ) ≃ sl ( 2 ; C ) we develop a new simple algebraic technique for complete classification of quantum deformations (the classical r-matrices) for real forms $${\mathfrak {o}}(3)$$ o (...

Full description

Bibliographic Details
Main Authors: J. Lukierski, V. N. Tolstoy
Format: Article
Language:English
Published: Springer 2017-04-01
Series:European Physical Journal C: Particles and Fields
Online Access:http://link.springer.com/article/10.1140/epjc/s10052-017-4786-9
id doaj-art-e306b78b57c74630b6a2415072319623
recordtype oai_dc
spelling doaj-art-e306b78b57c74630b6a24150723196232018-08-20T15:22:34ZengSpringerEuropean Physical Journal C: Particles and Fields1434-60441434-60522017-04-0177411210.1140/epjc/s10052-017-4786-9Quantizations of $$D=3$$ D = 3 Lorentz symmetryJ. Lukierski0V. N. Tolstoy1Institute for Theoretical Physics, University of WrocławInstitute for Theoretical Physics, University of WrocławAbstract Using the isomorphism $${\mathfrak {o}}(3;{\mathbb {C}})\simeq {\mathfrak {sl}}(2;{\mathbb {C}})$$ o ( 3 ; C ) ≃ sl ( 2 ; C ) we develop a new simple algebraic technique for complete classification of quantum deformations (the classical r-matrices) for real forms $${\mathfrak {o}}(3)$$ o ( 3 ) and $${\mathfrak {o}}(2,1)$$ o ( 2 , 1 ) of the complex Lie algebra $${\mathfrak {o}}(3;{\mathbb {C}})$$ o ( 3 ; C ) in terms of real forms of $${\mathfrak {sl}}(2;{\mathbb {C}})$$ sl ( 2 ; C ) : $${\mathfrak {su}}(2)$$ su ( 2 ) , $${\mathfrak {su}}(1,1)$$ su ( 1 , 1 ) and $${\mathfrak {sl}}(2;{\mathbb {R}})$$ sl ( 2 ; R ) . We prove that the $$D=3$$ D = 3 Lorentz symmetry $${\mathfrak {o}}(2,1)\simeq {\mathfrak {su}}(1,1)\simeq {\mathfrak {sl}}(2;{\mathbb {R}})$$ o ( 2 , 1 ) ≃ su ( 1 , 1 ) ≃ sl ( 2 ; R ) has three different Hopf-algebraic quantum deformations, which are expressed in the simplest way by two standard $${\mathfrak {su}}(1,1)$$ su ( 1 , 1 ) and $${\mathfrak {sl}}(2;{\mathbb {R}})$$ sl ( 2 ; R ) q-analogs and by simple Jordanian $${\mathfrak {sl}}(2;{\mathbb {R}})$$ sl ( 2 ; R ) twist deformation. These quantizations are presented in terms of the quantum Cartan–Weyl generators for the quantized algebras $${\mathfrak {su}}(1,1)$$ su ( 1 , 1 ) and $${\mathfrak {sl}}(2;{\mathbb {R}})$$ sl ( 2 ; R ) as well as in terms of quantum Cartesian generators for the quantized algebra $${\mathfrak {o}}(2,1)$$ o ( 2 , 1 ) . Finally, some applications of the deformed $$D=3$$ D = 3 Lorentz symmetry are mentioned.http://link.springer.com/article/10.1140/epjc/s10052-017-4786-9
institution Open Data Bank
collection Open Access Journals
building Directory of Open Access Journals
language English
format Article
author J. Lukierski
V. N. Tolstoy
spellingShingle J. Lukierski
V. N. Tolstoy
Quantizations of $$D=3$$ D = 3 Lorentz symmetry
European Physical Journal C: Particles and Fields
author_facet J. Lukierski
V. N. Tolstoy
author_sort J. Lukierski
title Quantizations of $$D=3$$ D = 3 Lorentz symmetry
title_short Quantizations of $$D=3$$ D = 3 Lorentz symmetry
title_full Quantizations of $$D=3$$ D = 3 Lorentz symmetry
title_fullStr Quantizations of $$D=3$$ D = 3 Lorentz symmetry
title_full_unstemmed Quantizations of $$D=3$$ D = 3 Lorentz symmetry
title_sort quantizations of $$d=3$$ d = 3 lorentz symmetry
publisher Springer
series European Physical Journal C: Particles and Fields
issn 1434-6044
1434-6052
publishDate 2017-04-01
description Abstract Using the isomorphism $${\mathfrak {o}}(3;{\mathbb {C}})\simeq {\mathfrak {sl}}(2;{\mathbb {C}})$$ o ( 3 ; C ) ≃ sl ( 2 ; C ) we develop a new simple algebraic technique for complete classification of quantum deformations (the classical r-matrices) for real forms $${\mathfrak {o}}(3)$$ o ( 3 ) and $${\mathfrak {o}}(2,1)$$ o ( 2 , 1 ) of the complex Lie algebra $${\mathfrak {o}}(3;{\mathbb {C}})$$ o ( 3 ; C ) in terms of real forms of $${\mathfrak {sl}}(2;{\mathbb {C}})$$ sl ( 2 ; C ) : $${\mathfrak {su}}(2)$$ su ( 2 ) , $${\mathfrak {su}}(1,1)$$ su ( 1 , 1 ) and $${\mathfrak {sl}}(2;{\mathbb {R}})$$ sl ( 2 ; R ) . We prove that the $$D=3$$ D = 3 Lorentz symmetry $${\mathfrak {o}}(2,1)\simeq {\mathfrak {su}}(1,1)\simeq {\mathfrak {sl}}(2;{\mathbb {R}})$$ o ( 2 , 1 ) ≃ su ( 1 , 1 ) ≃ sl ( 2 ; R ) has three different Hopf-algebraic quantum deformations, which are expressed in the simplest way by two standard $${\mathfrak {su}}(1,1)$$ su ( 1 , 1 ) and $${\mathfrak {sl}}(2;{\mathbb {R}})$$ sl ( 2 ; R ) q-analogs and by simple Jordanian $${\mathfrak {sl}}(2;{\mathbb {R}})$$ sl ( 2 ; R ) twist deformation. These quantizations are presented in terms of the quantum Cartan–Weyl generators for the quantized algebras $${\mathfrak {su}}(1,1)$$ su ( 1 , 1 ) and $${\mathfrak {sl}}(2;{\mathbb {R}})$$ sl ( 2 ; R ) as well as in terms of quantum Cartesian generators for the quantized algebra $${\mathfrak {o}}(2,1)$$ o ( 2 , 1 ) . Finally, some applications of the deformed $$D=3$$ D = 3 Lorentz symmetry are mentioned.
url http://link.springer.com/article/10.1140/epjc/s10052-017-4786-9
_version_ 1612687797664088064