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 (...
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Springer
2017-04-01
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| Series: | European Physical Journal C: Particles and Fields |
| Online Access: | http://link.springer.com/article/10.1140/epjc/s10052-017-4786-9 |
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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 |
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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 |