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

Abstract

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.