All basic quantizations of $$D=3$$ D = 3 , $$N=1$$ N = 1 Lorentz supersymmetry

Abstract

Abstract By the supersymmetrization of a simple algebraic technique proposed in Lukierski and Tolstoy (Eur Phys J C 77:226, 2017) we obtain the complete classification of all basic (nonisomorphic) quantum deformations for the orthosymplectic Lie superalgebra $$\mathfrak {osp}(1|2;{\mathbb {C}})$$ osp ( 1 | 2 ; C ) and its pseudoreal and real forms in terms of the classical r-matrices. In particular, we prove that pseudoreal compact form has only one quantum deformation (standart q-analog), and the $$D=3$$ D = 3 , $$N=1$$ N = 1 Lorentz supersymmetry, which is the non-compact real form of $$\mathfrak {osp}(1|2;{\mathbb {C}})$$ osp ( 1 | 2 ; C ) , has four different Hopf-algebraic quantum deformations: two standard q-analogs, and two (Jordanian and super-Jordanian) twist deformations. All basic Hopf-algebraic quantum deformations are presented in the explicit form.