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

TL;DR

This paper classifies all fundamental quantum deformations of the $D=3$, $N=1$ Lorentz supersymmetry algebra, revealing four distinct Hopf-algebraic structures including standard and twist deformations.

Contribution

It provides a complete classification of all basic quantum deformations of the $D=3$, $N=1$ Lorentz supersymmetry algebra using classical $r$-matrices, including explicit forms.

Findings

Four different quantum deformations identified for the Lorentz supersymmetry.

Only one quantum deformation for the pseudoreal compact form.

Explicit forms of all basic Hopf-algebraic quantum deformations provided.

Abstract

By the supersymmetrization of a simple algebraic technique proposed in \cite{LuTo2017} we obtain the complete classification of all basic (nonisomorphic) quantum deformations for the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2;\mathbb{C})$ 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 $D=3$, $N=1$ Lorentz supersymmetry, which is the non-compact real form of $\mathfrak{osp}(1|2;\mathbb{C})$, 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.