Classical r-matrices of two and three dimensional Lie super-bialgebras and their Poisson-Lie supergroups

TL;DR

This paper classifies all two and three dimensional coboundary Lie super-bialgebras, computes their classical r-matrices, and derives associated super Poisson structures on Poisson-Lie supergroups.

Contribution

It provides a complete classification of low-dimensional Lie super-bialgebras and explicitly constructs their classical r-matrices and Poisson-Lie supergroup structures.

Findings

Classified all 2D and 3D coboundary Lie super-bialgebras.

Derived classical r-matrices for these super-bialgebras.

Constructed super Poisson structures on corresponding supergroups.

Abstract

We obtain the classical r-matrices of two and three dimensional Lie super-bialgebras. We thus classify all two and three dimensional coboundary Lie super-bialgebras and their types (triangular, quasi-triangular, or factorable). Using the Sklyanin superbracket, we then obtain the super Poisson structures on the related Poisson-Lie supergroups.