Classification of real low dimensional Jacobi(generalized)-Lie bialgebras

TL;DR

This paper introduces a method to classify low-dimensional real Jacobi-Lie bialgebras using structure constants and automorphism groups, successfully classifying all two- and three-dimensional cases.

Contribution

It provides a systematic classification method for low-dimensional Jacobi-Lie bialgebras based on their structure constants and automorphisms.

Findings

Classified all real two-dimensional Jacobi-Lie bialgebras.

Classified all real three-dimensional Jacobi-Lie bialgebras.

Developed a new method using automorphism groups for classification.

Abstract

We describe the definition of Jacobi (generalized)-Lie bialgebras $(({\bf{g}},\phi_{0}),({\bf{g}}^{*},X_{0}))$ in terms of structure constants of the Lie algebras ${\bf{g}}$ and ${\bf{g}}^{*}$ and components of their 1-cocycles $X_{0}\in {\bf{g}}$ and $\phi_{0}\in {\bf{g}}^{*}$ in the basis of the Lie algebras. Then, using adjoint representations and automorphism Lie groups of Lie algebras, we give a method for classification of real low dimensional Jacobi-Lie bialgebras. In this way, we obtain and classify real two and three dimensional Jacobi-Lie bialgebras.