Classical r-matrices of real low-dimensional Jacobi-Lie bialgebras and their Jacobi-Lie groups

TL;DR

This paper classifies low-dimensional real Jacobi-Lie bialgebras, computes their classical r-matrices, and explores associated Jacobi structures and integrable systems, advancing the understanding of their algebraic and geometric properties.

Contribution

It provides a complete classification of two and three dimensional coboundary Jacobi-Lie bialgebras and introduces a new method for constructing classical integrable systems.

Findings

Classified all non-isomorphic real two and three dimensional coboundary Jacobi-Lie bialgebras.

Derived the generalized Sklyanin bracket formula for Jacobi structures.

Presented a novel approach to construct classical integrable systems using coboundary Jacobi-Lie bialgebras.

Abstract

In this research we obtain the classical r-matrices of real two and three dimensional Jacobi-Lie bialgebras. In this way, we classify all non-isomorphic real two and three dimensional coboundary Jacobi-Lie bialgebras and their types (triangular and quasitriangular). Also, we obtain the generalized Sklyanin bracket formula by use of which, we calculate the Jacobi structures on the related Jacobi-Lie groups. Finally, we present a new method for constructing classical integrable systems using coboundary Jacobi-Lie bialgebras.