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

Abstract

In this paper, we obtain the classical [Formula: see text]-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 quasi-triangular). 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.