Darboux families and the classification of real four-dimensional indecomposable coboundary Lie bialgebras

This work introduces a new concept, the so-called Darboux family, which is employed to determine, to analyse geometrically, and to classify up to Lie algebra automorphisms, in a relatively easy manner, coboundary Liebialgebras on real four-dimensional indecomposable Lie algebras. The Darboux family notion can be consideredas a generalisation of the Darboux polynomial for a vector field. The classification of $r$-matrices and solutions to classical Yang-Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed.