A Grassmann and graded approach to coboundary Lie bialgebras, their classification, and Yang-Baxter equations

We devise geometric, graded algebra, and Grassmann methods to study and to classify finite-dimensional coboundary Lie bialgebras. Mathematical structures on Lie algebras, like Killing forms, root decompositions, and gradations, are extended to their Grassmann algebras. The classification of real three-dimensional coboundary Lie bialgebras and gl2 up to Lie algebra automorphisms is retrieved throughout devised methods. The structure of modified classical Yang-Baxter equations on so(2,2) and so(3,2) are studied and r-matrices are found.