de Lucas, Javier; Wysocki, Daniel A Grassmann and graded approach to coboundary Lie bialgebras, their classification, and Yang-Baxter equations. (English) Zbl 1478.17021 J. Lie Theory 30, No. 4, 1161-1194 (2020). Summary: 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 \(\mathfrak{gl}_2\) up to Lie algebra automorphisms is retrieved throughout devised methods. The structure of modified classical Yang-Baxter equations on \(\mathfrak{so}(2,2)\) and \(\mathfrak{so}(3,2)\) are studied and \(r\)-matrices are found. Cited in 3 Documents MSC: 17B62 Lie bialgebras; Lie coalgebras 16T25 Yang-Baxter equations 17B22 Root systems 17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras Keywords:algebraic Schouten bracket; \(\mathfrak{g}\)-invariant metric; gradation; Grassmann algebra; Lie bialgebra; root decomposition; Killing form PDFBibTeX XMLCite \textit{J. de Lucas} and \textit{D. Wysocki}, J. Lie Theory 30, No. 4, 1161--1194 (2020; Zbl 1478.17021) Full Text: arXiv Link