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A Grassmann and graded approach to coboundary Lie bialgebras, their classification, and Yang-Baxter equations. (English) Zbl 1478.17021

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.

MSC:

17B62 Lie bialgebras; Lie coalgebras
16T25 Yang-Baxter equations
17B22 Root systems
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
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