Journal of Lie Theory 30 (2020), No. 4, 1161--1194
Copyright Heldermann Verlag 2020
A Grassmann and Graded Approach to Coboundary Lie Bialgebras, their Classification, and Yang-Baxter Equations
Javier de Lucas
Dept. of Mathematical Methods in Physics, University of Warsaw, 02-093 Warsaw, Poland
javier.de.lucas@fuw.edu.pl
Daniel Wysocki
Dept. of Mathematical Methods in Physics, University of Warsaw, 02-093 Warsaw, Poland
daniel.wysocki@fuw.edu.pl
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.
Keywords: Algebraic Schouten bracket, g-invariant metric, gradation, Grassmann algebra, Lie bialgebra, root decomposition, Killing form.
MSC: 17B62; 17B22, 17B40.
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