A Grassmann and Graded Approach to Coboundary Lie Bialgebras, their Classification, and Yang-Baxter Equations

J. de Lucas; D. Wysocki

doi:10.5802/jolt.1155

J. de Lucas; D. Wysocki

Journal of Lie Theory, Volume 30 (2020) no. 4, pp. 1161-1194

Abstract

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.

arXiv Zbl

DOI: 10.5802/jolt.1155

Classification: 17B62, 17B22, 17B40
Keywords: Algebraic Schouten bracket, g-invariant metric, gradation, Grassmann algebra, Lie bialgebra, root decomposition, Killing form

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     author = {J. de Lucas and D. Wysocki},
     title = {A {Grassmann} and {Graded} {Approach} to {Coboundary} {Lie} {Bialgebras,} their {Classification,} and {Yang-Baxter} {Equations}},
     journal = {Journal of Lie Theory},
     pages = {1161--1194},
     year = {2020},
     volume = {30},
     number = {4},
     doi = {10.5802/jolt.1155},
     zbl = {1478.17021},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1155/}
}

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J. de Lucas; D. Wysocki. A Grassmann and Graded Approach to Coboundary Lie Bialgebras, their Classification, and Yang-Baxter Equations. Journal of Lie Theory, Volume 30 (2020) no. 4, pp. 1161-1194. doi: 10.5802/jolt.1155

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