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

Abstract

This work pioneers the systematic study and classification (up to Lie algebra automorphisms) of finite-dimensional coboundary Lie bialgebras through Grassmann algebras. Several mathematical structures on Lie algebras, e.g. Killing forms or root decompositions, are extended to the Grassmann algebras of Lie algebras. This simplifies the description of the procedures and tools appearing in the theory of Lie bialgebras and originates novel techniques for its study and classification up to Lie algebra automorphisms. As a particular case, the classification of real three-dimensional coboundary Lie bialgebras is retrieved.

arXiv commentNew version has been significantly simplified and shortened by using new results on gradations. New applications have been added. 32 pages and 15 figures

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• Quantum $D=3$ Euclidean and Poincaré symmetries from contraction limits
• Darboux families and the classification of real four-dimensional indecomposable coboundary Lie bialgebras
• Stratified Lie systems: Theory and applications