Seminarium KMMF "Teoria Dwoistości"

A coboundary Lie bialgebra is a Lie algebra g equipped with a map δ : v ∈ g → [v, r]_S ∈Λ^2 g, where [·, ·]_S is the algebraic Schouten bracket on the Grassmann algebra Λg and r ∈ Λ^2 g is a solution of the modified classical Yang–Baxter equation (MCYBE), i.e.[v, [r, r]_S ]_S = 0 for any v ∈ g. The classification and properties of solutions of the MCYBE are well-studied mostly for semisimple Lie algebras or when dim g ≤ 3. To tackle non-semisimple and higher-dimensional cases, one needs new tools. In this talk, I will discussthe use of gradations on g and Λg in finding solutions and studying the structure of theMCYBE. Several examples will be presented to illustrate this approach.