Generalized classical Yang-Baxter equation and regular decompositions

Abstract

The focus of the paper is on constructing new solutions of the generalized classical Yang-Baxter equation (GCYBE) that are not skew-symmetric. Using regular decompositions of finite-dimensional simple Lie algebras, we construct Lie algebra decompositions of \(\mathfrak {g}(\!(x)\!) \times \mathfrak {g}[x]/x^m \mathfrak {g}[x]\). The latter decompositions are in bijection with the solutions to the GCYBE. Under appropriate regularity conditions, we obtain a partial classification of such solutions. The paper is concluded with the presentations of the Gaudin-type models associated to these solutions.