Generalized classical Yang-Baxter equation and regular decompositions

TL;DR

This paper develops a method to construct new, non-skew-symmetric solutions to the generalized classical Yang-Baxter equation using regular decompositions of Lie algebras, leading to partial classifications and related integrable models.

Contribution

It introduces a novel approach to generate and classify solutions of the GCYBE via regular decompositions of simple Lie algebras, extending the understanding of these solutions.

Findings

Constructed new solutions of GCYBE using Lie algebra decompositions.

Established a bijection between decompositions and solutions of GCYBE.

Presented Gaudin-type models associated with the solutions.

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.