Classification of quasi-trigonometric solutions of the classical Yang-Baxter equation

TL;DR

This paper classifies quasi-trigonometric solutions to the classical Yang-Baxter equation for simple Lie algebras, linking them to Dynkin diagram substructures and providing a quantization method.

Contribution

It provides a complete classification of quasi-trigonometric solutions using Dynkin diagram sub-diagrams and details their quantization process.

Findings

Complete list of quasi-trigonometric solutions derived

Classification based on Dynkin diagram sub-diagrams

Method for quantizing related Lie bialgebra structures

Abstract

It was proved by Montaner and Zelmanov that up to classical twisting Lie bialgebra structures on $\mathfrak{g}[u]$ fall into four classes. Here $\mathfrak{g}$ is a simple complex finite-dimensional Lie algebra. It turns out that classical twists within one of these four classes are in a one-to-one correspondence with the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. In this paper we give a complete list of the quasi-trigonometric solutions in terms of sub-diagrams of the certain Dynkin diagrams related to $\mathfrak{g}$. We also explain how to quantize the corresponding Lie bialgebra structures.