Classification of classical twists of the standard Lie bialgebra structure on a loop algebra

TL;DR

This paper classifies all classical twists of the standard Lie bialgebra structure on loop algebras using Belavin-Drinfeld quadruples, connecting solutions of the classical Yang-Baxter equation to known classifications.

Contribution

It provides a complete classification of classical twists of the Lie bialgebra structures on loop algebras via Belavin-Drinfeld quadruples, addressing open questions on quasi-trigonometric solutions.

Findings

Classification of twists via Belavin-Drinfeld quadruples

Reduction of the problem to known CYBE solutions

Resolution of open questions on quasi-trigonometric solutions

Abstract

The standard Lie bialgebra structure on an affine Kac-Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. In this paper we classify all classical twists of the induced Lie bialgebra structures in terms of Belavin-Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced bialgebra structures are defined by certain solutions of the classical Yang-Baxter equation (CYBE) with two parameters. Then, using the algebro-geometric theory of CYBE, based on torsion free coherent sheaves, we reduce the problem to the well-known classification of trigonometric solutions given by Belavin and Drinfeld. The classification of twists in the case of parabolic subalgebras allows us to answer recently posed open questions regarding the so-called quasi-trigonometric solutions of CYBE.