On the Classification of Lie Bialgebras by Cohomological Means

TL;DR

This paper classifies Lie bialgebra structures on simple Lie algebras using cohomological methods, providing new insights and solutions for Drinfeld-Jimbo Lie bialgebras over fields of characteristic zero.

Contribution

It introduces a cohomological framework for classifying Lie bialgebras, including a new interpretation of scalar multiples and solutions for non-split cases.

Findings

Classification achieved via faithfully flat cohomology over arbitrary rings.

Solved for Drinfeld-Jimbo Lie bialgebras over fields of characteristic zero.

Provided a new cohomological interpretation of scalar multiples of Lie bialgebras.

Abstract

We approach the classification of Lie bialgebra structures on simple Lie algebras from the viewpoint of descent and non-abelian cohomology. We achieve a description of the problem in terms faithfully flat cohomology over an arbitrary ring over $\mathbb{Q}$, and solve it for Drinfeld-Jimbo Lie bialgebras over fields of characteristic zero. We consider the classification up to isomorphism, as opposed to equivalence, and treat split and non-split Lie algebras alike. We moreover give a new interpretation of scalar multiples of Lie bialgebras hitherto studied using twisted Belavin-Drinfeld cohomology.