Lie Bialgebras, Fields of Cohomological Dimension at Most 2 and Hilbert's Seventeenth Problem

TL;DR

This paper classifies Lie bialgebra structures on simple Lie algebras of non-split type A using Belavin--Drinfeld cohomology, especially over fields with cohomological dimension at most two, and extends results to rational function fields over real closed fields.

Contribution

It introduces Belavin--Drinfeld cohomology sets for classifying Lie bialgebras and extends the classification to fields related to Hilbert's Seventeenth Problem.

Findings

Classified Lie bialgebra structures on simple Lie algebras of non-split type A.

Connected classifications to quaternion algebras and the Brauer group.

Extended results to rational function fields over real closed fields.

Abstract

We investigate Lie bialgebra structures on simple Lie algebras of non-split type $A$. It turns out that there are several classes of such Lie bialgebra structures, and it is possible to classify some of them. The classification is obtained using Belavin--Drinfeld cohomology sets, which are introduced in the paper. Our description is particularly detailed over fields of cohomological dimension at most two, and is related to quaternion algebras and the Brauer group. We then extend the results to certain rational function fields over real closed fields via Pfister's theory of quadratic forms and his solution to Hilbert's Seventeenth Problem.