Belavin-Drinfeld quantum groups and Lie bialgebras: Galois cohomology considerations

TL;DR

This paper connects Belavin-Drinfeld cohomologies with Galois cohomology to better understand quantum groups and Lie bialgebras over non-algebraically closed fields, confirming conjectures for classical types.

Contribution

It establishes a general framework linking Belavin-Drinfeld cohomologies to Galois cohomology, extending known results to broader classes of Lie algebras.

Findings

Unified the cohomologies with Galois cohomology for quantum groups.

Proved conjectures for classical Lie algebra types.

Generalized results to non-split simple Lie algebras.

Abstract

We relate the Belavin--Drinfeld cohomologies (twisted and untwisted) that have been introduced in the literature to study certain families of quantum groups and Lie bialgebras over a non algebraically closed field $\mathbb K$ of characteristic 0 to the standard non-abelian Galois cohomology $H^1(\mathbb K, \mathbf H)$ for a suitable algebraic $\mathbb K$-group $\mathbf H.$ The approach presented allows us to establish in full generality certain conjectures that were known to hold for the classical types of the split simple Lie algebras.