Classification of quantum groups and Lie bialgebra structures on $sl(n,\mathbb{F})$. Relations with Brauer group

TL;DR

This paper classifies Lie bialgebra structures on $sl(n,ield)$ over any characteristic 0 field, linking them to classical doubles, cohomology, and the Brauer group, revealing new structural insights.

Contribution

It introduces a comprehensive classification of Lie bialgebras on $sl(n,ield)$ using classical doubles and Belavin--Drinfeld cohomology, connecting to the Brauer group.

Findings

Classifies Lie bialgebra structures via classical doubles over various algebra extensions.

Introduces Belavin--Drinfeld cohomology for classification up to gauge equivalence.

Establishes a map between cohomology sets and the Brauer group for specific $r$-matrices.

Abstract

Given an arbitrary field $\mathbb{F}$ of characteristic 0, we study Lie bialgebra structures on $sl(n,\mathbb{F})$, based on the description of the corresponding classical double. For any Lie bialgebra structure $\delta$, the classical double $D(sl(n,\mathbb{F}),\delta)$ is isomorphic to $sl(n,\mathbb{F})\otimes_{\mathbb{F}} A$, where $A$ is either $\mathbb{F}[\varepsilon]$, with $\varepsilon^{2}=0$, or $\mathbb{F}\oplus \mathbb{F}$ or a quadratic field extension of $\mathbb{F}$. In the first case, the classification leads to quasi-Frobenius Lie subalgebras of $sl(n,\mathbb{F})$. In the second and third cases, a Belavin--Drinfeld cohomology can be introduced which enables one to classify Lie bialgebras on $sl(n,\mathbb{F})$, up to gauge equivalence. The Belavin--Drinfeld untwisted and twisted cohomology sets associated to an $r$-matrix are computed. For the Cremmer--Gervais $r$-matrix in $sl(3)$, we also construct a natural map of sets between the total Belavin--Drinfeld twisted cohomology set and the Brauer group of the field $\mathbb{F}$.