Classification of quantum groups and Belavin--Drinfeld cohomologies for orthogonal and symplectic Lie algebras

TL;DR

This paper advances the classification of quantum groups by computing Belavin-Drinfeld cohomology for non-skewsymmetric r-matrices associated with orthogonal and symplectic Lie algebras, extending previous work on simple complex Lie algebras.

Contribution

It provides explicit calculations of Belavin-Drinfeld cohomology for all relevant r-matrices in types B, C, and D, enhancing understanding of quantum group classifications.

Findings

Computed cohomology for all non-skewsymmetric r-matrices in types B, C, D

Extended classification framework for quantum groups related to these Lie algebras

Clarified the structure of Belavin-Drinfeld cohomology in new Lie algebra types

Abstract

In this paper we continue to study Belavin-Drinfeld cohomology introduced in arXiv:1303.4046 [math.QA] and related to the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra. Here we compute Belavin-Drinfeld cohomology for all non-skewsymmetric $r$-matrices from the Belavin-Drinfeld list for simple Lie algebras of type $B$, $C$, and $D$.