On classification of quantum groups and Belavin-Drinfeld twisted cohomologies

TL;DR

This paper advances the classification of quantum groups by analyzing Belavin-Drinfeld twisted cohomologies for specific Lie algebras and r-matrices, deepening understanding of their algebraic structures.

Contribution

It investigates twisted cohomologies for $sl(n)$ with Cremmer-Gervais r-matrices and for $o(n)$, extending previous classification frameworks.

Findings

Classification of twisted cohomologies for $sl(n)$ and $o(n)$.

Relationship between cohomology classes and Lie bialgebra structures.

Extension of quantum group classification to new algebraic cases.

Abstract

The present article is a continuation of QA/1303.4046, where we discussed the classification of quantum groups with quasi-classical limit $\mathfrak{g}$ and introduced a theory of Belavin-Drinfeld cohomology associated to any non-skewsymmetric $r$-matrix. Depending on the form of the corresponding double, there exists a one-to-one correspondence between gauge equivalence classes of Lie bialgebra structures on $\mathfrak{g}\otimes_{\mathbb{C}}\mathbb{K}$, where $\mathbb{K}=\mathbb{C}((\hbar))$, and untwisted or twisted cohomology classes. In the present paper we investigate twisted cohomologies for $sl(n)$ associated to generalized Cremmer-Gervais $r$-matrices, and twisted cohomologies for $o(n)$.