Classification of quantum groups and Belavin-Drinfeld cohomologies

Abstract: In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra $\mathfrak{g}$. This problem reduces to the classification of all Lie bialgebra structures on $\mathfrak{g}(\mathbb{K})$, where $\mathbb{K}=\mathbb{C}((\hbar))$. The associated classical double is of the form $\mathfrak{g}(\mathbb{K})\otimes_{\mathbb{K}} A$, where $A$ is one of the following: $\mathbb{K}[\epsilon]$, where $\epsilon^{2}=0$, $\mathbb{K}\oplus \mathbb{K}$ or $\mathbb{K}[j]$ where $j^{2}=\hbar$. The first case relates to quasi-Frobenius Lie algebras. In the second and third cases we introduce a theory of Belavin-Drinfeld cohomology associated to any non-skewsymmetric $r$-matrix from the Belavin-Drinfeld list. We prove a one-to-one correspondence between gauge equivalence classes of Lie bialgebra structures on $\mathfrak{g}(\mathbb{K})$ and cohomology classes (in case II) and twisted cohomology classes (in case III) associated to any non-skewsymmetric $r$-matrix.