  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.
