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 is reduced 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: ${\mathbb{K}[\varepsilon]}$, where ${\varepsilon^{2}=0}$, ${\mathbb{K}\oplus\mathbb{K}}$ or ${\mathbb{K}[j]}$, where ${j^{2}=\hbar}$. The first case is related 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 on the Belavin–Drinfeld list (Belavin and Drinfeld in Soviet Sci Rev Sect C: Math Phys Rev 4:93–165, 1984). 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.