The first example of a quantum group was discovered by P. Kulish and N. Reshetikhin. In their paper "Quantum linear problem for the sine-Gordon equation and higher representations" published in Zap. Nauchn. Sem. LOMI, 1981, Volume 101 (English version: Journal of Soviet Mathematics, 1983, 23:4), they found a new algebra which was later called Uq (sl2). Their example was developed independently by V. Drinfeld and M. Jimbo to a general notion of quantum group.
Recently, the so-called Belavin-Drinfeld cohomologies (twisted and untwisted) have been introduced in the literature to study and classify certain families of quantum groups and Lie bialgebras. Later, Pianzola and Stolin have interpreted non-twisted Belavin-Drinfeld cohomologies in terms of non-abelian Galois cohomology H1(F, H) for a suitable algebraic F-group H. Here F is an arbitrary field of zero characteristic. So, the untwisted case is now fully understood in terms of Galois cohomologies, while the twisted case has only been studied for the so-called ("standard") Drinfeld-Jimbo structure.
The aim of the talk is to establish a Galois cohomology interpretation for all twisted Belavin-Drinfeld cohomologies and thus, to present the classification of quantum groups in terms of Galois cohomologies. Our results show that there exist yet unknown quantum groups for Lie algebras of types An, D2n+1 and E6 .