Classification of Quantum Groups via Galois Cohomology

The first example of a quantum group was introduced by P.\ua0Kulish and N.\ua0Reshetikhin. In the paper Kulish et\ua0al. (J Soviet Math 23:2435–2441, 1983), they found a new algebra which was later called Uq(sl(2)). Their example was developed independently by V.\ua0Drinfeld and M.\ua0Jimbo, which resulted in the general notion of quantum group. Later, a complimentary approach to quantum groups was developed by L.\ua0Faddeev, N.\ua0Reshetikhin, and L.\ua0Takhtajan in Faddeev et\ua0al. (Leningr Math J 1:193–225, 1990). Recently, the so-called Belavin–Drinfeld cohomology (twisted and non-twisted) have been introduced in the literature to study and classify certain families of quantum groups and Lie bialgebras. Later, the last two authors interpreted non-twisted Belavin–Drinfeld cohomology 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. The non-twisted case is thus fully understood in terms of Galois cohomology. The twisted case has only been studied using Galois cohomology for the so-called (“standard”) Drinfeld–Jimbo structure. The aim of the present paper is to extend these results to all twisted Belavin–Drinfeld cohomology and thus, to present classification of quantum groups in terms of Galois cohomology and the so-called orders. Low dimensional cases sl(2) and sl(3) are considered in more details using a theory of cubic rings developed by B.\ua0N.\ua0Delone and D.\ua0K.\ua0Faddeev in Delone and Faddeev (The theory of irrationalities of the third degree. Translations of mathematical monographs, vol 10. AMS, Providence, 1964). Our results show that there exist yet unknown quantum groups for Lie algebras of the types An, D2n+1, E6, not mentioned in Etingof et\ua0al. (J Am Math Soc 13:595–609, 2000)