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Q A ] 2 8 A ug 2 01 9 Classification of Quantum Groups via Galois Cohomology
| Content Provider | Semantic Scholar |
|---|---|
| Author | Karolinsky, Eugene Pianzola, Arturo Stolin, Alexander |
| Copyright Year | 2019 |
| Abstract | The first example of a quantum group was introduced by P. Kulish and N. Reshetikhin. In the paper [17], they found a new algebra which was later called Uq(sl(2)). Their example was developed independently by V. Drinfeld and M. Jimbo, which resulted in the general notion of quantum group. Later, a complimentary approach to quantum groups was developed by L. Faddeev, N. Reshetikhin, and L. Takhtajan in [12]. Recently, the so-called Belavin–Drinfeld cohomology (twisted and nontwisted) 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 H(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. N. Delone and D. K. Faddeev in [5]. Our results show that there exist yet unknown quantum groups for Lie algebras of the types An, D2n+1, E6, not mentioned in [9]. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv-export-lb.library.cornell.edu/pdf/1806.05640 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
