Classification of quantum groups and Lie bialgebra structures on sl(n,F)sl(n,F). Relations with Brauer group

Given an arbitrary field FF of characteristic 0, we study Lie bialgebra structures on sl(n,F)sl(n,F), based on the description of the corresponding classical double. For any Lie bialgebra structure δ  , the classical double D(sl(n,F),δ)D(sl(n,F),δ) is isomorphic to sl(n,F)⊗FAsl(n,F)⊗FA, where A   is either F[ε]F[ε], with ε2=0ε2=0, or F⊕FF⊕F or a quadratic field extension of FF. In the first case, the classification leads to quasi-Frobenius Lie subalgebras of sl(n,F)sl(n,F). In the second and third cases, a Belavin–Drinfeld cohomology can be introduced which enables one to classify Lie bialgebras on sl(n,F)sl(n,F), up to gauge equivalence. The Belavin–Drinfeld untwisted and twisted cohomology sets associated to an r-matrix are computed. For the Cremmer–Gervais r  -matrix in sl(3)sl(3), we also construct a natural map of sets between the total Belavin–Drinfeld twisted cohomology set and the Brauer group of the field FF.