Abstract
We relate the Belavin–Drinfeld cohomologies (twisted and untwisted) that have been introduced in the literature to study certain families of quantum groups and Lie bialgebras over a non algebraically closed field mathbb {K} of characteristic 0 to the standard non-abelian Galois cohomology H^1(mathbb {K}, mathbf{H}) for a suitable algebraic mathbb {K}-group mathbf{H}. The approach presented allows us to establish in full generality certain conjectures that were known to hold for the classical types of the split simple Lie algebras.
Highlights
The appearance of Galois cohomology in the classification of certain quantum groups is one of the primary goals of this paper
The linearization problem is an extremely technical construction brought forward as a conjecture in the work of Drinfeld [5], and proved in the seminal work of Etingof and Kazhdan. An outline of this correspondence can be found in the Introductions of [9,11], wherein one can find an explanation of why the description of which Lie bialgebras structures exists on the Lie algebra g ⊗k k((t)), with g simple finite dimensional over an algebraically closed field k of characteristic 0, arise naturally in the classification of quantum groups
The methods that we describe open an avenue for further studies of Lie bialgebra structures over non-algebraically closed fields
Summary
The appearance of Galois cohomology in the classification of certain quantum groups is one of the primary goals of this paper. The linearization problem is an extremely technical construction brought forward as a conjecture in the work of Drinfeld [5] (see [3] and [4]), and proved in the seminal work of Etingof and Kazhdan (see [6,7]) An outline of this correspondence can be found in the Introductions of [9,11], wherein one can find an explanation of why the description of which Lie bialgebras structures exists on the Lie algebra g ⊗k k((t)), with g simple finite dimensional over an algebraically closed field k of characteristic 0, arise naturally in the classification of quantum groups. The methods that we describe open an avenue for further studies of Lie bialgebra structures over non-algebraically closed fields
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