Lie bialgebras, fields of cohomological dimension at most 2 and Hilbert's Seventeenth Problem
Journal article, 2017

We investigate Lie bialgebra structures on simple Lie algebras of non-split type A. It turns out that there are several classes of such Lie bialgebra structures, and it is possible to classify some of them. The classification is obtained using Belavin–Drinfeld cohomology sets, which are introduced in the paper. Our description is particularly detailed over fields of cohomological dimension at most two, and is related to quaternion algebras and the Brauer group. We then extend the results to certain rational function fields over real closed fields via Pfister's theory of quadratic forms and his solution to Hilbert's Seventeenth Problem.

Lie bialgebra


Compact type

Brauer group

Quantum group

Belavin-Drinfeld cohomology

Pfister form


Seidon Alsaody

Chalmers, Mathematical Sciences, Mathematics

Alexander Stolin

Chalmers, Mathematical Sciences, Algebra and geometry

University of Gothenburg

Journal of Algebra

0021-8693 (ISSN) 1090-266X (eISSN)

Vol. 478 368-394

Subject Categories



Basic sciences



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