Alsaody, Seidon; Pianzola, Arturo

On the Classification of Lie Bialgebras by Cohomological Means

Doc. Math. 24, 2583-2612 (2019)
DOI: 10.25537/dm.2019v24.2583-2612
Communicated by Nikita Karpenko


We approach the classification of Lie bialgebra structures on simple Lie algebras from the viewpoint of descent and non-abelian cohomology. We achieve a description of the problem in terms of faithfully flat cohomology over an arbitrary ring over \(\mathbb{Q}\), and solve it for Drinfeld-Jimbo Lie bialgebras over fields of characteristic zero. We consider the classification up to isomorphism, as opposed to equivalence, and treat split and non-split Lie algebras alike. We moreover give a new interpretation of scalar multiples of Lie bialgebras hitherto studied using twisted Belavin-Drinfeld cohomology.

Mathematics Subject Classification

17B62, 17B37, 20G10


Lie bialgebra, quantum group, faithfully flat descent, Galois cohomology


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Alsaody, Seidon
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada T6G 2G1
Pianzola, Arturo
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada T6G 2G1 and Centro de Altos Estudios en Ciencias Exactas, Avenida de Mayo 866, (1084), Buenos Aires, Argentina