Geometrization of solutions of the generalized classical Yang–Baxter equation and a new proof of the Belavin–Drinfeld trichotomy
[Géométrisation des solutions de l’équation de Yang–Baxter classique généralisée et nouvelle preuve de la trichotomie de Belavin–Drinfeld]
Abedin, Raschid1
1 ETH Zürich, Rämistrasse 101, 8092 Zurich (Switzerland)
Annales de l'Institut Fourier, Online first, 78 p.

We study the generalized classical Yang–Baxter equation (short: GCYBE) for central simple Lie algebras over fields of characteristic 0. Using a novel geometrization procedure, we assign to a solution of the GCYBE a cohomology free sheaf of Lie algebras on a projective curve. This assignment implies that all such solutions are algebraic in nature, i.e. they extend to rational functions on the product of two algebraic curves. Furthermore, the curves assigned to skew-symmetric solutions turn out to be either smooth, nodal, or cuspidal cubic plane curves. This results in a geometric trichotomy of skew-symmetric solutions of the GCYBE. Over the field of complex numbers, this geometric trichotomy implies the well-known Belavin–Drinfeld trichotomy, which states that skew-symmetric solutions of the GCYBE are either elliptic, trigonometric, or rational. As an interesting side result, we show that sheaves of Lie algebras with constant geometric fibers are locally free in the étale topology. This is used to classify such sheaves on the complex plane with at most one puncture.

Nous étudions l’équation de Yang–Baxter classique généralisée (abréviation anglaise : GCYBE) pour les algèbres de Lie simples centrales sur des corps de caractéristique 0. En utilisant une nouvelle procédure de géométrisation, nous attribuons à une solution de la GCYBE un faisceau sans cohomologie d’algèbres de Lie sur une courbe projective. Cette attribution implique que toutes ces solutions sont de nature algébrique, c’est-à-dire qu’elles s’étendent à des fonctions rationnelles sur le produit de deux courbes algébriques. De plus, les courbes assignées aux solutions antisymétrique s’avèrent être des courbes cubiques lisses, nodales ou cuspidales. Il en résulte une trichotomie géométrique des solutions antisymétrique de la GCYBE. Dans le domaine des nombres complexes, cette trichotomie géométrique implique la trichotomie bien connue de Belavin–Drinfeld, qui stipule que les solutions antisymétrique de la GCYBE sont soit elliptiques, trigonométriques ou rationnelles. Comme résultat secondaire intéressant, nous montrons que les faisceaux d’algèbres de Lie avec des fibres géométriques constantes sont localement libres dans la topologie étale. Ceci est utilisé pour classifier de telles faisceaux sur le plan complexe avec au plus un point supprimé.

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Révisé le :
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DOI : 10.5802/aif.3722
Classification : 17B38, 17B65, 14H70
Keywords: classical Yang–Baxter equation, sheaves of Lie algebras, infinite-dimensional Lie algebras
Mots-clés : géométrisation des solutions de l’équation de Yang–Baxter classique généralisée et nouvelle preuve de la trichotomie de Belavin–Drinfeld

Abedin, Raschid 1

1 ETH Zürich, Rämistrasse 101, 8092 Zurich (Switzerland) 
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Abedin, Raschid. Geometrization of solutions of the generalized classical Yang–Baxter equation and a new proof of the Belavin–Drinfeld trichotomy. Annales de l'Institut Fourier, Online first, 78 p.

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