Algèbres de Lie de dimension infinie et théorie de la descente
Mémoires de la Société Mathématique de France, no. 129 (2012) , 105 p.

Soit k un corps algébriquement clos de caractéristique zéro et soit R un anneau de polynômes de Laurent en deux variables sur k. La motivation principale derrière ce travail est une classe d’algèbres de Lie de dimension infinie sur k, appelées extended affine Lie algebras (EALAs). Ces algèbres correspondent à des torseurs sous des groupes algébriques linéaires sur R. Dans ce travail nous classifions les R-torseurs sous les groupes classiques de rang assez grand pour les types A extérieur, B,C,D et pour le type A intérieur sous des hypothèses plus fortes. Ainsi, nous pouvons déduire des résultats sur des EALAs. Nous obtenons aussi une réponse affirmative à une variante de la conjecture II de Serre pour l’anneau R  : tout R-torseur lisse sous un groupe semi-simple simplement connexe de rang assez grand de type classique B, C et D est trivial.

Let k be an algebraically closed field of characteristic zero and let R be the Laurent polynomial ring in two variables over k. The main motivation behind this work is a class of infinite dimensional Lie algebras over k, called extended affine Lie algebras (EALAs). These algebras correspond to torsors under algebraic groups over R. In this work we classify R-torsors under classical groups of large enough rank for outer type A and types B,C,D, as well as for inner type A under stronger hypotheses. We can thus deduce results on EALAs. We also obtain a positive answer to a variant of Serre’s Conjecture II for the ring R: every smooth R-torsor under a semi-simple simply connected R-group of large enough rank of classical type B,C,D is trivial.

DOI : 10.24033/msmf.440
Mot clés : Algèbre de Lie de dimension infinie, EALA, anneau de polynômes de Laurent, cohomologie galoisienne, groupe algébrique linéaire classique, algèbre d’Azumaya à involution, forme hermitienne, théorie de Witt triangulaire
Mots clés : Infinite Dimensional Lie Algebra, Extended Affine Lie Algebra (EALA), Laurent Polynomial Ring, Galois Cohomology, Classical Linear Algebraic Group, Azumaya Algebra with Involution, Hermitian Form, Triangular Witt Theory
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Steinmetz Zikesch, Wilhelm Alexander. Algèbres de Lie de dimension infinie et théorie de la descente. Mémoires de la Société Mathématique de France, Série 2, no. 129 (2012), 105 p. doi : 10.24033/msmf.440. http://numdam.org/item/MSMF_2012_2_129__1_0/

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