Algèbres de Lie de dimension infinie et théorie de la descente

Steinmetz Zikesch, Wilhelm Alexander

Mémoires de la Société Mathématique de France, no. 129 (2012) , 105 p.

Résumé
Abstract

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.

MR Zbl

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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     author = {Steinmetz Zikesch, Wilhelm Alexander},
     title = {Alg\`ebres de {Lie} de dimension infinie et th\'eorie de la descente},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {129},
     year = {2012},
     doi = {10.24033/msmf.440},
     mrnumber = {3059229},
     zbl = {1319.17014},
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     url = {http://www.numdam.org/item/MSMF_2012_2_129__1_0/}
}

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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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