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 Mémoires de la Société Mathématique de France 2012; 99 pp; softcover Number: 129 ISBN-13: 978-2-85629-349-2 List Price: US$48 Member Price: US$38.40 Order Code: SMFMEM/129 A note to readers: This book is in French. 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 the author classifies $$R$$-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. The author can thus deduce results on EALAs and 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. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in Lie Algebras. Table of Contents Introduction Généralités et préliminaires Les conjectures Le cas $${^1}A_{n-1}$$ et les groupes orthogonaux Le cas $$C_n$$, les autres groupes du type $$D_n$$ et le cas $$^{2}A{_n}$$ La conjecture B Bibliographie