Submission: 2010, Jan 15
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 linear algebraic groups over R. In this work we classify R-torsors under classical groups of large enough rank (and under stronger hypotheses for groups of interior type A) and obtain this way results on the above mentioned EALAs. We also obtain 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 or D is trivial.
2000 Mathematics Subject Classification:
Keywords and Phrases: Infinite Dimensional Lie Algebras, EALA, Laurent Polynomial Rings, Galois Cohomology, Torsors, Classical Linear Algebraic Groups, Azumaya Algebras with Involution, Hermitian Forms, Triangular Witt Theory
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