CONJUGACY THEOREMS FOR LOOP REDUCTIVE GROUP SCHEMES AND LIE ALGEBRAS

Abstract. The conjugacy of split Cartan subalgebras in the finite-dimensional simple case (Chevalley) and in the symmetrizable Kac-Moody case (Peterson-Kac) are fundamental results of the theory of Lie algebras. Among the Kac-Moody Lie algebras the affine algebras stand out. This paper deals with the problem of conjugacy for a class of algebras –extended affine Lie algebras – that are in a precise sense higher nullity analogues of the affine algebras. Unlike the methods used by Peterson-Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of Bruhat-Tits on buildings. The main ingredient of our conjugacy proof is the classification of loop torsors over Laurent polynomial rings, a result of its own interest