Conjugacy theorems for loop reductive group schemes and Lie algebras

V. Chernousov (1) , Philippe Gille (2) , Arturo Pianzola (1)

1 University of Alberta
2 ICJ - Institut Camille Jordan

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 J. Tits on buildings

Mots clés: Reductive group scheme; torsor; Laurent polynomials; non-abelian cohomology; building

Domaines: Anneaux et algèbres [math.RA]; Géométrie algébrique [math.AG]; Théorie des groupes [math.GR]

V. Chernousov, Philippe Gille, Arturo Pianzola. Conjugacy theorems for loop reductive group schemes and Lie algebras. 2011. ⟨hal-00625969v1⟩