Torsors, Reductive Group Schemes and Extended Affine Lie Algebras

Philippe Gille (1) , Arturo Pianzola (2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 University of Alberta

We give a detailed description of the torsors that correspond to multiloop algebras. These algebras are twisted forms of simple Lie algebras extended over Laurent polynomial rings. They play a crucial role in the construction of Extended Affine Lie Algebras (which are higher nullity analogues of the affine Kac-Moody Lie algebras). The torsor approach that we take draws heavily for the theory of reductive group schemes developed by M. Demazure and A. Grothendieck. It also allows us to find a bridge between multiloop algebras and the work of J. Tits on reductive groups over complete local fields.

Keywords: Extended Affine Lie Algebras; Reductive group scheme; torsor; multiloop algebra; Extended Affine Lie Algebras.

Domains: Rings and Algebras [math.RA]; Representation Theory [math.RT]; Algebraic Geometry [math.AG]

Philippe Gille, Arturo Pianzola. Torsors, Reductive Group Schemes and Extended Affine Lie Algebras. 2012. ⟨hal-00624106v2⟩