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Torsors, Reductive Group Schemes and Extended Affine Lie Algebras
Philippe Gille, Ecole Normale Supérieure, Paris, France, and Arturo Pianzola, University of Alberta, Edmonton, Canada
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Memoirs of the American Mathematical Society
2013; 112 pp; softcover
Volume: 226
ISBN-10: 0-8218-8774-2
ISBN-13: 978-0-8218-8774-5
List Price: US$73 Individual Members: US$43.80
Institutional Members: US\$58.40
Order Code: MEMO/226/1063

The authors 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 the authors take draws heavily from the theory of reductive group schemes developed by M. Demazure and A. Grothendieck. It also allows the authors to find a bridge between multiloop algebras and the work of F. Bruhat and J. Tits on reductive groups over complete local fields.

• Introduction
• Generalities on the algebraic fundamental group, torsors, and reductive group schemes
• Loop, finite and toral torsors
• Semilinear considerations
• Maximal tori of group schemes over the punctured line
• Internal characterization of loop torsors and applications
• Isotropy of loop torsors
• Acyclicity
• Small dimensions
• The case of orthogonal groups
• Groups of type $$G_2$$
• Case of groups of type $$F_4,$$ $$E_8$$ and simply connected $$E_7$$ in nullity $$3$$
• The case of $$\mathbf{PGL}_d$$
• Invariants attached to EALAs and multiloop algebras
• Appendix 1: Pseudo-parabolic subgroup schemes
• Appendix 2: Global automorphisms of $$G$$-torsors over the projective line
• Bibliography