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Torsors, reductive group schemes and extended affine Lie algebras

About this Title

Philippe Gille, UMR 8553 du CNRS, Ecole Normale Supérieure, 45 rue d’Ulm, 75005 Paris, France and Arturo Pianzola, Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 226, Number 1063
ISBNs: 978-0-8218-8774-5 (print); 978-1-4704-1063-6 (online)
Published electronically: May 23, 2013
Keywords:Reductive group scheme, torsor, multiloop algebra, extended affine Lie algebras
MSC: Primary 17B67, 11E72, 14L30, 14E20

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Generalities on the algebraic fundamental group, torsors, and reductive group schemes
  • Chapter 3. Loop, finite and toral torsors
  • Chapter 4. Semilinear considerations
  • Chapter 5. Maximal tori of group schemes over the punctured line
  • Chapter 6. Internal characterization of loop torsors and applications
  • Chapter 7. Isotropy of loop torsors
  • Chapter 8. Acyclicity
  • Chapter 9. Small dimensions
  • Chapter 10. The case of orthogonal groups
  • Chapter 11. Groups of type $G_2$
  • Chapter 12. Case of groups of type $F_4$, $E_8$ and simply connected $E_7$ in nullity 3
  • Chapter 13. The case of $\mathbf {PGL}_d$
  • Chapter 14. Invariants attached to EALAs and multiloop algebras
  • Chapter 15. Appendix 1: Pseudo-parabolic subgroup schemes
  • Chapter 16. Appendix 2: Global automorphisms of $G$-torsors over the projective line


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 F. Bruhat and J. Tits on reductive groups over complete local fields.

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