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Torsors, Reductive Group Schemes and Extended Affine Lie Algebras
 
Philippe Gille Ecole Normale Supérieure, Paris, France
Arturo Pianzola University of Alberta, Edmonton, Canada
Torsors, Reductive Group Schemes and Extended Affine Lie Algebras
Torsors, Reductive Group Schemes and Extended Affine Lie Algebras
eBook ISBN:  978-1-4704-1063-6
Product Code:  MEMO/226/1063.E
List Price: $73.00
MAA Member Price: $65.70
AMS Member Price: $43.80
Torsors, Reductive Group Schemes and Extended Affine Lie Algebras
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Torsors, Reductive Group Schemes and Extended Affine Lie Algebras
Torsors, Reductive Group Schemes and Extended Affine Lie Algebras
Philippe Gille Ecole Normale Supérieure, Paris, France
Arturo Pianzola University of Alberta, Edmonton, Canada
eBook ISBN:  978-1-4704-1063-6
Product Code:  MEMO/226/1063.E
List Price: $73.00
MAA Member Price: $65.70
AMS Member Price: $43.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2262013; 112 pp
    MSC: Primary 17; 11; 14;

    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.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Generalities on the algebraic fundamental group, torsors, and reductive group schemes
    • 3. Loop, finite and toral torsors
    • 4. Semilinear considerations
    • 5. Maximal tori of group schemes over the punctured line
    • 6. Internal characterization of loop torsors and applications
    • 7. Isotropy of loop torsors
    • 8. Acyclicity
    • 9. Small dimensions
    • 10. The case of orthogonal groups
    • 11. Groups of type $G_2$
    • 12. Case of groups of type $F_4,$ $E_8$ and simply connected $E_7$ in nullity $3$
    • 13. The case of $\mathbf {PGL}_d$
    • 14. Invariants attached to EALAs and multiloop algebras
    • 15. Appendix 1: Pseudo-parabolic subgroup schemes
    • 16. Appendix 2: Global automorphisms of $G$–torsors over the projective line
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2262013; 112 pp
MSC: Primary 17; 11; 14;

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.

  • Chapters
  • 1. Introduction
  • 2. Generalities on the algebraic fundamental group, torsors, and reductive group schemes
  • 3. Loop, finite and toral torsors
  • 4. Semilinear considerations
  • 5. Maximal tori of group schemes over the punctured line
  • 6. Internal characterization of loop torsors and applications
  • 7. Isotropy of loop torsors
  • 8. Acyclicity
  • 9. Small dimensions
  • 10. The case of orthogonal groups
  • 11. Groups of type $G_2$
  • 12. Case of groups of type $F_4,$ $E_8$ and simply connected $E_7$ in nullity $3$
  • 13. The case of $\mathbf {PGL}_d$
  • 14. Invariants attached to EALAs and multiloop algebras
  • 15. Appendix 1: Pseudo-parabolic subgroup schemes
  • 16. Appendix 2: Global automorphisms of $G$–torsors over the projective line
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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