Torsors, reductive group schemes and extended affine lie algebras / Philippe Gille, Arturo Pianzola.

Format:

Book

Author/Creator:

Gille, Philippe, 1968- author.

Pianzola, Arturo, 1955- author.

Series:

Memoirs of the American Mathematical Society ; Volume 226, Number 1063.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 226, Number 1063

Language:

English

Subjects (All):

Kac-Moody algebras.

Linear algebraic groups.

Geometry, Algebraic.

Physical Description:

1 online resource (124 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2013.

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""Chapter 1. Introduction""; ""Chapter 2. Generalities on the algebraic fundamental group, torsors, and reductive group schemes""; ""2.1. The fundamental group""; ""2.2. Torsors""; ""2.3. An example: Laurent polynomials in characteristic 0""; ""2.4. Reductive group schemes: Irreducibility and isotropy""; ""Chapter 3. Loop, finite and toral torsors""; ""3.1. Loop torsors""; ""3.2. Loop reductive groups""; ""3.3. Loop torsors at a rational base point""; ""3.4. Finite torsors""; ""3.5. Toral torsors""; ""Chapter 4. Semilinear considerations""; ""4.1. Semilinear morphisms""

""4.2. Semilinear morphisms""""4.3. Case of affine schemes""; ""4.4. Group functors""; ""4.5. Semilinear version of a theorem of Borel-Mostow""; ""4.6. Existence of maximal tori in loop groups""; ""4.7. Variations of a result of Sansuc""; ""Chapter 5. Maximal tori of group schemes over the punctured line""; ""5.1. Twin buildings""; ""5.2. Proof of Theorem 5.1""; ""Chapter 6. Internal characterization of loop torsors and applications""; ""6.1. Internal characterization of loop torsors""; ""6.2. Applications to (algebraic) Laurent series""; ""Chapter 7. Isotropy of loop torsors""

""7.1. Fixed point statements""""7.2. Case of flag varieties""; ""7.3. Anisotropic loop torsors""; ""Chapter 8. Acyclicity""; ""8.1. The proof""; ""8.2. Application: Witt-Tits decomposition""; ""8.3. Classification of semisimple â€?loop adjoint groups""; ""8.4. Action of _{ }(â??)""; ""Chapter 9. Small dimensions""; ""9.1. The one-dimensional case""; ""9.2. The two-dimensional case""; ""Chapter 10. The case of orthogonal groups""; ""Chapter 11. Groups of type â??""; ""Chapter 12. Case of groups of type â??, â?? and simply connected â?? in nullity 3""

""Chapter 13. The case of _{ }""""13.1. Loop Azumaya algebras""; ""13.2. The one-dimensional case""; ""13.3. The geometric case""; ""13.4. Loop algebras of inner type ""; ""Chapter 14. Invariants attached to EALAs and multiloop algebras""; ""Chapter 15. Appendix 1: Pseudo-parabolic subgroup schemes""; ""15.1. The case of _{ ,â??}""; ""15.2. The general case""; ""Chapter 16. Appendix 2: Global automorphisms of â€?torsors over the projective line""; ""Bibliography""

Notes:

"Volume 226, Number 1063 (fourth of 5 numbers)."

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-1063-X