Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 An Introduction to Affine Lie Algebras and the Associated Groups
- 2 Space of Vacua and its Propagation
- 3 Factorization Theorem for Space of Vacua
- 4 Fusion Ring and Explicit Verlinde Formula
- 5 Moduli Stack of Quasi-parabolic G-Bundles and its Uniformization
- 6 Parabolic G-Bundles and Equivariant G-Bundles
- 7 Moduli Space of Semistable G-Bundles Over a Smooth Curve
- 8 Identification of the Space of Conformal Blocks with the Space of Generalized Theta Functions
- 9 Picard Group of Moduli Space of G-Bundles
- Appendix A Dynkin Index
- Appendix B C-Space and C-Group Functors
- Appendix C Algebraic Stacks
- Appendix D Rank-Level Duality (A Brief Survey) (by Swarnava Mukhopadhyay)
- Bibliography
- Index
1 - An Introduction to Affine Lie Algebras and the Associated Groups
Published online by Cambridge University Press: 19 November 2021
Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 An Introduction to Affine Lie Algebras and the Associated Groups
- 2 Space of Vacua and its Propagation
- 3 Factorization Theorem for Space of Vacua
- 4 Fusion Ring and Explicit Verlinde Formula
- 5 Moduli Stack of Quasi-parabolic G-Bundles and its Uniformization
- 6 Parabolic G-Bundles and Equivariant G-Bundles
- 7 Moduli Space of Semistable G-Bundles Over a Smooth Curve
- 8 Identification of the Space of Conformal Blocks with the Space of Generalized Theta Functions
- 9 Picard Group of Moduli Space of G-Bundles
- Appendix A Dynkin Index
- Appendix B C-Space and C-Group Functors
- Appendix C Algebraic Stacks
- Appendix D Rank-Level Duality (A Brief Survey) (by Swarnava Mukhopadhyay)
- Bibliography
- Index
Summary
The aim of this chapter is first to set some basic notation and preliminaries (to be used through the book) and then recall the definition of affine Kac--Moody Lie algebras and their basic representation theory and to study the associated groups and their flag varieties. We define the loop group G((t)) (without the central extension) and its various subgroups. Then, we study the associated infinite Grassmannian and prove that it is a reduced ind-variety when G is a semisimple group. We show that G((t)) is a reduced affine ind-scheme. We further study the central extensions of G((t)).
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- Publisher: Cambridge University PressPrint publication year: 2021