The Resource Kac-Moody groups, their flag varieties, and representation theory, Shrawan Kumar, (electronic resource)

# Kac-Moody groups, their flag varieties, and representation theory, Shrawan Kumar, (electronic resource) Resource Information The item Kac-Moody groups, their flag varieties, and representation theory, Shrawan Kumar, (electronic resource) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Oklahoma Libraries.This item is available to borrow from all library branches.

Label
Kac-Moody groups, their flag varieties, and representation theory
Title
Kac-Moody groups, their flag varieties, and representation theory
Statement of responsibility
Shrawan Kumar
Creator
Subject
Genre
Language
eng
Summary
Kac-Moody Lie algebras 9 were introduced in the mid-1960s independently by V. Kac and R. Moody, generalizing the finite-dimensional semisimple Lie alge bras which we refer to as the finite case. The theory has undergone tremendous developments in various directions and connections with diverse areas abound, including mathematical physics, so much so that this theory has become a stan dard tool in mathematics. A detailed treatment of the Lie algebra aspect of the theory can be found in V. Kac's book [Kac-90l This self-contained work treats the algebro-geometric and the topological aspects of Kac-Moody theory from scratch. The emphasis is on the study of the Kac-Moody groups 9 and their flag varieties XY, including their detailed construction, and their applications to the representation theory of g. In the finite case, 9 is nothing but a semisimple Y simply-connected algebraic group and X is the flag variety 9 /Py for a parabolic subgroup p y C g
Member of
Action
digitized
OCLCE
1953-
Kumar, S.
Dewey number
512/.55
Index
index present
LC call number
QA252.3
LC item number
.K86 2002
Literary form
non fiction
Nature of contents
• dictionaries
• bibliography
Series statement
Progress in mathematics
Series volume
v. 204
• Kac-Moody algebras
• Representations of groups
• Flag manifolds
• Kac-Moody, Algèbres de
• Représentations de groupes
• Variétés de drapeaux (Géométrie)
• Lie-groepen
• Kac-Moody-Gruppe
• Darstellung <Mathematik>
• Fahnenmannigfaltigkeit
• Flag manifolds
• Kac-Moody algebras
• Representations of groups
Label
Kac-Moody groups, their flag varieties, and representation theory, Shrawan Kumar, (electronic resource)
http://dx.doi.org/10.1007/978-1-4612-0105-2
Instantiates
Publication
Antecedent source
file reproduced from original
Bibliography note
Includes bibliographical references (p. [559]-587) and indexes
Carrier category
online resource
Carrier category code
cr
Carrier MARC source
rdacarrier
Color
black and white
Content category
text
Content type code
txt
Content type MARC source
rdacontent
Contents
I. Kac-Moody Algebras: Basic Theory -- 1. Definition of Kac-Moody Algebras -- 2. Root Space Decomposition -- 3. Weyl Groups Associated to Kac-Moody Algebras -- 4. Dominant Chamber and Tits Cone -- 5. Invariant Bilinear Form and the Casimir Operator -- II. Representation Theory of Kac-Moody Algebras -- 1. Category $$\mathcal{O}$$ -- 2. Weyl-Kac Character Formula -- 3. Shapovalov Bilinear Form -- III. Lie Algebra Homology and Cohomology -- 1. Basic Definitions and Elementary Properties -- 2. Lie Algebra Homology of n-: Results of Kostant-Garland-Lepowsky -- 3. Decomposition of the Category $$\mathcal{O}$$ and some Ext Vanishing Results -- 4. Laplacian Calculation -- IV. An Introduction to ind-Varieties and pro-Groups -- 1. Ind-Varieties: Basic Definitions -- 2. Ind-Groups and their Lie Algebras -- 3. Smoothness of ind-Varieties -- 4. An Introduction to pro-Groups and pro-Lie Algebras -- V. Tits Systems: Basic Theory -- 1. An Introduction to Tits Systems -- 2. Refined Tits Systems -- VI. Kac-Moody Groups: Basic Theory -- 1. Definition of Kac-Moody Groups and Parabolic Subgroups -- 2. Representations of Kac-Moody Groups -- VII. Generalized Flag Varieties of Kac-Moody Groups -- 1. Generalized Flag Varieties: Ind-Variety Structure -- 2. Line Bundles on $${\mathcal{X}̂Y}$$ -- 3. Study of the Group $${\mathcal{U}̂ -- }$$ -- 4. Study of the Group $${\mathcal{G}̂{\min }}$$ Defined by Kac-Peterson -- VIII. Demazure and Weyl-Kac Character Formulas -- 1. Cohomology of Certain Line Bundles on $${Z_\mathfrak{w}}$$ -- 2. Normality of Schubert Varieties and the Demazure Character Formula -- 3. Extension of the Weyl-Kac Character Formula and the Borel-Weil-Bott Theorem -- IX. BGG and Kempf Resolutions -- 1. BGG Resolution: Algebraic Proof in the Symmetrizable Case -- 2. A Combinatorial Description of the BGG Resolution -- 3. Kempf Resolution -- X. Defining Equations of $$\mathcal{G}/\mathcal{P}$$ and Conjugacy Theorems -- 1. Quadratic Generation of Defining Ideals of $$\mathcal{G}/\mathcal{P}$$ in Projective Embeddings -- 2. Conjugacy Theorems for Lie Algebras -- 3. Conjugacy Theorems for Groups -- XI. Topology of Kac-Moody Groups and Their Flag Varieties -- 1. The Nil-Hecke Ring -- 2. Determination of $$\bar R$$ -- 3. T-equivariant Cohomology of $$\mathcal{G}/\mathcal{P}$$ -- 4. Positivity of the Cup Product in the Cohomology of Flag Varieties -- 5. Degeneracy of the Leray-Serre Spectral Sequence for the Fibration $${\mathcal{G}̂{\min }} \to {\mathcal{G}̂{\min }}/T$$ -- XII. Smoothness and Rational Smoothness of Schubert Varieties -- 1. Singular Locus of Schubert Varieties -- 2. Rational Smoothness of Schubert Varieties -- XIII. An Introduction to Affine Kac-Moody Lie Algebras and Groups -- 1. Affine Kac-Moody Lie Algebras -- 2. Affine Kac-Moody Groups -- Appendix A. Results from Algebraic Geometry -- Appendix B. Local Cohomology -- Appendix C. Results from Topology -- Appendix D. Relative Homological Algebra -- Appendix E. An Introduction to Spectral Sequences -- Index of Notation
Dimensions
unknown
Extent
1 online resource (xiii, 606 p.)
Form of item
online
Governing access note
Online full text is restricted to subscribers
Isbn
9781461266143
Level of compression
• lossless
• lossy
Media category
computer
Media MARC source
rdamedia
Media type code
c
Other control number
10.1007/978-1-4612-0105-2
Reformatting quality
• preservation
• access
Reproduction note
Electronic reproduction.
Specific material designation
remote
System control number
• (OCoLC)680416891
• (OCoLC)ocn680416891
System details
Master and use copy. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. Digital Library Federation, December 2002.
Label
Kac-Moody groups, their flag varieties, and representation theory, Shrawan Kumar, (electronic resource)
http://dx.doi.org/10.1007/978-1-4612-0105-2
Publication
Antecedent source
file reproduced from original
Bibliography note
Includes bibliographical references (p. [559]-587) and indexes
Carrier category
online resource
Carrier category code
cr
Carrier MARC source
rdacarrier
Color
black and white
Content category
text
Content type code
txt
Content type MARC source
rdacontent
Contents
I. Kac-Moody Algebras: Basic Theory -- 1. Definition of Kac-Moody Algebras -- 2. Root Space Decomposition -- 3. Weyl Groups Associated to Kac-Moody Algebras -- 4. Dominant Chamber and Tits Cone -- 5. Invariant Bilinear Form and the Casimir Operator -- II. Representation Theory of Kac-Moody Algebras -- 1. Category $$\mathcal{O}$$ -- 2. Weyl-Kac Character Formula -- 3. Shapovalov Bilinear Form -- III. Lie Algebra Homology and Cohomology -- 1. Basic Definitions and Elementary Properties -- 2. Lie Algebra Homology of n-: Results of Kostant-Garland-Lepowsky -- 3. Decomposition of the Category $$\mathcal{O}$$ and some Ext Vanishing Results -- 4. Laplacian Calculation -- IV. An Introduction to ind-Varieties and pro-Groups -- 1. Ind-Varieties: Basic Definitions -- 2. Ind-Groups and their Lie Algebras -- 3. Smoothness of ind-Varieties -- 4. An Introduction to pro-Groups and pro-Lie Algebras -- V. Tits Systems: Basic Theory -- 1. An Introduction to Tits Systems -- 2. Refined Tits Systems -- VI. Kac-Moody Groups: Basic Theory -- 1. Definition of Kac-Moody Groups and Parabolic Subgroups -- 2. Representations of Kac-Moody Groups -- VII. Generalized Flag Varieties of Kac-Moody Groups -- 1. Generalized Flag Varieties: Ind-Variety Structure -- 2. Line Bundles on $${\mathcal{X}̂Y}$$ -- 3. Study of the Group $${\mathcal{U}̂ -- }$$ -- 4. Study of the Group $${\mathcal{G}̂{\min }}$$ Defined by Kac-Peterson -- VIII. Demazure and Weyl-Kac Character Formulas -- 1. Cohomology of Certain Line Bundles on $${Z_\mathfrak{w}}$$ -- 2. Normality of Schubert Varieties and the Demazure Character Formula -- 3. Extension of the Weyl-Kac Character Formula and the Borel-Weil-Bott Theorem -- IX. BGG and Kempf Resolutions -- 1. BGG Resolution: Algebraic Proof in the Symmetrizable Case -- 2. A Combinatorial Description of the BGG Resolution -- 3. Kempf Resolution -- X. Defining Equations of $$\mathcal{G}/\mathcal{P}$$ and Conjugacy Theorems -- 1. Quadratic Generation of Defining Ideals of $$\mathcal{G}/\mathcal{P}$$ in Projective Embeddings -- 2. Conjugacy Theorems for Lie Algebras -- 3. Conjugacy Theorems for Groups -- XI. Topology of Kac-Moody Groups and Their Flag Varieties -- 1. The Nil-Hecke Ring -- 2. Determination of $$\bar R$$ -- 3. T-equivariant Cohomology of $$\mathcal{G}/\mathcal{P}$$ -- 4. Positivity of the Cup Product in the Cohomology of Flag Varieties -- 5. Degeneracy of the Leray-Serre Spectral Sequence for the Fibration $${\mathcal{G}̂{\min }} \to {\mathcal{G}̂{\min }}/T$$ -- XII. Smoothness and Rational Smoothness of Schubert Varieties -- 1. Singular Locus of Schubert Varieties -- 2. Rational Smoothness of Schubert Varieties -- XIII. An Introduction to Affine Kac-Moody Lie Algebras and Groups -- 1. Affine Kac-Moody Lie Algebras -- 2. Affine Kac-Moody Groups -- Appendix A. Results from Algebraic Geometry -- Appendix B. Local Cohomology -- Appendix C. Results from Topology -- Appendix D. Relative Homological Algebra -- Appendix E. An Introduction to Spectral Sequences -- Index of Notation
Dimensions
unknown
Extent
1 online resource (xiii, 606 p.)
Form of item
online
Governing access note
Online full text is restricted to subscribers
Isbn
9781461266143
Level of compression
• lossless
• lossy
Media category
computer
Media MARC source
rdamedia
Media type code
c
Other control number
10.1007/978-1-4612-0105-2
Reformatting quality
• preservation
• access
Reproduction note
Electronic reproduction.
Specific material designation
remote
System control number
• (OCoLC)680416891
• (OCoLC)ocn680416891
System details
Master and use copy. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. Digital Library Federation, December 2002.

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