Lie groups
 Author/Creator
 Bump, Daniel, 1952 author.
 Language
 English.
 Publication
 New York: Springer, [2013]
 Copyright notice
 ©2013
 Imprint
 Second edition.
 Physical description
 xiii, 551 pages 25 cm.
 Series
 Graduate texts in mathematics ; 225.
Access
Available online

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QA387 .B76 2013

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QA387 .B76 2013
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Contents/Summary
 Bibliography
 Includes bibliographical references (pages535544) and index.
 Contents

 Compact groups
 Haar measure
 Schur orthogonality
 Compact operators
 The PeterWeyl theorem
 Compact lie groups
 Lie subgroups of GL (n, C)
 Vector fields
 Leftinvariant vector fields
 The exponential map
 Tensors and universal properties
 The universal enveloping algebra
 Extension of scalars
 Representations of s1 (2, C)
 The universal cover
 The local Frobenius theorem
 Tori
 Geodesics and maximal tori
 The Weyl integration formula
 The root system
 Examples of root systems
 Abstract Weyl groups
 Highest weight vectors
 The Weyl character formula
 The fundamental group
 Noncompact lie groups
 Complexification
 Coxeter groups
 The borel subgroup
 The Bruhat decomposition
 Symmetric spaces
 Relative root systems
 Embeddings of lie groups
 Spin
 Duality and other topics
 Mackey theory
 Characters of GL (n, C)
 Duality between Sk and GL (n, C)
 The Jacobi Trudi identity
 Schur polynomials and GL (n, C)
 Schur polynomials and Sk
 The Cauchy identity
 Random matrix theory
 Symmetrie group branching rules and tableaux
 Unitary branching rules and tableaux
 Minors of Toeplitz matrices
 The involution model for Sk
 Some symmetric algebras
 Gelfand pairs
 Hecke algebras
 The philosophy of cusp forms
 Cohomology of grassmannians
 Appendix : Sage
 References
 Index.
 Summary
 "This book is intended for a one year graduate course on Lie groups and Lie algebras. The author proceeds beyond the representation theory of compact Lie groups (which is the basis of many texts) and provides a carefully chosen range of material to give the student the bigger picture. For compact Lie groups, the PeterWeyl theorem, conjugacy of maximal tori (two proofs), Weyl character formula and more are covered. The book continues with the study of complex analytic groups, then general noncompact Lie groups, including the Coxeter presentation of the Weyl group, the Iwasawa and Bruhat decompositions, Cartan decomposition, symmetric spaces, Cayley transforms, relative root systems, Satake diagrams, extended Dynkin diagrams and a survey of the ways Lie groups may be embedded in one another. The book culminates in a "topics" section giving depth to the student's understanding of representation theory, taking the FrobeniusSchur duality between the representation theory of the symmetric group and the unitary groups as a unifying theme, with many applications in diverse areas such as random matrix theory, minors of Toeplitz matrices, symmetric algebra decompositions, Gelfand pairs, Hecke algebras, representations of finite general linear groups and the cohomology of Grassmannians and flag varieties.
Subjects
 Subject
 Lie groups.
Bibliographic information
 Publication date
 2013
 Copyright date
 2013
 Responsibility
 Daniel Bump.
 Series
 Graduate texts in mathematics, 00725285 ; 225
 ISBN
 9781461480235
 146148023X