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Lie groups / Daniel Bump.



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Bump, Daniel, 1952- author.
Publication date:
Copyright date:
Second edition.
New York: Springer, [2013]
Copyright notice:
  • Book
  • xiii, 551 pages 25 cm.
Includes bibliographical references (pages535-544) and index.
  • Compact groups
  • Haar measure
  • Schur orthogonality
  • Compact operators
  • The Peter-Weyl theorem
  • Compact lie groups
  • Lie subgroups of GL (n, C)
  • Vector fields
  • Left-invariant 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.
"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 Peter-Weyl 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 Frobenius-Schur 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.
Graduate texts in mathematics, 0072-5285 ; 225
Graduate texts in mathematics ; 225.

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