This book is intended for a one-year graduate course on Lie groups and Lie algebras. The book goes 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. The book is organized to allow different paths through the material depending on one's interests. This second edition has substantial new material, including improved discussions of underlying principles, streamlining of some proofs, and many results and topics that were not in the first edition. For compact Lie groups, the book covers the PeteraWeyl theorem, Lie algebra, conjugacy of maximal tori, the Weyl group, roots and weights, Weyl character formula, the fundamental group and more. The book continues with the study of complex analytic groups and general noncompact Lie groups, covering the Bruhat decomposition, Coxeter groups, flag varieties, symmetric spaces, Satake diagrams, embeddings of Lie groups and spin. Other topics that are treated are symmetric function theory, the representation theory of the symmetric group, FrobeniusaSchur duality and GL(n) tnGL(m) duality with many applications including some in random matrix theory, branching rules, Toeplitz determinants, combinatorics of tableaux, Gelfand pairs, Hecke algebras, the qphilosophy of cusp formsq and the cohomology of Grassmannians. An appendix introduces the reader to the use of Sage mathematical software for Lie group computations.In the Dynkin diagram, there will be a double or triple bond in these examples, and we draw an arrow from the long root to ... The Dynkin diagram for the type A5 root system The root system of type An is associated with the Lie group SU(n + 1).
|Publisher||:||Springer Science & Business Media - 2013-10-01|