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Representation Theory: A First Course

Representation Theory: A First Course
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Field name Details
Dewey Class 512.55
Title Representation Theory ([EBook]) : A First Course / by William Fulton, Joe Harris.
Author Fulton, William. , 1939-
Added Personal Name Harris, Joe. , 1951- author.
Other name(s) SpringerLink (Online service)
Publication New York, NY : Springer , 2004.
Physical Details XV, 551 pages : online resource.
Series Graduate Texts in Mathematics, Readings in Mathematics 0072-5285 ; ; 129
ISBN 9781461209799
Summary Note The primary goal of these lectures is to introduce a beginner to the finite­ dimensional representations of Lie groups and Lie algebras. Since this goal is shared by quite a few other books, we should explain in this Preface how our approach differs, although the potential reader can probably see this better by a quick browse through the book. Representation theory is simple to define: it is the study of the ways in which a given group may act on vector spaces. It is almost certainly unique, however, among such clearly delineated subjects, in the breadth of its interest to mathematicians. This is not surprising: group actions are ubiquitous in 20th century mathematics, and where the object on which a group acts is not a vector space, we have learned to replace it by one that is {e. g. , a cohomology group, tangent space, etc. }. As a consequence, many mathematicians other than specialists in the field {or even those who think they might want to be} come in contact with the subject in various ways. It is for such people that this text is designed. To put it another way, we intend this as a book for beginners to learn from and not as a reference. This idea essentially determines the choice of material covered here. As simple as is the definition of representation theory given above, it fragments considerably when we try to get more specific.:
Contents note I: Finite Groups -- 1. Representations of Finite Groups -- 2. Characters -- 3. Examples; Induced Representations; Group Algebras; Real Representations -- 4. Representations of 15 and Other Exceptional Lie Algebras -- 23. Complex Lie Groups; Characters -- 24. Weyl Character Formula -- 25. More Character Formulas -- 26. Real Lie Algebras and Lie Groups -- Appendices -- A. On Symmetric Functions -- §A.1: Basic Symmetric Polynomials and Relations among Them -- §A.2: Proofs of the Determinantal Identities -- §A.3: Other Determinantal Identities -- B. On Multilinear Algebra -- §B.1: Tensor Products -- §B.2: Exterior and Symmetric Powers -- §B.3: Duals and Contractions -- C. On Semisimplicity -- §C.1: The Killing Form and Caftan’s Criterion -- §C.2: Complete Reducibility and the Jordan Decomposition -- §C.3: On Derivations -- D. Cartan Subalgebras -- §D.1: The Existence of Cartan Subalgebras -- §D.2: On the Structure of Semisimple Lie Algebras -- §D.3: The Conjugacy of Cartan Subalgebras -- §D.4: On the Weyl Group -- E. Ado’s and Levi’s Theorems -- §E.1: Levi’s Theorem -- §E.2: Ado’s Theorem --^F. Invariant Theory for the Classical Groups -- §F.1: The Polynomial Invariants -- §F.2: Applications to Symplectic and Orthogonal Groups -- §F.3: Proof of Capelli’s Identity -- Hints, Answers, and References -- Index of Symbols
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