The primary goal of these lectures is to introduce a beginner to the finite-dimensional representations of Lie groups and Lie algebras. Intended to serve non-specialists, the concentration of the text is on examples. The general theory is developed sparingly, and then mainly as useful and unifying language to describe phenomena already encountered in concrete cases. The book begins with a brief tour through representation theory of finite groups, with emphasis determined by what is useful for Lie groups. The focus then turns to Lie groups and Lie algebras and finally to the heart of the course: working out the finite dimensional representations of the classical groups. The goal of the last portion of the book is to make a bridge between the example-oriented approach of the earlier parts and the general theory.
By:
William Fulton,
Joe Harris
Imprint: Springer Verlag
Country of Publication: United States
Edition: 3rd Revised edition
Volume: v.129
Dimensions:
Height: 235mm,
Width: 155mm,
Spine: 29mm
Weight: 1.740kg
ISBN: 9780387974958
ISBN 10: 0387974954
Series: Graduate Texts in Mathematics
Pages: 566
Publication Date: 01 June 1999
Audience:
College/higher education
,
Professional and scholarly
,
Professional & Vocational
,
A / AS level
,
Further / Higher Education
Format: Paperback
Publisher's Status: Active
I: Finite Groups.- 1. Representations of Finite Groups.- 2. Characters.- 3. Examples; Induced Representations; Group Algebras; Real Representations.- 4. Representations of: $$ {\mathfrak{S}_d}$$ Young Diagrams and Frobenius’s Character Formula.- 5. Representations of $$ {\mathfrak{A}_d}$$ and $$ G{L_2}\left( {{\mathbb{F}_q}} \right)$$.- 6. Weyl’s Construction.- II: Lie Groups and Lie Algebras.- 7. Lie Groups.- 8. Lie Algebras and Lie Groups.- 9. Initial Classification of Lie Algebras.- 10. Lie Algebras in Dimensions One, Two, and Three.- 11. Representations of $$ \mathfrak{s}{\mathfrak{l}_2}\mathbb{C}$$.- 12. Representations of $$ \mathfrak{s}{\mathfrak{l}_3}\mathbb{C},$$ Part I.- 13. Representations of $$ \mathfrak{s}{\mathfrak{l}_3}\mathbb{C},$$ Part II: Mainly Lots of Examples.- III: The Classical Lie Algebras and Their Representations.- 14. The General Set-up: Analyzing the Structure and Representations of an Arbitrary Semisimple Lie Algebra.- 15. $$ \mathfrak{s}{\mathfrak{l}_4}\mathbb{C}$$ and $$ \mathfrak{s}{\mathfrak{l}_n}\mathbb{C}$$.- 16. Symplectic Lie Algebras.- 17. $$ \mathfrak{s}{\mathfrak{p}_6}\mathbb{C}$$ and $$ \mathfrak{s}{\mathfrak{p}_2n}\mathbb{C}$$.- 18. Orthogonal Lie Algebras.- 19. $$ \mathfrak{s}{\mathfrak{o}_6}\mathbb{C},$$$$ \mathfrak{s}{\mathfrak{o}_7}\mathbb{C},$$ and $$ \mathfrak{s}{\mathfrak{o}_m}\mathbb{C}$$.- 20. Spin Representations of $$ \mathfrak{s}{\mathfrak{o}_m}\mathbb{C}$$.- IV: Lie Theory.- 21. The Classification of Complex Simple Lie Algebras.- 22. $$ {g_2}$$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.
