Representation Theory: A First Course

ISBN-13: 9780387975276 / Angielski / Twarda / 1991 / 551 str.

W. Fulton; William Fulton; Joe Harris

Opis

Introducing finite-dimensional representations of Lie groups and Lie algebras, this example-oriented book works from representation theory of finite groups, through Lie groups and Lie algrbras to the finite dimensional representations of the classical groups.

Spis treści

I: Finite Groups.- 1. Representations of Finite Groups.- §1.1: Definitions.- §1.2: Complete Reducibility; Schur’s Lemma.- §1.3: Examples: Abelian Groups; $$ {\mathfrak_3}$$.- 2. Characters.- §2.1: Characters.- §2.2: The First Projection Formula and Its Consequences.- §2.3: Examples: $$ {\mathfrak_4}$$ and $$ {\mathfrak_4}$$.- §2.4: More Projection Formulas; More Consequences.- 3. Examples; Induced Representations; Group Algebras; Real Representations.- §3.1: Examples: $$ {\mathfrak_5}$$ and $$ {\mathfrak_5}$$.- §3.2: Exterior Powers of the Standard Representation of $$ {\mathfrak_d}$$.- §3.3: Induced Representations.- §3.4: The Group Algebra.- §3.5: Real Representations and Representations over Subfields of $$ \mathbb$$.- 4. Representations of: $$ {\mathfrak_d}$$ Young Diagrams and Frobenius’s Character Formula.- §4.1: Statements of the Results.- §4.2: Irreducible Representations of $$ {\mathfrak_d}$$.- §4.3: Proof of Frobenius’s Formula.- 5. Representations of $$ {\mathfrak_d}$$ and $$ G\left( {{\mathbb_q}} \right)$$.- §5.1: Representations of $$ {\mathfrak_d}$$.- §5.2: Representations of $$ G\left( {{\mathbb_q}} \right)$$ and $$ S\left( {{\mathbb_q}} \right)$$.- 6. Weyl’s Construction.- §6.1: Schur Functors and Their Characters.- §6.2: The Proofs.- II: Lie Groups and Lie Algebras.- 7. Lie Groups.- §7.1: Lie Groups: Definitions.- §7.2: Examples of Lie Groups.- §7.3: Two Constructions.- 8. Lie Algebras and Lie Groups.- §8.1: Lie Algebras: Motivation and Definition.- §8.2: Examples of Lie Algebras.- §8.3: The Exponential Map.- 9. Initial Classification of Lie Algebras.- §9.1: Rough Classification of Lie Algebras.- §9.2: Engel’s Theorem and Lie’s Theorem.- §9.3: Semisimple Lie Algebras.- §9.4: Simple Lie Algebras.- 10. Lie Algebras in Dimensions One, Two, and Three.- §10.1: Dimensions One and Two.- §10.2: Dimension Three, Rank 1.- §10.3: Dimension Three, Rank 2.- §10.4: Dimension Three, Rank 3.- 11. Representations of $$ \mathfrak{\mathfrak_2}\mathbb$$.- §11.1: The Irreducible Representations.- §11.2: A Little Plethysm.- §11.3: A Little Geometric Plethysm.- 12. Representations of $$ \mathfrak{\mathfrak_3}\mathbb,$$ Part I.- 13. Representations of $$ \mathfrak{\mathfrak_3}\mathbb,$$ Part II: Mainly Lots of Examples.- §13.1: Examples.- §13.2: Description of the Irreducible Representations.- §13.3: A Little More Plethysm.- §13.4: A Little More Geometric Plethysm.- III: The Classical Lie Algebras and Their Representations.- 14. The General Set-up: Analyzing the Structure and Representations of an Arbitrary Semisimple Lie Algebra.- §14.1: Analyzing Simple Lie Algebras in General.- §14.2: About the Killing Form.- 15. $$ \mathfrak{\mathfrak_4}\mathbb$$ and $$ \mathfrak{\mathfrak_n}\mathbb$$.- §15.1: Analyzing $$ \mathfrak{\mathfrak_n}\mathbb$$.- §15.2: Representations of $$ \mathfrak{\mathfrak_4}\mathbb$$ and $$ \mathfrak{\mathfrak_n}\mathbb$$.- §15.3: Weyl’s Construction and Tensor Products.- §15.4: Some More Geometry.- §15.5: Representations of $$ G\mathbb$$.- 16. Symplectic Lie Algebras.- §16.1: The Structure of $$ S\mathbb$$ and $$ \mathfrak{\mathfrak_2n}\mathbb$$.- §16.2: Representations of $$ \mathfrak{\mathfrak_4}\mathbb$$.- 17. $$ \mathfrak{\mathfrak_6}\mathbb$$ and $$ \mathfrak{\mathfrak_2n}\mathbb$$.- §17.1: Representations of $$ \mathfrak{\mathfrak_6}\mathbb$$.- §17.2: Representations of $$ \mathfrak{\mathfrak_2n}\mathbb$$ in General.- §17.3: Weyl’s Construction for Symplectic Groups.- 18. Orthogonal Lie Algebras.- §18.1: $$ S\mathbb$$ and $$ \mathfrak{\mathfrak_m}\mathbb$$.- §18.2: Representations of $$ \mathfrak{\mathfrak_3}\mathbb,$$$$ \mathfrak{\mathfrak_4}\mathbb,$$ and $$ \mathfrak{\mathfrak_5}\mathbb$$.- 19. $$ \mathfrak{\mathfrak_6}\mathbb,$$$$ \mathfrak{\mathfrak_7}\mathbb,$$ and $$ \mathfrak{\mathfrak_m}\mathbb$$.- §19.1: Representations of $$ \mathfrak{\mathfrak_6}\mathbb$$.- §19.2: Representations of the Even Orthogonal Algebras.- §19.3: Representations of $$ \mathfrak{\mathfrak_7}\mathbb$$.- §19.4. Representations of the Odd Orthogonal Algebras.- §19.5: Weyl’s Construction for Orthogonal Groups.- 20. Spin Representations of $$ \mathfrak{\mathfrak_m}\mathbb$$.- §20.1: Clifford Algebras and Spin Representations of $$ \mathfrak{\mathfrak_m}\mathbb$$.- §20.2: The Spin Groups $$ Spi\mathbb$$ and $$ Spi\mathbb$$.- §20.3: $$ Spi\mathbb$$ and Triality.- IV: Lie Theory.- 21. The Classification of Complex Simple Lie Algebras.- §21.1: Dynkin Diagrams Associated to Semisimple Lie Algebras.- §21.2: Classifying Dynkin Diagrams.- §21.3: Recovering a Lie Algebra from Its Dynkin Diagram.- 22. $$ $$and Other Exceptional Lie Algebras.- §22.1: Construction of $$ $$ from Its Dynkin Diagram.- §22.2: Verifying That $$ $$ is a Lie Algebra.- §22.3: Representations of $${{\mathfrak}_} $$.- §22.4: Algebraic Constructions of the Exceptional Lie Algebras.- 23. Complex Lie Groups; Characters.- §23.1: Representations of Complex Simple Groups.- §23.2: Representation Rings and Characters.- §23.3: Homogeneous Spaces.- §23.4: Bruhat Decompositions.- 24. Weyl Character Formula.- §24.1: The Weyl Character Formula.- §24.2: Applications to Classical Lie Algebras and Groups.- 25. More Character Formulas.- §25.1: Freudenthal’s Multiplicity Formula.- §25.2: Proof of (WCF); the Kostant Multiplicity Formula.- §25.3: Tensor Products and Restrictions to Subgroups.- 26. Real Lie Algebras and Lie Groups.- §26.1: Classification of Real Simple Lie Algebras and Groups.- §26.2: Second Proof of Weyl’s Character Formula.- §26.3: Real, Complex, and Quaternionic Representations.- 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.