Representation Theory

Produktbeschreibung

The primary goal of these lectures is to introduce the 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 a 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;
in particular, the symmetric groups are treated in some detail. 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 and exploring the related geometry. 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.  

Inhaltsverzeichnis

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.