Representation Theory A First Course

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.

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.