Description: 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$$G{L_2}left( {{mathbb{F}_q}} ight)$$.- 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.

Please Wait