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. Language: English