History and philosophy of the course
This course will be an introduction to Lie algebras in the context of linear algebraic groups, with emphasis on the classical complex matrix groups (the general and special linear group, orthogonal group, and symplectic group). It will cover material from Chapters 1, 2, 5, and 11 of the Goodman-Wallach book, with additional topics from Humphrey's book.
Topics will include elementary properties of linear algebraic groups, their finite-dimensional representations, and their Lie algebras (but no deep results from algebraic geometry or commutative algebra will be used). The Lie algebras of the classical groups will be studied using root systems and Weyl groups relative to a maximal torus. The complete reducibility of finite-dimensional representations will be proved and the Cartan-Weyl highest weight theory of irreducible finite-dimensional representations will be developed. For the classical simple Lie algebras explicit models for the irreducible representations will be constructed. The structure and classification of semisimple Lie algebras will be covered at the end of the course.
Text: Roe Goodman and Nolan R. Wallach, Representations and Invariants of the Classical Groups,
ISBN 0-521-66348-2, Cambridge University Press, 2003.
Supplementary Text: James E. Humphreys, Introduction to Lie Algebras and Representation Theory, ISBN 3-387-90052-7, Springer-Verlag, New York, 1987.