Summary: This course is the third quarter of the second year Algebra sequence. This quarter we will cover the basics of solvable and semisimple Lie algebras and their representation theory. It will also lay foundations for any further study of representation theory, Lie theory, Schubert geometry and many other related topics. We will focus on the structure of concrete examples of classical Lie algebras such as $gl_n$, $sl_n$, $sp_n$ and $so_n$. This will be followed by an account of the general theory of solvable and semisimple Lie algebras, which will in particular reduce the classification of the latter to the classification of finite root systems. We will then introduce enveloping algebras to classify the finite-dimensional complex representations of semisimple Lie algebras. Textbook and other reading material:
Additional Reading: Lie theory is an active area of reserach. This quarter every student will do a presentation on a research paper. Please look at the suggested papers or find some on your own. If you choose a paper on your own, be sure to discuss it with me asap.
Grading: The grade depends on weekly homework and the presentation. The homework will be approximately 5 problems per week.
* Lecture 1: Introduction to Lie algebras.
* Lecture 2: Examples. Ideals. Simple Lie algebras.