Lie Groups and Lie Algebras I
MIT's graduate course covers the foundations of Lie groups and Lie algebras, the first half of the year-long 18.745/18.755 sequence. Topics include the theorems of Engel and Lie, the universal enveloping algebra, the Poincare-Birkhoff-Witt theorem, free Lie algebras, and the Campbell-Hausdorff formula. The course continues into the classification and structure of finite dimensional complex simple Lie algebras, their finite dimensional representations, and the Weyl character formula. Materials come through MIT OpenCourseWare and typically include lecture notes and problem sets used in the actual graduate course. No certificate is offered, but all readings and assignments are freely available. This is advanced material suited to students with a solid background in abstract algebra and linear algebra, aiming to build the structural theory that underlies representation theory and mathematical physics.