MAT 552 Course Webpage Introduction to Lie groups and Lie algebras M W 2:40PM — 4PM, Physics P 127 Spring 2022

Course Announcements Announcements about the course will be posted here. Please check the site regularly for announcements (which will also be given in lecture and/or in recitation).

• I can no longer find the original reference that I used for properties of the BGG category (it is Artinian, and the Grothendieck group is generated by classes of Verma modules). However, a very similar reference is the following exposition by Atharva Korde at the University of British Columbia: The Weyl Character Formula. Another excellent exposition is a series of blog posts by Akhil Mathew: Weyl's character formula.
• Here is a link to an article of Borel computing the rational cohomology of the generalized flag variety of a simply connected, compact real Lie group (equivalently, a connected, semisimple complex Lie group). When combined with Hopf's theorem, this provides the link between the generators of the rational cohomology algebra of the Lie group and the "fundamental invariants" in the Harish-Chandra isomorphism.
• Jason Starr has another commitment on Monday, March 21st, so lecture is canceled that day.
• The question came up in lecture whether every compact real Lie group is "algebraic" in a natural way. This is proved by a theorem of Chevalley, cf. the answer by user BCnrd to this post on MathOverflow.
• Our textbook does not include a complete proof of the Baker-Campbell-Hausdorff Formula. There is a collection of many different proofs in this article by Michael Müger.
• In lecture on Wednesday, 26 January 2022, we used some results for connected, complex Lie groups that are due to Chevalley when the Lie group is further assumed to be a group object in the category of complex algebraic varieties (Chevalley's structure theorem). In the general setting, a good summary is given in the following article of Prof. Joerg Winkelmann: Complex analytic geometry of complex parallelizable manifolds.
• Prof. Abanov (SCGP) has an online list of the most commonly used homotopy groups of Lie groups and homogeneous manifolds.
• Pierre Cartier wrote a primer on Hopf algebras that includes a wonderful history of the topological study of Lie groups.
• Prof. Helgason (MIT) has made available several of his fantastic articles on the history of Lie groups under "Additional Readings" on his OCW website.
• Regarding the ongoing discussion in lecture about the cohomology of Lie groups and their flag manifolds, students might find the following MathOverflow question interesting (Borel's Theorem for other coefficient rings than Q). In particular, the article by Demazure linked in that post settles the question.

Course Description Description in the graduate bulletin. "An introduction to Lie groups and Lie algebras, as used in mathematics and physics. Basic facts about Lie groups and Lie algebras; classical groups. Structure theory of Lie algebras. Representation theory of sl (2,C). Classification of complex semisimple Lie algebras via their root systems. Examples of exceptional Lie algebras. Other possible topics include the representation theory compact Lie groups and semisimple Lie algebras (with a focus on analytic techniques)."

A Lie group is a differentiable manifold with a smooth group structure, and the induced structure on the tangent space of the manifold at the group identity is a Lie algebra. Lie groups arise as symmetry groups in many other mathematical subjects, which makes them central objects of study. This course studies Lie groups as important examples of differentiable manifolds, but we also explore the structure theory and classification of Lie groups and Lie algebras.

Prerequisites Students should have passed the graduate algebra sequence or its equivalent and understand the basics of differentiable manifolds.

Text There is no required textbook. The recommended textbook is An introduction to Lie groups and Lie algebras by Prof. Alexander Kirillov, Jr. For the theory of finite-dimensional complex linear representations of complex semisimple Lie algebras, I also recommend Representation theory, a first course. by William Fulton and Joe Harris. For the algebraic side, there are excellent books by Springer, by Humphreys, by Borel, and by Bourbaki.

Lectures The instructor for this course is Jason Starr. All instruction will occur in lectures. The tentative schedule is in the syllabus.

Lecture is held Mondays and Wednesdays, 2:40 PM — 4 PM in Physics P 127.

Office Hours. Here is a link to the current office hours.

Grading System Grades are based on class participation, on performance on assigned problem sets, and on a final 20-minute oral presentation on a topic related to the course and approved by the instructor.

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