Professor: Mark Haiman
Office: 855 Evans
Office hours: M 1:30-3:00 or by appointment
Phone: (510) 642-4318

This course is an introduction to Lie groups and their Lie algebras. This fall we will cover the basic definitions and structure theorems and the classification of semi-simple Lie algebras. On the syllabus below, I have also listed a few extra topics from which we may sample if time permits. My understanding from Robert Bryant, who will be teaching 261B this spring, is that he intends it to be a continuation of 261A, so we will leave some topics for next semester.

Prerequisites: Background in algebra and topology equivalent to 202A and 250A. Although 214 (Differential Manifolds) is the official prerequisite, I will review in the lectures those bits of differential geometry that we will need.

Textbook: V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations (Springer Graduate Texts in Mathematics, v. 102).

There is also a set of lecture notes from this course in previous years taken by Theo Johnson-Freyd and Anton Geraschenko. I plan to put a link to their notes here soon.

Some other useful references:

• Jeffrey M. Lee, Manifolds and Differential Geometry (Graduate Studies in Math, vol. 107) is a readable introductory textbook.
• V. V. Gorbatsevich, A. L. Onishchik and E. B. Vinberg, Foundations of Lie theory and Lie transformation groups; previously published as Lie groups and Lie algebras I (Encyclopaedia of Mathematical Sciences, vol. 20).
• A. L. Onishchik, and E. B. Vinberg, Lie groups and Lie algebras III: structure of Lie groups and Lie algebras (Encyclopaedia of Mathematical Sciences, vol. 41).
• N. Bourbaki, Lie Groups and Lie Algebras, Chapters 1-3, 4-6, 7-9.
• A. W. Knapp, Lie groups beyond an introduction, 2nd ed.

Syllabus:

• Introduction: groups and group actions in geometric categories. Examples: topological, algebraic and Lie groups (definitions sketched).
• Classical groups: GL_n, SL_n, symplectic and orthogonal groups—real and complex.
• Rudiments of differential geometry: the categories of smooth, real analytic, and complex analytic (holomorphic) manifolds; embeddings and immersions; tangent vectors, vector fields, differential of a map.
• Definition and elementary properties of real and complex Lie groups.
• Lie algebra of a Lie group, infinitesimal action Lie(G) → Vect(X) associated to an action of G on X, homomorphism Lie(G) → Lie(H) associated to a Lie group homomorphism. Examples.
• Closed subgroups of a Lie group.
• Universal enveloping algebra, Poincaré-Birkhoff-Witt theorem.
• Baker-Campbell-Hausdorff formula.
• Lie subgroups; Chevalley's theorem.
• Covering groups, Lie algebra-Lie group correspondence (statement; reduction to existence of a Lie group with a given Lie algebra).
• Structure theory of Lie algebras: solvable, nilpotent and semisimple Lie algebras, Lie and Engels' theorems, Cartan's criterion.
• Ext groups of Lie algebra representations, theorems of Weyl, Levi and Ado. Proof of the Lie algebra-Lie group correspondence completed.
• Representation theory of sl₂
• Classification of complex semisimple Lie algebras and Lie groups

• Representation theory of complex semisimple Lie groups and algebras.
• Compact real Lie groups; classification and representations.
• Construction of the complex semisimple Lie groups as algebraic groups.
• The flag variety, Borel-Weil-Bott construction of representations.

Homework Assignments:

• Set 1 [Problem 1 corrected 9/20, problem 2 modified 12/6]
• Set 2 [Correction: Problem 9 should say "discrete normal subgroup." Oops: Problem 12 is a duplicate of Problem 6, part (a)]
• Set 3
• Set 4 [Oops: Problems 1, 3, 4 and 5 are duplicates from Set 3]

Grading: Based on homework assignments. No exams will be given.