Faculty of Science Course Syllabus
Department of Mathematics & Statistics

MATH 4057/5057: Lie Theory (Winter 2024)

Instructor: Theo Johnson-Freyd, theojf@dal.ca.
Lectures: Tuesdays and Thursdays, 10:05-11:25 Atlantic Time, LSC C202.
Office Hours: Wednesdays, 14:35-16:55 Atlantic Time, Chase 214.
Course website: http://categorified.net/24Winter5057/.

This course is governed by the academic rules and regulations set forth in the University Calendar and by the Senate.

Course description

A Lie group is a group (a collection of symmetries) which is also a manifold (a smooth space). Lie groups arise whenever an object has continuous families of automorphisms. Finite groups are notoriously hard to classify. This is because groups exist in order to act on things, but a generic finite group may not have any easily accessible actions. Every Lie group, on the other hand, has a nice small linear space that it act on: its tangent space at the identity. This linear space furthermore comes with an important algebraic structure: the first derivative of the group multiplication. This algebraic structure makes it into a Lie algebra.

A nontrivial application of the theory of ODEs says even more: the Lie algebra of a Lie group almost completely determines the Lie group. This makes the analysis and classification of Lie groups much more accessible than the analysis and classification of discrete groups. In fact, for compact connected Lie groups, the analysis and classification reduces to a completely solved combinatorial problem. The goal of this course is to explain this classification.

Course materials and delivery

This will be a fast-paced lecture course, following closely the book:

• Berkeley Lectures on Lie Groups and Quantum Groups, by R. Borcherds, M. Haiman, T. Johnson-Freyd, N. Reshetikhin, and V. Serganova, http://categorified.net/LieQuantumGroups.pdf.

Our goal is to get through the first six chapters, i.e. Part I, of this book. This goal is ambitious.

Lectures will be given in person. (The occasional lecture may be delivered over Zoom.) This is a very small class. Although attendance is not mandatory, your presence or absence will certainly be noticed by all participants. Lectures will not be recorded.

Course assessment and requirements

A problem set will be due every two weeks. The problem sets are hard: expect to dedicate a lot of time to them, and work with your classmates on them.

There will be a short oral final exam based on the problem sets.

Component weighting

Homework: 70%.
Final exam: 30%.

Conversion of numerical grades to Final Letter Grades follows the Dalhousie Common Grade Scale. Note that for graduate students, any grade below B- is recorded as an F.

Lecture topics by week

1. Definition of Lie group. The classical Lie groups.
2. Manifolds and vector fields
3. Definition of Lie algebra
4. Relationship between Lie algebras and Lie groups
5. Universal enveloping algebras
6. Structure theory of Lie algebras
7. Structure theory continued: nilpotency, solvability, and semisimplicity
8. Representation theory of sl(2)
9. Cartan subalgebras
10. Root systems
11. Cartan matrices and Dynkin diagrams
12. Weyl character formula