Math 261AB: Lie Groups—Fall-Spring 2015-16

Announcements

Professor

Time and place

Course content and prerequisites

Textbook

Syllabus

Homework and grades

Announcements

(1/15) Welcome to the Spring semester of Lie Groups! Check this page periodically for updates.

Professor

Mark Haiman,

855 Evans Hall, office hours MWF 11-12 or by appointment.

Time and place

MWF 10-11am, 2 Evans Hall.

Course content and prerequisites

This is a two-semester course in Lie groups, Lie algebras, and their representations, covering fundamentals in the fall, and more advanced material, including open research problems, in the spring. This spring I plan to concentrate on the areas where geometry and representation theory of semi-simple (or more generally, reductive) Lie groups meet algebra and combinatorics. Topics will include algebraic groups and Hopf algebras, flag varieties and the Borel-Weil-Bott construction, and quantum groups and canonical bases.

The spring semester should be of interest to a variety of students with some prior knowledge of Lie groups and Lie algebras, not limited to those continuing from 261A this past fall.

The official but not really necessary prerequisite for this course is Math 214 (Differential Geometry). In practice a strong general background in basic algebra and topology is sufficient for 261A, and 261A or equivalent is sufficient for 261B.

Textbook

Main course text

V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations. Available online from computers on campus or using the library proxy server.

We will use Vardarajan in the spring semester for structure of reductive and compact Lie groups and their representations. All material on other topics will be presented in the lectures.

Other suggested references

Introduction to differential geometry:

Jeffrey M. Lee, Manifolds and Differential Geometry, for an introduction to that subject.

General references on Lie groups and algebras:

N. Bourbaki, Lie Groups and Lie Algebras, Chapters 1-3, 4-6, 7-9, (a comprehensive reference).
N. Jacobson, Lie Algebras
V. V. Gorbatsevich, A. L. Onishchik and E. B. Vinberg, Lie groups and Lie algebras I, available online

Algebraic groups and related topics

A. Borel, Linear Algebraic groups, available online
A. L. Onishchik and E. B. Vinberg, Lie Groups and Algebraic Groups, available online

Syllabus

Spring

The first portion completes the foundational material we began in the fall and can serve as review for students joining the course this spring:

Representation theory of sl₂ reviewed.
Classification of complex semisimple Lie algebras and Lie groups, and compact real Lie groups.
Representation theory of semisimple complex and compact real Lie groups and algebras.
Cohomology of Lie algebra representations. Theorems of Weyl, Levi and Ado. Proof of the Lie algebra to Lie group correspondence.

After that:

Algebraic groups and Hopf algebras.
Construction of reductive algebraic groups from their representations.
Flag varieties and the Borel-Weil-Bott construction.
Generalizations of Borel-Weil-Bott with applications to combinatorics and geometry and open problems.
Quantum groups, crystals, and canonical bases. More open problems.

Other possible topics, depending on time: algebraic groups over Z and finite Chevalley groups; Hecke algebras and Kazhdan-Lusztig theory.

Fall

Groups and group actions in geometric categories: topological, algebraic and Lie groups.
Classical groups: GL_n, SL_n, symplectic and orthogonal groups, both real and complex.
Basic differential geometry: smooth, real analytic, and holomorphic manifolds; embeddings and immersions; tangent vectors, vector fields, differential of a map.
Definition and basic properties of real and complex Lie groups.
Lie algebra of a Lie group. Exponential map. Infinitesimal action Lie(G)→Vect(X) associated to G action on a manifold. Lie(G) as a functor.
Closed subgroups of a Lie group.
Universal enveloping algebra, Poincaré-Birkhoff-Witt theorem.
Baker-Campbell-Hausdorff formula.
Lie subgroups. Chevalley's theorem.
Covering groups. Statement of the correspondence between Lie groups and Lie algebras.
Structure theory of Lie algebras. Solvable, nilpotent and semisimple Lie algebras. Lie and Engels' theorems. Cartan's criterion.
Representation theory of sl₂.

Homework and grades

Grades will be based on homework sets assigned periodically. I will try to keep them more current this spring than I did in the fall, and there will probably be somewhat fewer problems overall.

For a grade of A, you should do about half of the homework problems, including some of the more challenging ones; or proportionally fewer for a lower passing grade.

Alternatively, you can earn a grade of A by studying any open research problem connected to the subject matter. You don't have to solve the problem, but should make a serious attempt to understand it and work out some special cases.

Spring Problem Sets

TBA

Fall Problem Sets

Problem Set 1 on Lectures 1-33. Due on last day of class, Friday Dec. 4. Here is the LaTeX source of PS 1 in case you want to compose solutions in TeX and find this helpful.
Problem Set 2 on Lectures 34-40, with LaTeX source. Due on the last day of final exam week, Friday Dec. 18.