Math 261A—Lie Groups—Fall 2010
Lectures: MWF 10:00-11:00am, Room 45 Evans
Course Control Number: 54521
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
groups and Lie algebras I (Encyclopaedia of Mathematical Sciences,
- 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,
- A. W. Knapp, Lie
groups beyond an introduction, 2nd ed.
Possible additional topics if there is time:
- Introduction: groups and group actions in geometric categories.
Examples: topological, algebraic and Lie groups (definitions
- Classical groups: GLn, SLn,
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.
- 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.
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