MAT 552  Introduction to Lie groups and Lie algebras   Spring 2015 

Mon & Wed 11:30am - 12:50pm in Math 4-130

Instructor: Ljudmila Kamenova

Office: Math Tower 3-115
Office hours: Tuesdays 1-4pm; drop by or send me an e-mail anytime

This course will cover the basic theory of Lie groups and Lie algebras. This material is frequently used by matematicians in various fields. It will be useful to graduate students in the fields of algebra, topology, geometry, dynamics and theoretical physics. Some of the topics are: classical groups (e.g., the general linear, orthogonal and symplectic Lie groups), representations, characters, Campbell-Hausdorff's formula, Poincare-Birkhoff-Witt's theorem, Schur's lemma, Peter-Weyl's theorem, Lie's theorem (about representations of a solvable Lie algebra), Engel's theorem, Cartan's criterion of solvability, Dynkin diagrams, Weyl chambers, classification of simple Lie algebras, etc.

Recommended Text: An Introduction to Lie Groups and Lie Algebras, by A. Kirillov Jr., Cambridge University Press (2008)

You can find a preliminary version of this book here.

Additional references:

Prerequisites and grading: We will assume the material covered in the algebra core courses MAT534 and MAT535, as well as basic differential geometry.

Grades will be based upon class participation.

Tentative Syllabus (subject to change):

Week 1: 2.1 - 2.6, Definition of Lie groups, subgroups, cosets, group actions on manifolds, homogeneous spaces.

Week 2: 2.7 - 3.6, Classical groups, exponential and logarithmic maps, Lie bracket, Lie algebras, subalgebras, ideals, stabilizers, center

Week 3: 3.7 - 3.10, Baker-Campbell-Hausdorff formula, Lie's Theorems

Week 4: 4.1 - 4.4, Representations, operations on representations, irreducible representations, Schur's lemma

Week 5: 4.5 - 4.7, Unitary representations and complete reducibility, representations of finite groups, Haar measure on compact Lie groups, characters

Week 6: 4.7, 5.1 - 5.3, Peter-Weyl theorem, universal enveloping algebra and Poincare-Birkoff-Witt, commutants

Week 7: 5.4 - 5.8, Solvable and nilpotent Lie algebras (with Lie/Engel theorems), semisimple and reductive algebras, invariant bilinear forms, Killing form, Cartan criteria

Week 8: 5.9 - 6.3, Jordan decomposition, complex semisimple Lie algebras, compact groups/algebras, complete reducibility of representations

Week 9: 4.8, 6.4 - 6.6, Toral subalgebras, Cartan subalgebras, root systems

Week 10: 6.7 - 7.3, Regular elements and Cartan subalgebras, abstract root systems, Weyl group, rank 2 root systems

Week 11: 7.4 - 7.7, Positive roots, simple roots, weight lattice, root lattice, Weyl chambers, simple reflections

Week 12: 7.8 - 7.10, Dynkin diagrams, classification of root systems, classification of semisimple Lie algebras

Week 13: 8.1 - 8.2, Representations of semisimple Lie algebras, weight decomposition, characters, highest weight representations, Verma modules

Week 14: 8.3 - 8.6, Classification of irreducible finite-dimensional representations, BGG resolution, Weyl character formula

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