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.
You can find a preliminary version of this book here.
Grades will be based upon class participation.
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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