MAT 552 Syllabus Introduction to Lie groups and Lie algebras Spring 2022

What follows is a tentative syllabus. The pace of these topics (as well as any additional topics to be covered) will be finalized in the first weeks of the semester.

• Week 1. Section 2.1 - 2.6. Lie groups and differentiable actions.
• Week 2. Section 2.7 - 3.6. Lie algebras and classical groups.
• Week 3. Section 3.7 - 3.10. Baker-Campbell-Hausdorff and Lie's Theorems.
• Week 4. Section 4.1 - 4.4. Representations of Lie groups and Lie algebras.
• Week 5. Section 4.5 - 4.7. Haar measure and characters.
• Week 6. Section 4.7, 5.1 - 5.3. The Peter-Weyl Theorem and Poincare-Birkhoff-Witt.
• Week 7. Section 5.4 - 5.8. Structure theory of Lie algebras and the Killing form.
• Week 8. Section 5.9 - 6.3. Finite-dimensional complex-linear representations of complex semisimple Lie algebras.
• Week 9. Section 4.8, 6.4 - 6.6. Cartan subalgebras and root systems.
• Week 10. Section 6.7 - 7.3. Abstract root systems, Coxeter groups, Weyl groups.
• Week 11. Section 7.4 - 7.7. The weight lattice and the root lattice.
• Week 12. Section 7.8 - 7.10. Classification of root systems and complex semisimple Lie algebras.
• Week 13. Section 8.1 - 8.2. Highest weight representations and Verma modules.
• Week 14. Section 8.3 - 8.6. Bernstein-Gelfand-Gelfand resolution and the Weyl character formula.

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