MAT 552 Syllabus Introduction to Lie groups and Lie algebras Spring 2025

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.

Below is the schedule for the course topics, aligned with the academic calendar.

• Week 1: January 27 (Mon) & January 29 (Wed) - Sections 2.1 – 2.6: Lie groups and differentiable actions.
• Week 2: February 3 (Mon) & February 5 (Wed) - Sections 2.7 – 3.6: Lie algebras and classical groups.
• Week 3: February 10 (Mon) & February 12 (Wed) - Sections 3.7 – 3.10: Baker-Campbell-Hausdorff and Lie's Theorems.
• Week 4: February 17 (Mon) & February 19 (Wed) - Sections 4.1 – 4.4: Representations of Lie groups and Lie algebras.
• Week 5: February 24 (Mon) & February 26 (Wed) - Sections 4.5 – 4.7: Haar measure and characters.
• Week 6: March 3 (Mon) & March 5 (Wed) - Sections 4.7, 5.1 – 5.3: The Peter-Weyl Theorem and Poincare-Birkhoff-Witt.
• Week 7: March 10 (Mon) & March 12 (Wed) - Sections 5.4 – 5.8: Structure theory of Lie algebras and the Killing form.
• Spring Recess: March 17 (Mon) & March 19 (Wed) - No classes.
• Week 8: March 24 (Mon) & March 26 (Wed) - Sections 5.9 – 6.3: Finite-dimensional complex-linear representations of complex semisimple Lie algebras.
• Week 9: March 31 (Mon) & April 2 (Wed) - Sections 4.8, 6.4 – 6.6: Cartan subalgebras and root systems.
• Week 10: April 7 (Mon) & April 9 (Wed) - Sections 6.7 – 7.3: Abstract root systems, Coxeter groups, Weyl groups.
• Week 11: April 14 (Mon) & April 16 (Wed) - Sections 7.4 – 7.7: The weight lattice and the root lattice.
• Week 12: April 21 (Mon) & April 23 (Wed) - Sections 7.8 – 7.10: Classification of root systems and complex semisimple Lie algebras.
• Week 13: April 28 (Mon) & April 30 (Wed) - Sections 8.1 – 8.2: Highest weight representations and Verma modules.
• Week 14: May 5 (Mon) & May 7 (Wed) - Sections 8.3 – 8.6: Bernstein-Gelfand-Gelfand resolution and the Weyl character formula.

Note: This schedule is tentative and subject to adjustments based on the course's progression. Refer to the Stony Brook University Academic Calendar for official dates.

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