Lie theory

In this course we will familiarize ourselves with the basic theory of Lie groups and Lie algebras. This subject is an appealing mix of group theory, geometry, algebra, and analysis. Please see the syllabus for a more detailed description.

• When: Friday 10:10 am - 12:00 pm
• Where: Math 528
• Primary textbooks: [Ada82] J. Frank Adams. Lectures on Lie Groups. University of Chicago Press, Chicago, 1982.
  [Kir08] A. A. Kirillov. Introduction to Lie Groups and Lie Algebras, volume 113 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2008.
  Other sources: [FH91] William Fulton and Joe Harris. Representation Theory: A First Course, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, 1991.
  [CSM95] Roger Carter, Graeme Segal, and Ian Macdonald. Lectures on Lie Groups and Lie Algebras, volume 32 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1995.
  Historical accounts: [Bor01] Armand Borel. Essays on the History of Lie Groups and Algebraic Groups, volume 21 of History of Mathematics. American Mathematical Society, Providence, RI, 2001.
  [Haw00] Thomas Hawkins. Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869–1926. Springer, New York, 2000.
  [JP21] Lizhen Ji and Athanase Papadopoulos. Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics. Springer, Cham, 2021.
• Syllabus
Here is a link to a course by John Morgan which covers the "Lie groups" portion of this course in much more detail.

Schedule

Date	Speaker	Topic	Notes
January 31	Caleb	Historical overview of Lie theory	Notes
February 7	Richard	The exponential map and the Lie bracket	Notes
February 7	Chang	The Lie correspondence	Notes
February 14	Dany	Basics of representation theory	Notes
February 14	Liam	The Peter-Weyl theorem	Notes
February 21	Angelina	Algebraic topology and the Lefschetz fixed point theorem	Notes
February 21	Liam	Maximal tori in Lie groups	Notes
February 28	Yan	Roots and the Weyl group	Notes
February 28	Chang	The Weyl integration formula	Notes
March 7	Dany	Representations of sl(2, C)	Notes
March 7	Owen	Representations of sl(n, C)	Notes
March 14	Angelina	Basic structure of Lie algebras	See [Kir08], Ch. 5
March 14	Richard	The Cartan decomposition and the Killing form	Notes
March 21	No class	Spring break - no class
March 28	Yan	The Weyl group, root systems, and Dynkin diagrams
March 28	Richard	Classification of root systems and semisimple Lie algebras
April 4	Liam	Theorem of the highest weight, Verma modules
April 4	Yan	Weyl character formula
April 11	Dany	Borel - Weil theorem
April 11	Owen	Exceptional Lie groups
April 18	Angelina	Simple groups of Lie type
April 18	Chang	Deligne-Lustzig theory
April 25	TBD	Algebraic groups
April 25	Owen	TBD
May 2	TBD	TBD
May 2	TBD	TBD