# MATH 534: Lie groups.

Instructor: Julia Gordon.
Where: Math building, Room 225
When: MWF 1-2pm.
My office: Math 217.

e-mail: gor at math dot ubc dot ca

The textbook: Onischik and Vinberg, "Lie groups and algebraic groups".

### Presentations:

(Please note the room changes! We got kicked out of MATX 1102!)
• Friday November 30, 1-2pm, Math 225 (as usual):
Aurel Meyer, Galois cohomology and algebraic groups.
• Wednesday December 5, 1:30-3:30pm: in Math 104
Alex Duncan, Finite groups of Lie type.
Jun Ho Hwang, Cohomology of Grassmannians.
• Thursday December 6, 2-4pm: in Math 104
Andrew Morrison, Geometric plethysm.
Michele Klaus, McKay correspondence.
• Friday December 7, 10am -12 in Math 103
Hesam Abbaspour, Existence of compact real forms for complex Lie groups.
David Steinberg, Riemannian metrics on Lie groups, and Cartan's theorem.

### Homework assignments

• Problem Set 1 (Due Fri Sept 14)
• Problem Set 2 (Due Wednesday Sept 26)
• Problem set 3 (Due Wednesday Oct 10)
• Problem Set 4 (Due Monday October 22) (a confusing typo in problem 1 is now corrected). Another correction: in problem 2(b), we need to assume n>2.
• Problem Set 5 (Due Fri Nov 2)
• Problem Set 6 (Due Mon Nov 19)
• Final assignment

• ### Lecture Notes.

• We have covered:
• Chapter 1 of Onischik-Vinberg in detail + Sections 7, 8 of Bump. That includes defintions of real and complex Lie groups and their actions on manifolds, the tangent algebra, the exponential map, connectedness and simple connectedness, solvalbe Lie groups and Lie algebras, Lie's theorem.
• Review of root systems and Lie algebras: From Chapter 4 of Onischik-Vinberg, Sections 4.1.4, 4.1.5, 4.1.6, 4.2.1, 4.2.5 (not including extended Dynkin diagrams), 4.2.7, and statements of the main theorems of 4.3.2 and 4.3.3. We did all this without proofs, and with one modification: instead of using t -- the tangent algebra of a maximal torus, we just used a "toral subalgebra" of the Lie algebra (without knowing that it corresponds to a torus in G). An alternative reference for this is for example Humphreys, "Introduction to Lie algebras and Representation theory". We skipped entirely the rest of 4.3.1 -- 4.3.3 (since it is all about proving existence/uniqueness theorems for Lie algebras).
• Skipped Chapter 2 -- algebraic varieties.
• In the second half of the semester: Algebraic groups (Chapters 3 and 4.1 -- 4.3 of the textbook).
• definition of an algebraic group;
• complexification and the notion of a real form of a complex Lie group;
• Jordan decomposition;
• Tori and rational structures on their characters;
• Solvable algebraic groups; Borel's fixed point theorem;
• Borel subgroups; the variety G/B;
• Weyl group and Weyl chambers;
• reductive algebraic groups;
• Irreducuibility and connectedness;
• Existence of a faithful representation for connected semisimple Lie groups
• Algebraicity of connected semisimple complex Lie groups;
• Root and weight lattices;
• Classification of semisimple connected complex Lie groups (or linear algebraic groups)
• Week 1 (Lectures 1-2). (by Aurel M.)
• Lecture 3. (by Andrew M.)
• Lecture 4. (by Andrew M.)
• Lie bracket and the exponential map handout
• Lectures 13-14 (notes by Michele K.)
• Example: root system for sl(3)