Lie Groups and Lie Algebras: Fall 2025
                            
                                Tuesday and Thursday 1:10 -- 2:25
                    
                                Instructor:  John Morgan,  615 Mathematics
                                                    jmorgan@math.columbia.edu

                                Office hours Tuesday 2:30 pm - 4 pm

                                TA:       Wenqi Li
                                               Office Hours 10 am - 1 pm Wednesday in the Help Room
                                              wl2935@columbia.edu
                                               

                                Course Website:  www.math.columbia.edu/~jmorgan


                           

Prerequisites: I will assume basic knowledge in the following areas:

(i) linear algebra including matrices, tensor products and endomorphisms,
(ii) point-set topology,
(iii) differentiable manifolds, includinginverse and implicit function theorems,
vector fields and their integral  curves,       
(iv) group theory, subgroups, cosets, normal subgproups and quotient groups,
(v) algebra including associative algebras, ideals and quotients.

On several occasions I will use more advanced topics out of these areas; e.g. Frobenius's theorem
on integrating distributions to give foliations and various results from algebra about completions; e.g., the
ell-adic topology. I will provide  introductory notes on the course website for these topics. 
 

Course Structure and Grades:

The only exam in the course will be an in-class final exam at the end on the semester.

Every week or two as the course proceeds, I will assign homework with a due date.
The homework will be graded by the TA and returned to you.
The TA will also hold review sessions each week to go over the homework and answer
questions about the lectures.

Grades will be assigned based performance on the homework and final exam
with the final having largest weight.



Reading Material:

I will post notes of the lectures on the class website at this website.
There will also notes on some more advanced topics that will be used with only brief,
if any explanation, in the course

The homework will be posted on the course website.

 
The Course is divided into 6 parts. 

I. The objects, Lie Groups and Lie Algebras, and some of the first relations
    between them including Lie's basic theorems.

II. The Universal Enveloping Algebra and the Baker-Campbell-Hausdorff formula.
    Proof that finite dimensional real Lie Groups are naturally real analytic objects.

III. Compact Lie groups and their maximal tori; the Weyl group. Complete Reducibility of representations of        compact Lie groups.                          

iV. Root Systems and Dynkin Diagrams

V. Complex Semi-simple Lie groups and relation to compact Lie groups.

VI. Classification of compact groups/semi-simple complex groups.




Lecture Notes.

Primer on Integral Curves for Vector Fields can be found here.
Primer on Poincare' duality and the Lefschetz fixed point Theorem can be found here.
Primer on measure and integration on compact Lie groups can be found here.  




Notes for Lecture 1 can be found here. 
Notes for Lecture 2 can be found here.
Notes for Lecture 3 can be found here.
Notes for Lecture 4 can be found here.
Notes for Lecture 5 can be found here.
Notes for Lecture 6 can be found here.
Notes for Lecture 7 can be found here.
Notes for Lecture 8 can be found here.
Notes for Lecture 9 can be found here.
Notes for Lecture 10 can be found here.
Notes for Lecture 11 can be found here.
Notes for Lecture 12 can be found here.




Problem Sets. (Hand in homework in class on due date.)

Problem Set 1 can be found here.  Due Tuesday, Sept. 16
Problem Set 2 can be found here.  Due Thursday, Sept. 25
Problem Set 3 can be found here.  Due Thursday, Oct.2
Problem Set 4 can be found here.  Due Tuesday, Oct. 14
Problem Set 5 can be found here.  Due Tuesday, Oct. 21
Problem Set 6 can be found here.
Problem Set 7 can be found here.
Problem Set 8 can be found here.
Problem Set 9 can be found here.
Problem Set 10 can be found here