Instructor

• Xiang Tang, Cupples I, 112B, xtang@math.wustl.edu
  

Office Hour

• Thursday 2-4 or by appointment
  

Preview

• This is a general introduction course to Lie algebras and Lie groups. We will essentially follow the book (references 1) by Varadarajan. We plan to discuss the following three topics this semester. 

 Homework

• There will be monthly due homework set.

 Exams

• No midterms or finals.

 Presentation

• In the final week, each student will be given 20 minutes to present a topic related to Lie algebras and Lie groups. See detailed schedule.
  

Grade

• Grade is assigned according to homework scores and final presentation.  

Reference

• Lie groups, Lie algebras, and their representations,  V. S. Varadarajan
• Representation theory, W. Fulton and J. Harris
• Lie algebras and Lie groups, J. Serre
• Representations of Compact Lie groups,  T. Brocker and T. Dieck

Plan of this semester

• Part I: Lie groups and Lie algebras

 Week 1(Jan 13-19)

1. Definition of Lie groups and Examples (Sec  2.1)
2. Definition of Lie algebras and Examples (Sec 2.2)
3. Lie algebra of a Lie group and enveloping algebra (Sec 2.3-2.4)

 Week 2(Jan 20-26)

      Jan 21, MLK day, no class

1. Subgroups and subalgebras (Sec 2.5)
2. Locally isomorphic groups (Sec 2.6)
   

Week 3(Jan 27-Feb 2)

1. Homomorphisms (Sec. 2.7)
2. Closed Lie subgroups and homogeneous spaces I (Sec. 2.8)
3. Closed Lie subgroups and homogeneous spaces II(Sec. 2.8)
   

Week 4(Feb 3-Feb 9)

1. Exponential map (Sec. 2.10)
2. Taylor Series Expansions on a Lie group (Sec. 2.12)
3. Adjoin Representations (Sec. 2.13)

Week 5(Feb 10-Feb 16)

1. Differential of exponential map (Sec. 2.14)
2. Baker-Campbell-Hausdorff formula (Sec. 2.15)
3. Transformation groups I (Sec. 2.16)

• Part II. Structure theory

Week 6(Feb 17-Feb 23)

1. Transformation groups II (Sec. 2.16)
2. Universal enveloping algebra I (Sec. 3.2)
3. Universal enveloping algebra II (Sec. 3.3)

Week 7(Feb 24-Mar 1)

1.  Enveloping algebra of a Lie group (Sec. 3.4)
2. Nilpotent Lie algebra and Lie group (Sec. 3.5-3.6)
3. Solvable Lie algebra and Lie group (Sec. 3.7)
   

Week 8(Mar 2-Mar 8)

1. Radical and Nil radical (Sec. 3.8)
2. Cartan-Killing’s form (Sec. 3.9)
3. Semi-simple Lie algebra (Sec. 3.10)

Week 9(Mar 9-Mar 15)

       Spring break, No class

Week 10(Mar 16-Mar 22)

1. Casimir element (Sec. 3.11)
2. Weyl’s theorem and Levi decomposition (Sec. 3.13-14)
3. Lie third theorem and Ado’s theorem (Sec. 3.15-17)

      Presentation topics are due.

• Part III. Representation theory 
  

Week 11(Mar 23-Mar 29)

1. Cartan subalgebra (Sec. 4.1)
2. Representations of sl(2, C) (Sec. 4.2)
3. Structure theory I (Sec. 4.3)

Week 12(Mar 30-Apr 5)

1. Structure theory II (Sec. 4.3)
2. Structure theory III (Sec. 4.3)
3. Classical Lie algebras I (Sec. 4.4)

Presentation topic is determined.

Week 13(Apr 6-Apr 12)

1. Classical Lie algebras II (Sec. 4.4)
2. Determination of Simple Lie algebras (Sec. 4.5)
3. Representation of highest weight I (Sec. 4.6)

 
Week 14(Apr 13-Apr 19)

1. Representation of highest weight II (Sec. 4.6)
2. Representation of simple Lie algebra (Sec. 4.7)
3. Construction of a semimplie Lie algebra from its Cartan matrix (Sec. 4.8)

Week 15(Apr 20-Apr 25)

1. Compact and semisimple Lie group I. (Sec. 4.11)
2. Compact and semisimple Lie group II. (Sec. 4.11)
3. Maximal tori (Sec. 4.12)