Math 371 - Lie Theory
Sevak Mkrtchyan
456 Herman Brown Hall
e-mail: (first name).(last name)@rice.edu

Course Description

Study of classical groups as symmetries of Euclidean spaces. Geometry of complex numbers and quaternions, rotations and reflections of R^n, the orthogonal, unitary and sympletic groups. Tangent spaces to matrix groups, Lie algebras and exponential map. If time permits: the structure of Lie algebras and the matrix logarithm.
The following is the schedule for what I have covered and plan to cover each day. It will be updated regularly during the semester.

Lectures

TuTh 10:40-12:05PM in HBH 227

Office Hours:

M 11-12,W 3-4PM (456 Herman Brown Hall)

Textbook

The textbook for the course is: Naive Lie Theory by John Stillwell.
Other recommended elementary textbooks on the subject are
Matrix Groups by Morton Curtis
Introduction to Lie Algebras by Erdmann and Wildon

Grades

Your grade in the class will be based on the following weights:
50% - Homework assignments
50% - Any exams and projects/presentations.

Homework

Homework will be assigned on Owl Space and collected in class at the beginning of class on Tuesdays. No late homework will be accepted. The lowest homework grade will be dropped.

This is a schedule for what I have covered and what I plan to cover in class each day. This section will be updated regularly during the semester.

Date
 Topic
 Homework
 Due
Tue, August 24 Introduction: Groups of symmetries, Lie groups, Lie algebras, Group actions, Representations theory.
Thu, August 26 The group of rotational symmetries of R^2. Hwk1 Aug 31
Tue, August 31 Introductoin to quaternions
Thu, September 2 Rotations of S^3 in terms of quaternions Hwk2 Sep 7
Tue, September 7 Cosets, Normal Subgroups, The Hopf Fibration
Thu, September 9 The group of isometries of R^n fixing the origin is O(n). Intro to C^n Hwk3 Sep 14
Tue, September 14 Inner product on C^n, the groups U(n) and SU(n).
Thu, September 16 Simplicity of SO(3). Direct product of groups. The map SU(2)xSU(2)->SO(4) Hwk4 Sep 21
Tue, September 21 Path connectedness. SO(2) and SU(2) are path connected.
Thu, September 23 SO(n) and SU(n) are path connected. Introduction to H^n and Sp(n). Hwk5 Sep 30
Tue, September 28 Maximal Tori in Matrix Lie Groups
Thu, September 30 Centers of the Matrix Lie Groups Hwk6 Oct 05
Tue, October 5 Introduction to tangent spaces. The tangent space at 1 to unit quaternions.
Thu, October 7 The Lie bracket in the tangent space at identity of SU(2). Midterm Oct 19
Tue, October 12 No class - mid term break.
Thu, October 14 The exponential of a matrix. Convergence of the series.
Tue, October 19 The Lie algebras so(n),su(n),u(n),sp(n). Hwk7 Oct 26
Thu, October 21 Dimensions of so(n),u(n),su(n),sp(n).
Tue, October 26 Complexification of Lie algebras. The groups O(n,C) and SO(n,C).
Thu, October 28 Symplectic forms.
Tue, November 2 The symplectic groups Sp(2n,R) and Sp(2n,C). Sp(n) as the intersection of U(n) and Sp(2n,C) Hwk8 Nov 9
Thu, November 4 Midterm problem solutions
Tue, November 9 Simplicity of sl(n,C), so(3). Non-trivial ideals in so(4) Hwk9 Nov 16
Thu, November 11 Simplicity of so(n) for n>4
Tue, November 16 Matrix logarithm. Exp maps the Lie algebra to the matrix Lie group.
Thu, November 18 Exp from gl(n,C) to GL(n,C) is onto. Sequential tangents are tangents. Hwk10 Nov 23
Tue, November 23 Log is onto a neighborhood. A path-connected Lie group is generated by a neighborhood.
Thu, November 25 No class - Thanksgiving break.
Tue, November 30 Campbell-Baker-Hausdorff theorem
Tue, December 2 Representation theory of sl(2,C). Final exam Dec 15


You are expected to attend every class and arrive on time. It is your responsibility to be informed of any announcements made in class.

Disability Support

Any student with a documented disability seeking academic adjustments or accommodations is requested to speak with me during the first two weeks of class. All such discussions will remain as confidential as possible. Students with disabilities will need to also contact Disability Support Services in the Allen Center.

Any student with a disability requiring accommodations in this course is encouraged to contact me after class or during office hours. Additionally, students will also need to contact Disability Support Services in the Allen Center.

If you have a documented disability that will impact your work in this class, please contact me to discuss your needs. Additionally, you will need to register with the Disability Support Services Office in the Allen Center.