• Instructor: Professor Joel Roberts

office: Vincent 351
phone: 625-1076
e-mail: roberts@math.umn.edu
http://www.math.umn.edu/~roberts
 
• Schedule:
  • TuTh     12:20p.m. to 1:10 p.m.   Vincent 311
  • Friday   10:10a.m. to 11:00a.m.    Vincent 364
     
• Office hours:
  • TuTh   10:40a.m. to 11:40 a.m.
  • Mon.   11:00a.m. to 12 noon.
  • Wed.   2:30p.m. to 3:20p.m.
     
• Subject matter: We will begin with Part II of the text (Lie groups and Lie algebras) and continue with Part III (The classical Lie algebras and their representations). Our plan is to cover chapters 7 through 15, thus including the irreducible representations of sl_n but not the representations of the other classical simple Lie algebras. If time permits, we will study just enough of the material in Part I (Finite groups) of the text in order to understand H. Weyl's construction of the irreducible representations of sl_n.
  The tentative plan for Spring semester includes the remainder of Part III, as well as Part IV (Lie theory).

 
• Prerequisite: The official prerequisite is Math 8301/8302 (Manifolds and topology). Some knowledge about manifolds is definitely needed, and knowledge about group theory and linear algebra is also very useful.

 
• Required work: There will be homework assignments, as follows (the due dates are subject to revision):
   

  	topics	approximate due date	assigned exercises
  #1	Chapter 7	September 23	##7.1, 7.2, 7.3, 7.11, 7.14
  #2	Chapter 8	October 12	##8.1, 8.10, 8.24, 8.26, 8.40
  #3	Chapter 9	November 30	Any 4 of the following: ##9.3, 9.6, 9.7, 9.16, 9.23
  	We'll skip Chapter 10
  #4	Chapters 11, 12 & 13	optional	##13.8, 13.9, 13.10, 13.11; misc. problems from class


 
• General description: A Lie groupis a group which is also a C manifold or a complex manifold. In the two cases we speak of real Lie groups and complex Lie groups respectively. Matrix groups are the most immediate examples and are also very important in the general theory.
   
  We can associate a Lie algebrato a given Lie group. In this case of matrix groups the Lie algebras are matrix algebras where the operation is the Lie bracket: [X,Y] = XY - YX. To construct the Lie bracket in general we let the group Gact on itself by conjugation. Taking the differential, we get a homomorphism of Lie groups, from Gto the group Aut(g) of linear automorphisms, where g := T_eG is the tangent space of Gat the identity. Taking the differential again(!), we obtain the Lie bracket, a bilinear pairing from g to itself with special additional properties.
   
  It is a surprising and useful fact that properties of the Lie algebra determine properties of the Lie group, at least to a considerable extent. Similarly for homomorphisms of Lie algebras vis-à-vishomomorphisms of Lie groups. It turns out to be possible to classify the simple complex Lie algebras, and thus to classify the simple complex Lie groups (at least the simply connected ones, anyway). This classification theory will have to wait until next semester; this semester we will make considerable progress in studying the irreduciblerepresentations. Next semester that will be completed, and those methods will then be applied to the classification theory.
   
  The theory of Lie groups has connections with various branches of mathematics such as differential geometry, differential equations, automorphic forms, etc., and also with theoretical physics. Unfortunately, we really won't have time to study those applications in our introductory course.

Last updated December 15, 1999.

Back to the class homepage.