Weekly outline

  • General

    • Instructors: Wushi Goldring and Rikard B√∂gvad 
    • Time Period: Fall 2017, ABCD
    • Registration: We will pass out a sheet of paper during the first meeting of class so that Master's students can sign up.
    • Office Hours: W.G. Thursdays 9-10. No office hours Nov 16.
    • Course Room: 306 in House 6. UPDATE: All remaining lectures will be in Room 34, House 5
    • Course meeting times: Thursdays 10:15-12:00. UPDATE: The first meeting will be on Thursday September 7, the last meeting will be on December 7.
    • Texts: See suggested texts in Syllabus. We hope to post specific portions of different references on the course webpage. The primary text will be Knapp's Lie groups Beyond an introduction
    • Grading MSc students: Assessed based on ca. 4-6 homework assignments; no written exam.  UPDATE: All homework problems will count equally. We plan to use the following grading scheme: 90%-100% A, 80%-90% B, 70%-80% C, 60%-70% D, 50%-60% E, <50% F. 
    • Grading PhD students: Pass/Fail. In addition to homework, PhD students will be asked to make a short oral presentation (ca. 20 minutes) .For the oral presentations by PhD students, an excellent presentation will raise the grade by one letter (e.g. from C up to B), an OK presentation will maintain the same grade and a poor presentation will lower the grade by one letter (A to B, B to C etc.)  Tentatively, after the oral presentation, you will need an A or B to pass (meaning that if you have at least a B, you certainly pass; if the grades become very low for some reason, then we may re-evaluate whether C should pass too)
  • Overview

    It is not an exaggeration to say that algebraic groups play a key role in almost every area of mathematics. There has also been a deep interplay between algebraic groups and several areas in physics. Some key aspects of the theory of algebraic groups arose from physics. In turn, algebraic groups have had numerous applications to physics.  


    In many subjects, algebraic groups are increasingly appearing at the forefront, as in number theory, algebraic geometry, representation theory, Hodge theory, etc. In other areas algebraic groups remain hidden in the background, but even there one may argue that their importance will come to light in the future.   


    Besides their ubiquity, an additional attraction of algebraic groups is that one can  work with them in practice without getting too entangled in the proofs of technical results about them. 


    The aim of this course will be to provide an appreciation for algebraic groups and connected structures such as root data and flag varieties by focusing on examples and applications, while keeping the prerequisites and technicalities to a minimum.  


    • Topic 1: Introduction and outline

      We will begin with an overview of algebraic groups. We will outline some of the different aspects of the theory that will be discussed in the course. We will mention some of the applications that algebraic groups have had, and some of the ways they are used in research today.

      We hope to mention the following aspects:

      1. Structure theory: different kinds of groups and subgroups. This leads to 

      2. Root data

      3. Classification

      3. Connection with Lie algebras

      4. Specialization to different fields. Two examples: Lie groups over R and C, finite groups when the field is finite.

      5. Representation theory: Theory of the highest weight. One of the best ways to construct representations is by geometric methods:

      6. Connection with flag varieties


      • Topic 2: Examples

        We will illustrate some of the things discussed in the outline in some elementary examples. We will focus on the example GL(n), the group of n by n invertible matrices. In this example, many of the associated objects can be described explicitly in terms of matrices and linear algebra with a minimum amount of background knowledge.


        For G=GL(n), we will describe Borel subgroups, a maximal torus, semisimple and unipotent elements, root subgroups, the Lie algebra, root spaces, the adjoint action, topological properties of GL(n, C) and GL(n R), number of points of GL(n, F_p). We will describe the characters, cocharacters, roots and coroots of GL(n) relative to the diagonal maximal torus; this will give us an example of a root datum for GL(n).

        If we have time, we can discuss some representations of GL(n) as well, such as the symmetric and exterior powers.

        Much of the rest of the course will focus on explaining how these objects make sense for a general class of groups, of which GL(n) is the prototypical example. The class is that of connected reductive groups, notions that will be defined later in the course.


         

        • Topic 3: Root data, root systems, Weyl group

          1. Definition of root data

          2. Two definitions of root systems, as given by Knapp and Springer

          3. Example of root datum of GL(n) and its diagonal subgroup

          4. Statement of the classification of root systems.

          5. Structure associated with root systems: a. Weyl group and Weyl chambers, b. regular elements, c. positive roots, d. simple roots, e. Dynkin diagram

          6. Further examples of the above.



          • Topic 4: Connected Reductive groups and their classification by Root data

            1. Definition of Zariski topology. Discussion of connectedness in Zariski topology versus classical topology of G(R), G(C).

            2. Definition of reductive and semisimple groups. The radical and unipotent radical of an algebraic group

            3. Maximal tori in a reductive group.

            4. The root datum of (G,T), where G is connected, reductive over an algebraically closed field and T is a maximal torus.

            5. Classification: a. Existence: Every root datum arises from some (G,T). b. Uniqueness: (G,T) is determined up to isomorphism by its root datum. 

            • Oral Presentations by PhD students

              Place: 

              • All presentations in Room 34, House 5

              Time:  

              • Wednesday Dec. 6, 15:30-17:35
              • Thursday Dec. 7, 15:00-17:00 
              • Friday Dec. 8, 14:30-17:00

              Attendance: 

              • PhD students: If you don't have a time-conflict, you should attend your fellow students' presentations.
              • MSc/BSc/other students: You are very welcome to attend any number of presentations that you want, but you are in no way obliged to do so. For example, if you need the time to study for other courses, or for your job, feel free to choose to do that instead.