Linear Algebraic Groups (MA5113)

Prof. Dr. Gregor Kemper

 

Schedule

Content

In mathematics, many of the groups that appear naturally have a nice description as matrix groups - they are linear algebraic groups. Examples are GL_n, the group of all diagonal matrices D_n, the group of all upper triangular matrices U_n, groups like SO_n or O_n consisting of linear morphisms respecting a bilinear form. Even all finite groups fall into this class. But on the other hand this class of groups is still small enough that a very rich structure theory exists.

To study these groups, we combine methods from many different areas in mathematics: Group theory (obviously), Topology (by endowing linear algebraic groups with the Zariski topology), Algebraic geometry (by using the description of the group structure via polynomials) and Analysis/Riemannian geometry (by defining an important invariant, the associated Lie algebra, in terms of derivations and tangent spaces).

The following topics will be discussed: Definition of linear algebraic groups, connected components, actions and representations, Lie algebras, quotients, Jordan decomposition, solvable, nilpotent and unipotent groups, tori, Weyl groups and other topics if time permits (like semi-simple or reductive groups).

Literature

J. E. Humphreys: Linear algebraic groups
A. Borel: Linear algebraic groups
T. A. Springer: Linear algebraic groups

Requirements

Algebra (MA2101) is necessary and some basic knowledge about commutative algebra or algebraic geometry will be helpful.

Exercises

Exercise sheets will be handed out each Wednesday during the lecture and solutions are to be returned during the Wednesday lecture one week later.

• Sheet 1 (correction for E2a: Replace "morphism of algebraic groups" by "morphism of affine varieties")
• Sheet 2
• Sheet 3
• Sheet 4
• Sheet 5
• Sheet 6 (correction for E23: Assume G abelian for the first part as well)
• Sheet 7
• Sheet 8
• Sheet 9
• Sheet 10
• Sheet 11
• Sheet 12
• Sheet 13
• Sheet 14

Solutions

Here are hand-written solutions to the exercises. Even though they are prepared with some care, beware that there may be omitted arguments (in particular some of the calculations) and these notes may contain mistakes! Usually new solutions will be uploaded each Thursday.

• Solutions Sheet 1
• Solutions Sheet 2
• Solutions Sheet 3
• Solutions Sheet 4
• Solutions Sheet 5 (Update: E20 now with matrices in SL_2)
• Solutions Sheet 6 (Update: E23 now corrected)
• Solutions Sheet 7
• Solutions Sheet 8
• Solutions Sheet 9 (Update: E34 now with a second alternative solution)
• Solutions Sheet 10
• Solutions Sheet 11
• Solutions Sheet 12
• Solutions Sheet 13
• Solutions Sheet 14
• Additional notes on an exercise discussed October 13
• Additional notes on an exercise discussed November 17
• Additional exercises (without solutions) discussed during classes October 20 to November 10
• Additional exercises (without solutions) discussed during classes November 24 to December 15
• Additional exercises (without solutions) discussed during classes January 12 to January 26
• Christmas Special (additional exercises to various topics) and its solutions

Exams

Office Hours: