# Linear Algebraic Groups

**Time and venue:** We, 12-13:30, Fr, 12:15-13:45, MI 02.08.011

Tutor group (P. Hamacher): Fr 10-12, MI 02.08.020

**Content:** Linear algebraic groups are subgroups of the group GL_n of invertible n x n-matrices that are defined by polynomial equations. In particular, they are affine algebraic varieties. Examples are the groups of upper triangular or diagonal matrices, or the orthogonal or symplectic group. Linear algebraic groups have a rich structure theory that is one of the main goals of this lecture.

In the lecture we will discuss the following topics:

Definition of linear algebraic groups, connected components, actions and representations, Lie algebras, quotients, Jordan decomposition, solvable, nilpotent and unipotent groups, tori, Weyl groups, roots and root systems, structure of linear algebraic groups.

**Books on the subject:**

Borel: Linear algebraic groups

Humphreys: Linear algebraic groups

Springer: Linear algebraic groups

**Prerequisites:** Algebra 1, 2. Some knowledge of algebraic geometry is helpful, but not required.

**Exercises:**

Exercise sheet 1

Exercise sheet 2

Exercise sheet 3

Exercise sheet 4

Exercise sheet 5

Exercise sheet 6

Exercise sheet 7

Exercise sheet 8

Exercise sheet 10

Exercise sheet 11

Exercise sheet 12

Exercise sheet 13