Cornell Math - MATH 767, Spring 2000

MATH 767 — Spring 2000
Linear Algebraic Groups

Linear groups are subgroups of GL_n(F) for some field F and some n. Linear algebraic groups are those subgroups that are defined by polynomial equations in the matrix entries and hence are endowed with the structure of algebraic variety. A typical example is the special linear group SL_n(F), which is defined by the single polynomial equation det(A) = 1. Other examples are orthogonal and symplectic groups.

The course will give an introduction to the theory of linear algebraic groups, presupposing only a standard first course in algebra at the graduate level (groups, algebras, tensor products...). Important notions to be discussed will be unipotent and semisimple elements, Jordan-Chevalley decomposition, tori, connected solvable groups, the theorems of Lie and Kolchin, parabolic subgroups, Lie algebras of linear algebraic groups, reductive groups and their root systems.

The goal of the course will be a survey of the structure theory of reductive groups over algebraically closed fields. (All examples mentioned in the first paragraph are reductive.) In particular, we will see that these groups always possess a BN-pair and hence give rise to a building. This makes it possible to interpret some of the algebraic notions geometrically. There will be some overlap here with the course on buildings given by Ken Brown in Fall 1999 (which is not a prerequisite).