活动地点：

活动类型：学术报告

主讲人：Prof. J.S. Milne

活动时间：

活动内容：

 

(Totally 40 lectures.)

Algebraic groups are groups of matrices determined by polynomial conditions. For example, the group of matrices of determinant 1 and the orthogonal group of a symmetric bilinear form are both algebraic groups. The elucidation of the structure of algebraic groups and the classification of them were among the great achievements of twentieth century mathematics(Borel, Chevalley and others). Algebraic groups are used in most branches of mathematics, and since the famous work of Hermann Weyl in the 1920s they have also played a vital role in quantum mechanics and other branches of physics (usually as Lie groups).

Arithmetic groups are groups of matrices with integer entries. They are an important source of discrete groups acting on manifolds, and recently they have appeared as the symmetry groups of several string theories in physics.

The course will provide an introductory overview of the subject. I plan to cover the following topics:
-- The definition and basic theory of algebraic groups.
-- Algebraic groups and Lie algebras.
-- The classification of semisimple Lie algebras.
-- The classification of semisimple algebraic groups.
-- Algebraic groups and Lie groups.
-- Algebraic groups and algebras with involution.
-- Arithmetic subgroups of algebraic groups.

BOOKS
I will provide notes as the course proceeds, so no text book will be needed. Nevertheless, the following references will be useful (they are more advanced than the course, which may serve as an introduction to them).

T.A. Springer, Linear Algebraic Groups, 2nd edition, Birkhauser, 1998. James E. Humphreys, Introduction to Lie Algebras and Representation  Theory, Springer, 1972. A. Borel, Introduction aux groupes arithmetiques, Hermann, 1969.

PREREQUISITES
-- A standard course on algebra, for example, a good knowledge of the book Algebra by M. Artin.
-- Some knowledge of the language of algebraic geometry, for example, the first few sections and section 9 of my notes Algebraic Geometry
(available on my website www.jmilne.org/math/.)