Program
 Linear algebraic groups Student No.： 50 Time： Wed/Fri 13:00-14:50, 2015-03-04~ 2015-06-19(except for public holidays) Instructor： Shenghao Sun  [Tsinghua University] Place： Conference Room 1, Floor 1, Jin Chun Yuan West Building Starting Date： 2015-3-4 Ending Date： 2015-6-19

Description:

Groups are used to describe symmetries of mathematical objects. Studying groups from the representation-theoretic point of view leads to the notion of algebraic groups.

We present the classical theory of linear algebraic groups (mainly due to A. Borel and C. Chevalley), over an algebraically closed field of characteristic zero. Then we study their representation theory, such as the Tannakian formalism, and the relation with Lie algebras. Finally we work out some examples of linear algebraic groups, arising from algebraic geometry, e.g. from Hodge theory and Galois representations.

Prerequisite:

A one-semester course on Algebraic Geometry, or equivalent, is required.

It is preferable that you have taken Lie groups and Lie algebras; we will briefly go over some results when needed from this course without proof.

Reference:

Humphreys, Linear algebraic groups, GTM 21

Brocker and tom Dieck, Representations of compact Lie groups, GTM 98

Borel, Linear algebraic groups, GTM 126

Milne, Algebraic groups, Lie groups, and their arithmetic subgroups, available on his webpage

Chevalley et al., Classification des groups algebriques semi-simples, Coll. Works, Vol. 3, Springer

For Lie algebras:

Bourbaki, Groupes et algebres de Lie, Ch. 1,6,7,8, Springer

Serre, Algebres de Lie semi-simple complexes

Serre, Lie algebras and Lie groups (1964 Harvard lecture notes), Springer

Seminaire "Sophus LIE", 1954-56, available on numdam.org