pdf file for the current version (3.00)
This work is a modern exposition of the theory of algebraic
groups (affine group schemes), Lie algebras, Lie groups, and their arithmetic subgroups.
Contents
I: Basic Theory of Algebraic Groups
Definition;
examples;
some basic constructions;
affine groups and Hopf algebras;
affine groups and affine group schemes;
group theory: subgroups and quotient groups;
representations of affine groups;
group theory: the isomorphism theorems;
recovering a group from its representations; Jordan decompositions;
dharacterizations of categories of representations;
finite flat affine groups;
the connected components of an algebraic group;
groups of multiplicative type; tori;
unipotent affine groups;
solvable affine groups;
the structure of algebraic groups;
the spin groups;
the classical semisimple groups;
the exceptional semisimple groups;
tannakian categories.
II: Lie Algebras and Algebraic Groups
The Lie algebra of an algebraic group;
Lie algebras and algebraic groups;
nilpotent and solvable Lie algebras;
unipotent algebraic groups and nilpotent Lie algebras;
semisimple Lie algebras and algebraic groups;
semisimplicity of representations
III: The Structure of Semisimple Lie Algebras and Algebraic Groups in Characteristic Zero
Root systems and their classification;
structure of semisimple Lie algebras and their representations;
structure of semisimple algebraic groups and their representations;
real Lie algebras and real algebraic groups;
reductive groups.
IV: Lie groups
Lie groups; Lie groups and algebraic groups; compact topological groups
V: The Structure of Reductive Groups: the split case
Split reductive groups: the program;
the root datum of a split reductive group;
Borel fixed point theorem and applications;
parabolic subgroups and roots;
root data and their classification;
construction of split reductive groups: the existence theorem;
Construction of isogenies of split reductive groups: the isogeny theorem;
representations of split reductive groups.
VI: The Structure of Reductive Groups: general case
The cohomology of algebraic groups: applications;
Classical groups and algebras with involution;
relative root systems and the anisotropic kernel
VII: Arithmetic groups
History
v0.00 (May 22, 2005; 219 pages). Posted as Algebraic Groups and Arithmetic Groups (AAG).
v1.00 (April 27, 2009; 192 pages). First version of expanded notes (first two chapters only).
v2.00 (April 27, 2010; 378 pages). Posted new version of all six chapters.
v3.00 (April 1, 2011; 422 pages). Major revision (now seven chapters).