Affine Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups - J.S. Milne   Top
Course Notes
Group Theory
Fields and Galois Theory
Algebraic Geometry
Algebraic Number Theory
Modular Functions and Modular Forms
Elliptic Curves -- see books.
Abelian Varieties
Lectures on Etale Cohomology
Class Field Theory
Complex Multiplication
Basic Theory of Affine Group Schemes
Lie Algebras, Algebraic Groups, and Lie Groups
Reductive Groups
iAG Algebraic Groups (algebraic group schemes over fields) v2.00, 20 December 2015, 528 pages. pdf
AGS Basic Theory of Affine Group Schemes v1.00; 11 March 2012; 275 pages pdf
LAG Lie Algebras, Algebraic Groups, and Lie Groups v2.00; 5 May 2013; 186 pages pdf
RG Reductive Groups v1.00; 11 March 2012; 77 pages pdf

The goal of this project is to make it possible for everyone to learn the essential theory of algebraic group schemes (especially reductive groups), Lie algebras, Lie groups, and arithmetic subgroups with the minimum of prerequisites and the minimum of effort. In particular, it should not be necessary to learn the subject twice, once using 1950s style algebraic geometry, and then again using modern algebraic geometry. Nor should it be necessary to read EGA first.

Of course, there is considerable overlap between the notes. My hope is to publish iAG soon and LAG eventually.

iAG: Algebraic Groups: An introduction to the theory of algebraic group schemes over fields

These notes represent my attempt to write a modern successor to the three standard works, all titled "Linear Algebraic Groups," by Borel, Humphreys, and Springer. More specifically, it is a comprehensive introduction to the theory of algebraic group schemes over fields, based on modern algebraic geometry, but with minimal prerequisites.
v1.00 (September 20, 2014). First version on the web (331 pages).
v1.20 (January 29, 2015). Revised Parts A,B (373 pages).
v2.00 (December 20, 2015). Completely rewritten and completed (528 pages).
These notes have been rewritten for publication. I don't plan to update the notes on this page. See Books.

AGS: Basic Theory of Affine Group Schemes

This is a modern exposition of the basic theory of affine group schemes. Although the emphasis is on affine group schemes of finite type over a field, we also discuss more general objects, for example, affine group schemes not of finite type and base rings not fields. "Basic" means that we do not investigate the detailed structure of reductive groups using root data except in the final survey chapter. Prerequisites have been held to a minimum: all the reader really needs is a knowledge of some basic commutative algebra and a little of the language of algebraic geometry. The first 17 chapters are fairly complete, but require revision; the final chapter has yet to be written.

LAG: Lie Algebras, Algebraic Groups, and Lie Groups

These notes are an introduction to Lie algebras, algebraic groups, and Lie groups in characteristic zero, emphasizing the relationships between these objects visible in their categories of representations. Eventually these notes will consist of three chapters, each about 100 pages long, and a short appendix. The first chapter (Lie algebras) is mostly complete, the second (algebraic groups) treats only semisimple groups in detail, the third (Lie groups) has yet to be written, and the appendix (a survey of arithmetic subgroups) is complete. The notes will be largely independent of AGS.

v2.00 (May 5, 2013; 186 pages). Major revision.

RG: Reductive Groups

Eventually it is intended that these notes will provide a detailed exposition of the theory of reductive algebraic groups (in about 300 pages). At present only the first chapter on split reductive groups over arbitrary fields exists. The notes will refer to AGS for proofs, but otherwise will be largely independent.


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). Posted as Algebraic Groups, Lie Groups, and their Arithmetic Subgroups (ALA).
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).
v1.00 (March 11, 2012; 494 pages). Major revision; split into three (AGS, LAG, and RG).
September 20, 2014. Added iAG.

Algebraic Groups, Lie Groups, and their Arithmetic Subgroups

This work has been replaced by the above three, and will not be revised or corrected.
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.


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; characterizations 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