# Algebraic groups and arithmetic groups

### These are old notes that are not being updated. For new versions, see ala

pdf (for printing on a4 paper 29.7x21 cm)
These notes provide an introductory overview of the theory
of algebraic groups, Lie algebras, Lie groups, and arithmetic groups.

v0.0 Posted during the lectures Feb 28, 2005 --- May 7, 2005.

v1.0, May 22, 2005. Minor corrections and revisions; added table of contents and index of definitions.

v1.01, June 4, 2006. Fixed problem with diagrams, which caused small changes in pagination.

### Contents

- Overview and examples
- Definition of an affine algebraic group
- Linear representations
- Matrix groups
- Example: the spin group
- Group theory
- Finite (etale) algebraic groups
- The connected components of an algebraic group
- Diagonalizable groups; tori
- Jordan decompositions
- Solvable algebraic groups
- The Lie algebra of an algebraic group: basics
- The Lie algebra of an algebraic group (continued)
- Semisimple algebraic groups and Lie algebras
- Reductive algebraic groups
- Split reductive groups: the program
- The root datum of a split reductive group
- Generalities on root data
- Classification of semisimple root data
- The construction of all split reductive groups
- Borel fixed point theorem and applications
- Parabolic subgroups and roots
- Representations of split reductive groups
- Tannaka duality
- Algebraic groups over R and C; relation to Lie groups
- The cohomology of algebraic groups; applications
- Classical groups and algebras with involution
- Arithmetic subgroups