Algebraic groups and arithmetic groups

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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.


  1. Overview and examples
  2. Definition of an affine algebraic group
  3. Linear representations
  4. Matrix groups
  5. Example: the spin group
  6. Group theory
  7. Finite (etale) algebraic groups
  8. The connected components of an algebraic group
  9. Diagonalizable groups; tori
  10. Jordan decompositions
  11. Solvable algebraic groups
  12. The Lie algebra of an algebraic group: basics
  13. The Lie algebra of an algebraic group (continued)
  14. Semisimple algebraic groups and Lie algebras
  15. Reductive algebraic groups
  16. Split reductive groups: the program
  17. The root datum of a split reductive group
  18. Generalities on root data
  19. Classification of semisimple root data
  20. The construction of all split reductive groups
  21. Borel fixed point theorem and applications
  22. Parabolic subgroups and roots
  23. Representations of split reductive groups
  24. Tannaka duality
  25. Algebraic groups over R and C; relation to Lie groups
  26. The cohomology of algebraic groups; applications
  27. Classical groups and algebras with involution
  28. Arithmetic subgroups