Mathematics 261AB Fall 2011/Spring 2012

Professor: Richard Borcherds

Office hours: Tu Th 2:00-3:30 927 Evans Hall

This class meets in 255 DWINELLE from 3:30-5:00 Tu Th. This is the course home page (address The course control number is 54548.

Catalogue Description: Mathematics 261AB

Three hours of lecture per week. Prerequisites: 214. Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall.

Textbook and course notes:

The course will be using notes, posted online below. Those who want a textbook can use Representation theory: a first course by Fulton and Harris. ISBN 978-0387974958

Background reading

The older references are the foundational papers on Lie groups, and are sometimes hard to read.
  1. A. Borel Essays in the history of Lie groups and algebraic groups ISBN 978-0-8218-0288-5 Covers the history.
  2. Elie Cartan Sur la structure des groupes de transformations finis et continus Cartan's famous 1894 thesis, cleaning up Killing's work on the classification Lie algebras.
  3. T. Hawkins Emergence of the theory of Lie groups ISBN 978-0-387-98963-1 Covers the early history of the work by Lie, Killing, Cartan and Weyl, from 1868 to 1926.
  4. N. Jacobson, Lie algebras ISBN 978-0486638324 A good reference for all proofs about finite dimensional Lie algebras
  5. Wilhelm Killing, "Die Zusammensetzung der stetigen endlichen Transformations-gruppen" 1888-1890 part 1 part 2 part 3 part 4 Killing's classification of simple Lie complex Lie algebras.
  6. S. Lie, F. Engel "Theorie der transformationsgruppen" 1888 Volume 1 Volume 2 Volume 3 Lie's monumental summary of his work on Lie groups and algebras.
  7. Claudio Procesi, Lie Groups: An Approach through Invariants and Representations, ISBN 978-0387260402. Similar to the course, with more emphasis on invariant theory.
  8. J.-P. Serre, Lie algebras and Lie groups ISBN 978-3540550082 Covers most of the basic theory of Lie algebras.
  9. J.-P. Serre, Complex semisimple Lie algebras ISBN 978-3-540-67827-4 Covers the classification and representation theory of complex Lie algebras.
  10. Hermann Weyl, Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. 1925-1926 I, II, III. Weyl's paper on the representations of compact Lie groups, giving the Weyl character formula.
  11. H. Weyl The classical groups ISBN 978-0-691-05756-9 A classic, describing the representation theory of lie groups and its relation to invariant theory

Examinations: None

Lectures and homework:

Lectures 1-56 in one pdf file
  1. Lecture 1 Overview
  2. Lecture 2 Overview
  3. Lecture 3 Overview
  4. Lecture 4 Lie algebras
  5. Lecture 5 The Poincare-Birkhoff-Witt theorem
  6. Lecture 6 Proof of the Poincare-Birkhoff-Witt theorem, and Hopf algebras
  7. Lecture 7 Hopf algebras and the Steenrod algebra
  8. Lecture 8 The exponential map
  9. Lecture 9 The Baker-Campbell-Hausdorff formula
  10. Lecture 10 Nilpotent groups and Lie algebras; Engel's theorem
  11. Lecture 11 Lie's theorem about solvable Lie algebras
  12. Lecture 12 Picard-Vessiot theory
  13. Lecture 13 Classification of Lie groups in dimensions at most 3
  14. Lecture 14 The Killing form and Cartan's criterion
  15. Lecture 15 Cartan subalgebras
  16. Lecture 16 General linear groups: Iwasawa decomposition
  17. Lecture 17 General linear groups: root spaces
  18. Lecture 18 General linear groups: Symmetric spaces
  19. Lecture 19 Orthogonal groups: small dimensional examples and symmetric spaces
  20. Lecture 20 Orthogonal groups: root systems
  21. Lecture 21 Clifford algebras
  22. Lecture 22 Structure of Clifford algebras
  23. Lecture 23 Clifford groups and spin groups
  24. Lecture 24 Spin representations
  25. Lecture 25 Symplextic groups
  26. Lecture 26 Pfaffians and dominoes
  27. Lecture 27 G2 and the octionions
  28. Lecture 28 Dynkin diagrams and root systems
  29. Lecture 29 Quivers and tilting
  30. Lecture 30 Representation theory
  31. Lecture 31 Representations of finite groups
  32. Lecture 32 Examples fo character tables
  33. Lecture 33 Characters of SU(2)
  34. Lecture 34 Schur indicator
  35. Lecture 35 Representations of the Lie algebra sl2: Casimir and complete reducibility
  36. Lecture 36 Irreducible finite dimensinoal representations of sl2 and Verma modules
  37. Lecture 37 Infinite dimensional representations of SL2(R)
  38. Lecture 38 Serre relations
  39. Lecture 39 Serre relations (continued)
  40. Lecture 40 The Weyl groups of exceptional groups
  41. Lecture 41 Invariants of Weyl groups
  42. Lecture 42 Invariants of Lie groups and Hilbert's finiteness theorem
  43. Lecture 43 Examples of representations: minuscule and SL3
  44. Lecture 44 The Schur index for compact groups
  45. Lecture 45 Weyl character formula
  46. Lecture 46 Examples of character calculations for rank 2 groups
  47. Lecture 47 The Freudenthal formula and calculations with E8
  48. Lecture 48 The Jacobi triple product identity
  49. Lecture 49 Symmetric functions
  50. Lecture 50 Representations of the symmetric group
  51. Lecture 51 Schur-Weyl duality
  52. Lecture 52 Littlewood-Richardson rule
  53. Lecture 53 Construction of a Lie algebra from a root lattice
  54. Lecture 54 Construction for types B, C, F, G
  55. Lecture 55 Construction of (exceptional) simple real Lie algebras
  56. Lecture 56 Examples of discrete subgroups of Lie groups

Links related to the course: