Lectures on Lie Groups

These notes are based on lectures given in Math 273 at UNC in the Spring of 2002. The first seventeen sections deal with the general theory of Lie groups. We discuss integration on a Lie group, the Lie algebra, and general results on representations. We present some classical results on compact Lie groups, such as the Peter-Weyl theorem, on the completeness of the matrix entries of irreducible unitary representations of a compact Lie group G in L2(G).

Sections 18-34 concentrate on the unitary groups U(n). Topics discussed include the classification of irreducible unitary representations of U(n), involving the notion of roots and weights, and some of their properties. We also treat the decomposition of the k-fold tensor product of Cn into irreducible spaces for U(n), and the duality of the symmetric group Sk that arises here, and also classical character formulas and some of their implications for harmonic analysis on U(n).

Sections 35-38 extend some of the general results of Sections 19-21 to the setting of general compact Lie groups, particularly discussing roots of their Lie algebras and weights of their representations. Sections 39-44 provide a more detailed extension of such results from the setting of U(n) to the setting of the orthogonal groups SO(n) and cerain two-fold covers, denoted Spin(n), which are introduced in Section 42, via use of Clifford algebras, which are introduced in Section 41.

These notes end with several appendices, presenting some background material and also some material complementary to that in the main body of the notes.

These notes will serve to prepare the reader for more advanced texts, such as Noncommutative Harmonic Analysis.

Contents

  1. Definition and basic examples
  2. The matrix exponential and other functions of matrices
  3. Quaternions and the group Sp(n)
  4. Integration on a Lie group
  5. Representations of a group
  6. Weyl orthogonality
  7. The Peter-Weyl theorem
  8. Characters and central functions
  9. Comments on representations of finite groups
  10. The convolution product and group algebras
  11. Approximate identities and the Peter-Weyl theorem in general
  12. Lie algebras
  13. Lie algebra representations
  14. The adjoint representation
  15. The Campbell-Hausdorff formula
  16. More Lie group - Lie algebra connections
  17. Enveloping algebras
  18. Representations of SU(2) and related groups
  19. Representations of U(n), I: roots and weights
  20. Representations of U(n), II: some basic examples
  21. Representations of U(n), III: identification of highest weights
  22. Connections between representations of U(n), SU(n), and Gl(n,C)
  23. Decomposition of Sk tensor bar(S)l
  24. Commutants, double commutants, and dual pairs
  25. The first fundamental theorem of invariant theory
  26. Decomposition of Tk(Cn)
  27. The Weyl integration formula
  28. The character formula
  29. Examples of characters
  30. Duality and the Frobenius character formula
  31. Integral of |Tr gk|2 and variants
  32. The Laplace operator on U(n)
  33. The heat equation on U(n)
  34. The Harish-Chandra/Itzykson-Zuber integral
  35. Roots and weights for general compact Lie groups
  36. Roots and weights for compact G, II: injections of su(2) into g
  37. The Weyl group
  38. A generating function
  39. Representations of SO(n), n<6
  40. Representations of SO(n), general n
  41. Clifford algebras
  42. The groups Spin(n)
  43. Spinor representations
  44. Weight spaces for the spinor representations

Appendices

  1. Flows and vector fields
  2. Lie brackets
  3. Frobenius' theorem
  4. Variation of flows
  5. Lie algebras of matrix groups
  6. The Poincare-Birkhoff-Witt theorem
  7. Analytic continuation from U(n) to Gl(n,C), another approach
  8. The complexification of a general compact Lie group
  9. Exterior algeba
  10. Simplicity of M(n,F)
  11. Two-step nilpotent Lie algebras