Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 261A: Lie Groups
Approved: 2009-07-01, Dmitry Fuchs

ATTENTION:

Effective 09-10, 261B will be taught irregularly. 261A will continue to be taught every other Winter.

Suggested Textbook: (actual textbook varies by instructor; check your instructor)

Course Description:

Combined syllabus.

Suggested Schedule:

I. Lie groups, Lie algebras, and basic relations between them.

  • Definition of a Lie groups. Examples (classical groups.)
  • Products, coverings, Lie subgroups, quotients (homogeneous spaces). Examples.
  • One-parameter subgroups, exponential map TeG → G (the case of matrices).
  • Commutator operation in TeG. Formal definition of a Lie algebra. Construction G → Lie G. Review of examples.
  • Homomorphisms ϕ: G1 → G2 and dϕ: Lie G1 → Lie G2 – all relations between them. Representation of Lie groups and Lie algebras. (The goal here is to reduce the theory of Lie groups to the theory of Lie algebras; the latter will be handled in Part II.)
  • Relation between Lie subgroups of G and Lie subalgebras of Lie G. Here (or before) a theorem is proved: every closed subgroup of a Lie group is a Lie subgroup.
  • From Lie G to G: Campbell-Hausdorff, local Lie groups, Ado theorem (without a proof, at least here).

II. Theory of Lie algebras.

  • Universal enveloping algebras and the PBW theorem (this may be postponed or/and dissolved in the future lectures.)
  • Nilpotent Lie algebras and nil-representations. Engel’s Theorem. Solvable Lie groups. Lie’s Theorem. Solvable and nilpotent Lie groups.
  • Radical and nil-radical. Semisimple Lie algebras ( = radical is 0).
  • The Killing form. Cartan criteria for solvability and semisimplicity. (Technically, this is the main result which requires a rather long proof. Surprisingly, the best proof is given in the Bourbaki.) Semisimplicity/simplicity. Reductive Lie algebras.
  • General theory of semisimple Lie algebras. Casimir element. Some cohomology (necessary for Representation Theory): H1 and H2 .
  • Representations of semisimple Lie algebras. Weyl’s Theorem (they are semisimple).
  • Representation theory for sl(2, C).

III. Structure theory.

  • Cartan subalgebras. Roots and root spaces. Cartan matrices and Dynkin diagrams. The Weyl group. A survey of the classical Lie algebras. The classification of simple complex Lie algebras.
  • Real Lie groups. Compactness and maximal compact subgroup (via the Killing form). Topology of a real Lie group. Complexification, compact form. Maximal tori.

IV Representation theory.

  • Representations: weights, highest weights. Classification of irreducible representations of complex semisiple Lie algebras/ Lie groups.
  • Characters. Character formulas.

V. Kac-Moody and Virasoro Lie algebras.

  • Definition of a Kac-Moody Lie algebra (generality: the Cartan matrix is integral, symmetrizable, diagonal entries are all 2, non-diagonal entries are non-positive. Special cases: finite-dimensional (semi-)simple Lie algebras; affine Lie algebras. Highest weight representations of Kac-Moody Lie algebras. Verma modules. Category

O. BGG resolutions. Kac-Kazhdan theorem (characters of irreducible highest weight representations). Virasoro algebra and its representations. Verma and Fock modules.

Additional Notes:

Of existing textbooks, I prefer V.S.Varadarajan’s ”Lie groups, Lie Algebras, and Their Representations”. (This is a well written book following, mainly, the classical works of E.Cartan.) As an additional source I used ”Lie groups and Algebraic Groups” by E.B.Vinberg and A.L.Onishchik, ”Infinite-dimensional Lie Algebras” by V.G.Kac and some survey articles.