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.
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.