# 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)

**Prerequisites:**

MAT 215A; MAT 240A; MAT 250A; MAT 250B; Or the equivalent, or consent of 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): H
^{1 }and H^{2 }. - 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.