# 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 261B: 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.