Suggested Topics for Presentations
Notes: You are free to choose any topic related to infinite-dimensional Lie algebras as long as it was not covered in full in the lectures. You may combine or split any of the following suggested topics. Please discuss your choices with the instructor so that we avoid duplication. With one exception, all books cited were available at first posting. If a book is no longer available most likely one of your classmates has it. Please share them as some books are used for multiple topics.
· Lie-Cartan algebras of vector fields and their representations on differential forms. Fuks86, Ch. 1, Sec. 1, 2.
· Cohomology of Lie algebras; derivations, central extensions, and extensions of modules. Fuks86, Ch. 1, Sec. 3, 4.
· Loop algebras and loop groups. Guest97, Sec. 11.
· Kac-Moody algebras; Cartan matrix, Serre relations, Weyl group. Kac90, Ch. 1, 3.
· Weyl-Kac character formula and applications to combinatorial identities. Kac90, Ch. 10, Exercises in Ch. 12.
· Weyl-Kac character formula for sl(2)^ and theta functions. Kac-Raina87, Ch. 11.
· Construction and unitarity of the discrete series representations of the Virasoro algebra. Kac-Raina87, Ch. 12, Sec. 3.
· Lie conformal algebras. Kac98, Sec. 2.1-2.7; Kac97.
· Vertex algebras, modular forms, and the Monster finite group. Introductions of Gannon06 and Frenkel-Lepowsky-Meurman88.
· Modular forms and functions. Gannon06, Ch. 2.
· Lattices and theta functions; the E8 and Leech lattices.Conway-Sloane99, Ch. 4.
· Wakimoto representations of sl(2)^. Wakimoto86; Frenkel02.
· Super boson-fermion correspondence and sums of squares. Kac98, Sec. 5.8.
· Hamiltonian equations and Lie algebras. Guest97, Sec. 2-7.
· Soliton equations. Miwa-Jimbo-Date00; Filippov00; Hirota04; etc.
· Vertex operator realizations of affine Kac-Moody algebras. Kac90, Ch. 14; Kac98, Sec. 5.6.
· Frenkel-Kac construction of sl(2)^ and the nonlinear Schrödinger equation. Kac90, Ch. 14.
· The Lie algebra W1+infinity and its representations. Kac98, Sec. 5.3; Kac-Radul93.
· Highest weight modules for the W1+infinity algebra and the bispectral problem. Bakalov-Horozov-Yakimov96.