For the part on infinite-dimensional Lie algebras, I will discuss mostly infinite-dimensional Heisenberg algebras, affine Lie algebras the Virasoro algebras and their representations. Vertex operator algebras and their representations will be introduced as natural structures on representations of these infinite-dimensional Lie algebras. Various formulations of Vertex operator algebras, modules and intertwining operators, including the component formulation, the formal variable formulation, the complex variable formulation, the conformal geometric formulation and the D-module formulation, will be discussed.

Prerequisites: I will assume that the students have some basic knowledge in algebra and complex variables, as covered in the first-year graduate courses.

References:

Books

• E. Frenkel and D. Ben-Zvi, Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, Vol. 88, Amer. Math. Soc., Providence, 2001.

• I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Appl. Math., Vol. 134, Academic Press, Boston, 1988.

• Y.-Z. Huang, Two-dimensional Conformal Field Theory and Vertex Operator Algebras, Progress in Math., Vol. 148, Birkhauser, Boston, 1997.

• J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Math., Vol. 227, Birkhauser, Boston, 2003.

• R. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA 83 (1986), 3068--3071.

• I. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Memoirs Amer. Math. Soc. 104 (1993).

• Y.-Z. Huang and J. Lepowsky, On the D-module and formal variable approaches to vertex algebras, in: Topics in Geometry: In Memory of Joseph D'Atri, ed. S. Gindikin, Progress in Nonlinear Differential Equations, Vol. 20, Birkhuser, Boston, 1996, 175--202.