Introduction to vertex operator algebras II

This is the second of three lectures on introduction to vertex operator algebras. In this lecture, we shall continue Professor Dong’s lecture to present more fundamental properties of vertex operator algebras. From the mathematical point of view, a vertex operator algebra formally resembles a Lie algebra because the Jacobi identity is used as one of the main axioms. For the Lie algebra aspect of vertex operator algebras, the notion of contragredient module [FHL] and the notion of tensor product ([HL1-4], [Li4]) have been developed. On the other hand, from the physical point of view, a vertex operator algebra looks like a commutative associative algebra with identity because roughly speaking, a vertex operator algebra is a sort of quantization of the commutative associative algebra of observables in conformal field theory [BPZ]. For the associative algebra aspect of vertex operator algebras, it has been proved [FHL] that the tensor product of any finitely many vertex operator algebras has a natural vertex operator algebra structure. In concrete examples, for a fixed level l, one of the generalized Verma modules, called the vacuum representation for any affine Lie algebra g̃, has a natural vertex operator algebra structure ([FZ], [Li2], [Lia]) and the universal enveloping algebra [FZ] of the vertex operator algebra is a certain completion of the universal enveloping algebra of g̃. This fact together with some facts on tensor products ([HL1-4], [Li4], [KL0-2],...) strongly indicates that a vertex operator algebra is analogous to a quasi-Hopf algebra or a quantum group.