The notion of commutative associative algebra in a braided tensor
category is analogous to the classical notion of commutative
associative algebra, but based on the braided tensor category
structure. Under suitable conditions, the module category of a vertex
operator algebra has a natural vertex tensor category structure, as
developed in joint work with Yi-Zhi Huang. This is a subtle
enhancement, requiring complex variables, of braided tensor category
structure. I will sketch joint work with Huang and Alexander Kirillov
Jr., based on this structure, relating commutative associative
algebras in the braided tensor category of modules for a suitable
vertex operator algebra V to vertex operator algebras containing V as
a subalgebra (extensions of V). This talk will be introductory.
