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.