On axiomatic approaches to intertwining operator algebras

Abstract

We study intertwining operator algebras introduced and constructed by Huang. In the case that the intertwining operator algebras involve intertwining operators among irreducible modules for their vertex operator subalgebras, a number of results on intertwining operator algebras were given in [Y.-Z. Huang, Generalized rationality and a “Jacobi identity” for intertwining operator algebras, Selecta Math. (N.S.) 6 (2000) 225–267] but some of the proofs were postponed to an unpublished monograph. In this paper, we give the proofs of these results in [Y.-Z. Huang, Generalized rationality and a “Jacobi identity” for intertwining operator algebras, Selecta Math. (N.S.) 6 (2000) 225–267] and we formulate and prove results for general intertwining operator algebras without assuming that the modules involved are irreducible. In particular, we construct fusing and braiding isomorphisms for general intertwining operator algebras and prove that they satisfy the genus-zero Moore–Seiberg equations. We show that the Jacobi identity for intertwining operator algebras is equivalent to generalized rationality, commutativity and associativity properties of intertwining operator algebras. We introduce the locality for intertwining operator algebras and show that the Jacobi identity is equivalent to the locality, assuming that other axioms hold. Moreover, we establish that any two of the three properties, associativity, commutativity and skew-symmetry, imply the other (except that when deriving skew-symmetry from associativity and commutativity, more conditions are needed). Finally, we show that three definitions of intertwining operator algebras are equivalent.