Intertwining operators among twisted modules associated to not-necessarily-commuting automorphisms

We introduce intertwining operators among twisted modules or twisted intertwining operators associated to not-necessarily-commuting automorphisms of a vertex operator algebra. Let V be a vertex operator algebra and let g1, g2 and g3 be automorphisms of V. We prove that for g1-, g2- and g3-twisted V-modules W1, W2 and W3, respectively, such that the vertex operator map for W3 is injective, if there exists a twisted intertwining operator of type (W3W1W2) such that the images of its component operators span W3, then g3=g1g2. We also construct what we call the skew-symmetry and contragredient isomorphisms between spaces of twisted intertwining operators among twisted modules of suitable types. The proofs of these results involve careful analysis of the analytic extensions corresponding to the actions of the not-necessarily-commuting automorphisms of the vertex operator algebra.