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We introduce intertwining operators among twisted modules or twistedintertwining operators associated to not-necessarily-commuting automorphisms ofa vertex operator algebra. Let $V$ be a vertex operator algebra and let$g_{1}$, $g_{2}$ and $g_{3}$ be automorphisms of $V$. We prove that for$g_{1}$-, $g_{2}$- and $g_{3}$-twisted $V$-modules $W_{1}$, $W_{2}$ and$W_{3}$, respectively, such that the vertex operator map for $W_{3}$ isinjective, if there exists a twisted intertwining operator of type${W_{3}\choose W_{1}W_{2}}$ such that the images of its component operatorsspan $W_{3}$, then $g_{3}=g_{1}g_{2}$. We also construct what we call theskew-symmetry and contragredient isomorphisms between spaces of twistedintertwining operators among twisted modules of suitable types. The proofs ofthese results involve careful analysis of the analytic extensions correspondingto the actions of the not-necessarily-commuting automorphisms of the vertexoperator algebra.

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