## Abstract

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 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} is injective, if there exists a twisted intertwining operator of type (W_{3}W_{1}W_{2}) such that the images of its component operators span W_{3}, then g_{3}=g_{1}g_{2}. 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.

Original language | English (US) |
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Pages (from-to) | 346-380 |

Number of pages | 35 |

Journal | Journal of Algebra |

Volume | 493 |

DOIs | |

State | Published - Jan 1 2018 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

## Keywords

- Orbifold conformal field theory
- Twisted intertwining operator
- Vertex operator algebra