# Generalized vertex algebras and duality

## Abstract

Now we return to the setting of Chapter 5. The axioms for a generalized vertex operator algebra do not quite apply to the structure of Ω_{*}, the vacuum space (3.29) (for the Heisenberg algebra (**ĥ**_{ * })_{ z }) equipped with the relative vertex operators *Y*_{*}. In fact, the finite-dimensionality axiom (6.4) and the boundedness axiom (6.5) fail to hold in general, since *L* is not necessarily positive definite. Moreover, the Jacobi identity (5.11) (Theorem 5.1) differs from the Jacobi identity (6.12) by the insertion of the factor *c*(*ā*, *b*). Of course, a suitable group *G* would also have to be defined. We would like to extend our axioms so as to include this structure. Then the corresponding generalization of the duality arguments of Chapter 7 would provide an alternate duality-based proof of the Jacobi identity (5.11) (recall Remark 7.18), and in particular, of Theorems 8.6.1 and 8.8.23 of [FLM3]. (The arguments in Chapter 7 above, in the Appendix of [FLM3] and in [FHL] are not sufficiently general to give a duality-based proof of Theorem 8.8.23 of [FLM3] — the special case of (5.11) in which **h**_{ * } = 0 and 〈*ā*, *b*〉 ? ∈ **Z** — because the rational lattice in that theorem is not assumed even or integral, and has an alternating form *c*(·,·); in the proof of Proposition 7.16, the vector *v3* is arbitrary.)

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