On the concepts of intertwining operator and tensor product module in vertex operator algebra theory

Details On the concepts of intertwining operator and tensor product module in vertex operator algebra theory On the concepts of intertwining operator and tensor product module in vertex operator algebra theory

2004-09-20

arxiv.org/abs/math/0409364v1

Mathematics

Quantum Algebra

30 pages

Scientific paper

We produce counterexamples to show that in the definition of the notion of intertwining operator for modules for a vertex operator algebra, the commutator formula cannot in general be used as a replacement axiom for the Jacobi identity. We further give a sufficient condition for the commutator formula to imply the Jacobi identity in this definition. Using these results we illuminate the crucial role of the condition called the ``compatibility condition'' in the construction of the tensor product module in vertex operator algebra theory, as carried out in work of Huang and Lepowsky. In particular, we prove by means of suitable counterexamples that the compatibility condition was indeed needed in this theory.

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