On replacement axioms for the Jacobi identity for vertex algebras and their modules

TL;DR

This paper explores alternative axioms for vertex algebras and their modules, introducing a new property called weak skew-associativity and showing its equivalence to the Jacobi identity under certain conditions.

Contribution

It introduces weak skew-associativity as a new axiom and proves its equivalence to the Jacobi identity, expanding the foundational understanding of vertex algebra axioms.

Findings

Weak skew-associativity completes the $ ext{S}_3$-symmetry of axioms.

Jacobi identity is equivalent to weak associativity or weak skew-associativity under certain conditions.

Results generalize vertex algebra axioms without requiring a vacuum vector.

Abstract

We discuss the axioms for vertex algebras and their modules, using formal calculus. Following certain standard treatments, we take the Jacobi identity as our main axiom and we recall weak commutativity and weak associativity. We derive a third, companion property that we call "weak skew-associativity." This third property in some sense completes an $\mathcal{S}_{3}$-symmetry of the axioms, which is related to the known $\mathcal{S}_{3}$-symmetry of the Jacobi identity. We do not initially require a vacuum vector, which is analogous to not requiring an identity element in ring theory. In this more general setting, one still has a property, occasionally used in standard treatments, which is closely related to skew-symmetry, which we call "vacuum-free skew-symmetry." We show how certain combinations of these properties are equivalent to the Jacobi identity for both vacuum-free vertex algebras and their modules. We then specialize to the case with a vacuum vector and obtain further replacement axioms. In particular, in the final section we derive our main result, which says that, in the presence of certain minor axioms, the Jacobi identity for a module is equivalent to either weak associativity or weak skew-associativity. The first part of this result has previously appeared and been used to show the (nontrivial) equivalence of representations of and modules for a vertex algebra. Many but not all of our results appear in standard treatments; some of our arguments are different from the usual ones.