关于顶点代数及其模的雅可比恒等式的替换公理
On replacement axioms for the Jacobi identity for vertex algebras and their modules
摘要：
我们用形式演算讨论顶点代数及其模的公理。按照一定的标准处理，我们把雅可比恒等式作为我们的主要公理，我们回忆弱交换性和弱结合性。我们推导出第三个伴随性质，我们称之为“弱斜结合性”。这第三个性质在某种意义上完成了公理的$\mathcal{S}_{3}$ -对称性，它与已知的雅可比恒等式的$\mathcal{S}_{3}$ -对称性有关。我们一开始不需要真空向量，这类似于在环论中不需要单位元。在这种更一般的情况下，仍然有一个性质，偶尔在标准处理中使用，它与斜对称密切相关，我们称之为“无真空斜对称”。“我们展示了这些性质的某些组合如何等价于真空自由顶点代数及其模的雅可比恒等式。然后，我们专门的情况下，真空向量，并获得进一步的替换公理。特别地，在最后一节中，我们得到了我们的主要结果，它说，在某些小公理的存在下，一个模的雅可比恒等式等价于弱结合性或弱斜结合性。这一结果的第一部分以前已经出现，并被用来显示（非平凡的）表示的等价和模块的顶点代数。我们的许多（但不是所有）结果都出现在标准处理中;我们的一些论点与通常的论点不同。
英文摘要：
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.
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