77 pages, based in part on the lecture given by the first author at the Harvard CMSA Math-Science Literature Lecture Series on May 22, 2020. Minor corrections and references updated
本文首先简要回顾了顶点算子代数（VOA）理论创立之前及之后不久的时期。这一回顾旨在强调促成VOA理论诞生的一些重要发展。随后，本文描述了VOA领域在过去15年中基于VOA与交换代数之间富有成效的类比和联系所取得的进展。首先，存在几个从VOA到交换代数的函子，允许使用交换代数的方法来解决VOA问题。为了说明这一点，我们提出了一种使用经典不变量理论方法来描述轨道和余集的方法。这对于近期解决物理学界当前关注的Gaiotto和Rapčák猜想至关重要。我们还将该学科中的一些旧猜想用交换代数重新表述，并给出了这些猜想的推广。此外，我们还概述了拓扑VOA（TVOA）理论及其在BRST上同调理论和共形弦理论中的应用，这些基于90年代的工作。我们构造了一个从TVOA到Batalin-Vilkovisky代数（配备了由二阶微分算子实现的某种奇泊松结构的超交换代数）的函子，并给出了许多有趣的应用。本文部分基于第一作者于2020年5月22日在Harvard CMSA Math-Science Literature Lecture Series上发表的演讲。
英文摘要：
This paper begins with a brief survey of the period prior to and soon after the creation of the theory of vertex operator algebras (VOAs). This survey is intended to highlight some of the important developments leading to the creation of VOA theory. The paper then proceeds to describe progress made in the field of VOAs in the last 15 years which is based on fruitful analogies and connections between VOAs and commutative algebras. First, there are several functors from VOAs to commutative algebras that allow methods from commutative algebra to be used to solve VOA problems. To illustrate this, we present a method for describing orbifolds and cosets using methods of classical invariant theory. This was essential in the recent solution of a conjecture of Gaiotto and Rapčák that is of current interest in physics. We also recast some old conjectures in the subject in terms of commutative algebra and give some generalizations of these conjectures. We also give an overview of the theory of topological VOAs (TVOAs), with applications to BRST cohomology theory and conformal string theory, based on work in the 90's. We construct a functor from TVOAs to Batalin-Vilkovisky algebras -- supercommutative algebras equipped with a certain odd Poisson structure realized by a second order differential operator -- and present a number of interesting applications. This paper is based in part on the lecture given by the first author at the Harvard CMSA Math-Science Literature Lecture Series on May 22, 2020.
On double Poisson structures on commutative algebras
研究了交换代数上的双Poisson结构（a la Van den Bergh），主要结果表明，在大于1的克鲁尔维数的多项式代数上不存在非平凡的双Poisson结构.对于一般的交换代数A，这对可能的双泊松结构施加了重要的限制。奇异的双泊松结构表现出的情况下，多项式代数上的一个单一的发电机，以前认为范登伯格。
