Algebraic operads

操作数是一种代数结构，它结合了熟悉的代数运算的抽象性质，如数字的加法和乘法，以及不那么基本的运算，如函数的组合和矩阵的乘法，这些运算总是相关联的，扩展到非相关联的运算，如向量场的李括号，以及在当代纯数学和应用数学中出现的其他更奇特的运算。在操作数理论中，重点是操作本身，而不是由操作组合的参数。因此，某种类型的代数是相应操作符上的模块；例如，关联代数是关联操作符上的模块。***操作数理论是近40年来在代数拓扑和同调代数问题的基础上发展起来的，但它也与发展良好的非结合代数和泛代数理论有着密切的联系。在这一时期的后半段，歌剧理论的领导者是Jeal-Louis Loday，他从20世纪90年代初的调查论文“La renaissance des oprades”开始，并在2012年以他的综合专著“Algebraic operads”（与Bruno Vallette合作）达到高潮。***在过去几年中，我与我的研究合作者介绍了一个新的发展，即计算机代数在操作数理论问题中的应用，特别是使用计算线性代数、交换代数和对称群的表示理论，对各种类型的代数操作数的参数化族进行分类。我们在这个方向上的第一个主要成功是我与Vladimir Dotsenko一起解决了Loday关于参数化单关系操作的问题。我们能够证明，除了一些不太重要的情况外，这个类中唯一的常规操作符是众所周知的关联操作符、泊松操作符、莱布尼茨操作符和津比尔操作符。我们即将出版的书“代数操作：一个算法伴侣”（CRC出版社）的最后两章给出了这些方法的两个进一步的例子，一个应用于满足三次关系的二元操作的操作，另一个应用于满足二次关系的三元操作的操作。***这些方法同样适用于多个操作的操作符；例如，两个二进制操作。实际上，对于具有两个或多个操作的操作数，或者对于n bbbb2的n元操作，几乎没有做过什么工作。这些领域有许多有待解决的问题。这项研究的一个意想不到的结果是，计算数据导致了我们的猜想，即在许多由参数定义的大型操作数类中，“几乎所有”（技术上是Zariski密集子集）操作数都是幂零的：一旦达到一定的密度（度），每个组合都是零。换句话说，只有“测度0”（Zariski闭）子集是“显著的”

Operads are algebraic structures that incorporate the abstract properties of familiar algebraic operations, such as addition and multiplication of numbers, as well as less elementary operations such as composition of functions and multiplication of matrices, which are always associative, extending to operations which are not associative, such as Lie brackets of vector fields, and other more exotic operations which arise in contemporary pure and applied mathematics. In the theory of operads, the focus is on the operations themselves, not on the arguments which are being combined by the operations. Thus an algebra of a certain type is a module over the corresponding operad; for example, an associative algebra is a module over the associative operad. *** The theory of operads developed during the last 40 years out of problems in algebraic topology and homological algebra, but it also has close connections with the well-developed theories of nonassociative algebra and universal algebra. The leader of the theory of operads during the second half of this period was Jeal-Louis Loday, starting with his survey paper "La renaissance des opérades" from the early 1990's, and culminating with his comprehensive monograph "Algebraic Operads" (joint with Bruno Vallette) in 2012.*** A new development, which I have introduced during the last few years with my research collaborators, is the application of computer algebra to problems in operad theory, and in particular the use of computational linear algebra, commutative algebra, and representation theory of the symmetric group, to classify parametrized families of algebraic operads of various types. Our first major success in this direction was my solution with Vladimir Dotsenko of Loday's problem on parametrized one-relation operads. We were able to show that apart from a few less significant cases, the only regular operads in this class are the well-known associative, Poisson, Leibniz, and Zinbiel operads. The last two chapters of our forthcoming book "Algebraic Operads: An Algorithmic Companion" (CRC Press) present two further examples of these methods, one application to operads with a binary operation satisfying cubic relations, and another to operads with a ternary operation satisfying quadratic relations. *** These methods apply equally well to operads with more than one operation; for example, two binary operations. In fact, very little work has been done on operads with two or more operations, or with n-ary operations for n > 2. These promise to be exciting areas with many open problems. One unexpected result of this research is the computational data leading to our conjecture that in many large classes of operads defined by parameters, "almost all" (technically, a Zariski dense subset) of the operads are nilpotent: as soon as a certain arity (degree) is reached, every composition is zero. In other words, only a "measure 0" (Zariski closed) subset are "significant".**