Algebraic operads

代数运算

• 批准号: RGPIN-2016-03725
• 负责人: Bremner, Murray
• 金额: $ 1.09万
• 依托单位: University of Saskatchewan
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=646210
• 关键词: Algebraic; operads

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".**

运算是代数结构，它结合了熟悉的代数运算的抽象属性，例如数字的加法和乘法，以及不太基本的运算，例如函数的组合和矩阵的乘法，这些运算总是结合的，扩展到非结合的运算，例如向量场的李括号，以及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，而不是操作所组合的参数。 因此，某种类型的代数是相应操作数的模；例如，关联代数是关联操作数上的模块。 *** 运算理论是在过去 40 年中针对代数拓扑和同调代数问题而发展起来的，但它也与成熟的非结合代数和普适代数理论有着密切的联系。 这一时期后半期运算理论的领导者是 Jeal-Louis Loday，他从 1990 年代初期的调查论文“La renaissance des opérades”开始，到 2012 年他的综合专着“代数运算”（与 Bruno Vallette 合着）达到顶峰。*** 我在过去几年中与我的研究合作者介绍的一项新发展是计算机代数的应用到 运算理论中的问题，特别是使用计算线性代数、交换代数和对称群表示论，对各种类型的代数运算的参数化族进行分类。 我们在这个方向上的第一个重大成功是我与 Vladimir Dotsenko 一起解决了参数化单关系运算的洛迪问题。我们能够证明，除了一些不太重要的情况外，此类中唯一的常规操作是著名的结合操作、泊松操作、莱布尼茨操作和津比尔操作。 我们即将出版的书《代数运算：算法伴侣》（CRC Press）的最后两章进一步介绍了这些方法的两个示例，一个应用于满足三次关系的二元运算的运算，另一个应用于满足二次关系的三元运算的运算。 *** 这些方法同样适用于具有多个操作的操作数；例如，两个二元运算。事实上，在具有两个或多个运算的操作数或 n > 2 的 n 元运算上所做的工作非常少。这些有望成为令人兴奋的领域，并存在许多悬而未决的问题。 这项研究的一个意想不到的结果是计算数据导致我们推测，在由参数定义的许多大类操作数中，“几乎所有”（技术上是 Zariski 稠密子集）的操作数都是幂零的：一旦达到一定的数量（度），每个组合都为零。 换句话说，只有“测量 0”（Zariski 闭合）子集是“显着的”。**