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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675477
• 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

歌剧是代数结构，它结合了熟悉的代数操作的抽象特性，例如增加数量的加法和倍增，以及诸如基本操作（例如功能的组成和综合物的繁殖），这些操作始终是关联的，不断扩展到非关联的操作，例如，纯粹的数学和其他越来越多的运算，这些操作和其他纯净的操作和其他peristers and puroise and peristers and puroise and earise and earise in Arise and Arise and arise in ar arise and arise and arise and arise。在运营商的理论中，重点是运营本身，而不是由操作组合的论点。某种类型的代数是相应运算符上的模块。例如，关联代数是关联运算符上的模块。 ***在过去40年中，运营商的理论出于代数拓扑和同源代数的问题而发展，但它与发达的非缔约代数和通用代数的理论也有密切的联系。在此期间的后半段，运营商理论的领导者是Jeal-Louis Loday，首先是他的调查论文“ La Renaissance desopérades”，从1990年代初期开始，并以他的全面专题为“代数行动”（与Bruno Vallette的联合”（与Bruno vallette的联合）达到了最新的启动。在运算符理论中，特别是使用计算线性代数，交换代数和对称群体的表示理论来对各种类型的代数运算符的参数化家族进行分类。在这个方向上，我们的第一个主要成功是我解决了Loday问题的Vladimir Dotsenko在参数化的一关系算子方面的问题。我们能够证明，除了几个不太重要的情况外，该班级唯一的常规运营商是著名的协会，Poisson，Leibniz和Zinbiel运营商。我们即将出版的著作《 Algebraic Operads：算法伴侣》（CRC Press）的最后两章提出了这些方法的另外两个示例，其中一个应用于具有二元操作的运营商满足立体关系的操作员，另一个应用于操作员，并对经营者进行了满足二次关系的三元操作。 ***这些方法同样适用于多个操作的运营商；例如，两个二进制操作。实际上，对具有两个或多个操作的运营商或N- ARY操作的运营商几乎没有完成工作。这些承诺将成为令人兴奋的领域，并有许多开放的问题。这项研究的一个意外结果是计算数据导致我们的概念，在许多由参数定义的大量运算符中，几乎几乎