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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=708757
• 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

运算符是一种代数结构，它包含了常见代数运算的抽象属性，例如数的加法和乘法，以及不太基本的运算，例如函数的合成和矩阵的乘法，它们总是关联的，扩展到不关联的运算，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异的运算。 在运算理论中，重点是运算本身，而不是运算所结合的论点。 因此，一个特定类型的代数是对应运算元上的模;例如，一个结合代数是结合运算元上的模。 在过去的40年里，运算理论的发展是出于代数拓扑和同调代数的问题，但它也与非结合代数和泛代数的成熟理论有着密切的联系。 这一时期后半期的歌剧理论的领导者是Jeal-Louis Loday，从20世纪90年代初的调查论文“La renaissance des opérades”开始，并在2012年与Bruno Vallette联合撰写了他的综合专著“代数运算”。 一个新的发展，我已经介绍了在过去几年中与我的研究合作者，是应用计算机代数问题的运算理论，特别是使用计算线性代数，交换代数，和代表性理论的对称群，分类参数化家庭的代数运算的各种类型。 我们在这个方向上的第一个重大成功是我的解决方案与弗拉基米尔Dotsenko的Loday的问题参数化的一个关系运算。我们能够证明，除了少数不太重要的情况下，唯一的正规运算在这一类是众所周知的联想，泊松，莱布尼茨和津比尔运算。 我们即将出版的书的最后两章“代数运算：一个代数伴侣”（CRC出版社）提出了这些方法的两个进一步的例子，一个应用程序的运算与二元运算满足三次关系，另一个操作与三元运算满足二次关系。 这些方法同样适用于具有多个操作的操作数;例如，两个二进制操作。事实上，很少有工作已经做了两个或两个以上的操作，或与n元操作n > 2。 这些领域有希望成为令人兴奋的领域，但也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导致我们猜想的计算数据，在许多由参数定义的大类运算中，