Algebraic operads

运算是一种代数结构，它包含了常见代数运算的抽象性质，如数字的加法和乘法，以及较不基本的运算，如函数的合成和矩阵的乘法，这些运算总是结合的，扩展到非结合的运算，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。在歌剧理论中，重点是操作本身，而不是操作结合的论点。因此，某种类型的代数是对应算符上的模；例如，结合代数是结合算符上的模。算子论是近40年来在代数拓扑学和同调代数问题的基础上发展起来的，但它也与发展成熟的非结合代数和泛代数理论有着密切的联系。这一时期后半期歌剧理论的领军人物是让-路易斯·洛代，他从1990年初的《S的文艺复兴》一文开始，到2012年与布鲁诺·瓦莱特合著的综合性专著《代数歌剧》达到顶峰。*最近几年我和我的研究合作者介绍了一个新的发展，那就是计算机代数在歌剧理论问题中的应用，特别是使用计算线性代数、交换代数和对称群的表示理论来分类各种类型的代数歌剧的参数化族。我们在这个方向上取得的第一个重大成功是我与弗拉基米尔·多森科(Vladimir Dotsenko)就参数化单关系歌剧提出的问题的解决方案。我们能够证明，除了几个不太重要的案例外，这类歌剧中唯一的常规歌剧是著名的联合歌剧、泊松歌剧、莱布尼茨歌剧和津比尔歌剧。我们即将出版的《代数运算：算法伴侣》(CRC Press)一书的最后两章提供了这些方法的另外两个例子，一个应用于具有满足三次关系的二元运算的算子，另一个应用于具有满足二次关系的三元运算的算子。*这些方法同样适用于具有一个以上操作的操作员；例如，两个二元操作。事实上，关于具有两个或更多操作的歌剧，或者n&gt；2的n元操作的工作很少。这些领域有望成为令人兴奋的领域，有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，这导致了我们的猜测，在许多由参数定义的大类歌剧中，几乎所有的歌剧(从技术上讲，是扎里斯基稠密子集)是幂零的：一旦达到一定的程度，每一首曲子都是零。换句话说，只有“度量0”(Zariski闭合)子集才是“重要的”。**

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".**