Algebraic operads

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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".**