Categories, Hopf Algebras, and Algebraic Combinatorics

本项目涉及范畴论和Hopf代数理论，以及代数组合学的研究。这项研究主要是由组合性质的具体问题推动的，但它也解决了对范畴论和一般代数感兴趣的问题，这些问题反过来应该在其他地方找到应用。PI最近完成了一篇800页的专著（与印度理工学院孟买分校的Swapneel Mahajan合著），在此基础上，为研究组合学中出现的一大类Hopf代数奠定了坚实的概念框架。这个项目花了大约7年的时间。这一漫长的旅程开启了一扇通往大量新问题和令人兴奋的探索途径的大门。这项工作的一个非常重要的发现是Hopf代数的一般构造的存在性，它同时包含了上面提到的组合Hopf代数，以及简单李代数（量子群）的包络代数的变形。这种联系还没有被深入探讨，但它确实值得深入探讨。因此，这个项目的一个重要目标将是在组合霍普夫代数的世界和抽象霍普夫代数的世界之间架起一座桥梁。最近在后者（Andruskiewitsch， Schneider等人）方面取得了重要进展，这对这一努力应该是互利的。这项工作的核心是Joyal的物种概念。它为范畴思想在代数组合中的应用提供了正确的框架。Fock函子理论的第一阶段是在PI与Mahajan的工作中发展起来的，它将Hopf单群在种中的设置与分级Hopf代数的设置联系起来。该理论的扩展将被探讨。代数组合学的具体应用将是本课题的主要重点。使用代数技术来研究具体的组合问题，由Rota、Stanley和其他许多人开创，在过去的二十年中已经取得了坚实的基础。这个项目通过利用范畴论的概念和思想来构建这个方向。分类思想，如果运用得当，可以非常有力和清晰。相反，本项目研究的具体问题将为代数和范畴论的新发展提供动力和指导。PI将向研究生介绍这一领域的研究，并在欧洲、拉丁美洲和印度进行国际接触和合作。

This project involves research in category theory and the theory of Hopf algebras, as well as algebraic combinatorics. The research is largely motivated by concrete questions of a combinatorial nature, but it also addresses questions of interest in category theory and general algebra, which in turn should find applications elsewhere. The PI has recently completed an 800 page monograph (coauthored with Swapneel Mahajan from IIT Mumbai), on which a solid conceptual framework for the study of a large class of Hopf algebras that arise in combinatorics was laid out. This project took about 7 years of dedicated work. This long journey has opened the door to a wealth of new questions and exciting avenues to explore. A very important finding of this work is the existence of a general construction of Hopf algebras which includes at the same time the combinatorial Hopf algebras alluded to above, and the deformations of the enveloping algebras of simple Lie algebras (quantum groups). This connection has not been explored in depth yet, but it certainly deserves to be. Thus, an important goal of this project will be to bridge between the world of combinatorial Hopf algebras and that of abstract Hopf algebras. There have been recent important advances in the latter (by Andruskiewitsch, Schneider, and others) which should be mutually beneficial to this endeavor. Central to this work is Joyal's notion of species. It provides the right framework for the application of categorical ideas to algebraic combinatorics. The theory of Fock functors, whose first stages are developed in the PI's work with Mahajan, links the setting of Hopf monoids in species with that of graded Hopf algebras. Extensions of the theory will be explored. Concrete applications to algebraic combinatorics will be a theme of major focus for this project.The use of algebraic techniques in order to study concrete combinatorial problems, pioneered by Rota, Stanley, and many others, has gained solid ground over the past two decades. This project builds in this direction by exploiting notions and ideas from category theory. Categorical ideas, properly employed, can be very powerful and clarifying. Conversely, the concrete questions studied in this project will serve as motivation and guide for new developments in algebra and category theory. The PI will introduce graduate students to this area of research and make international contacts and collaborations in Europe, Latin America, and India.