Categories, Hopf Algebras, and Algebraic Combinatorics

这个项目涉及范畴理论和Hopf代数理论的研究，以及代数组合学的研究。这项研究在很大程度上是由组合性质的具体问题推动的，但它也解决了范畴理论和普通代数中感兴趣的问题，这反过来又应该在其他地方找到应用。PI最近完成了一本800页的专著(与来自印度理工学院孟买的Swapneel Mahajan合著)，在这本专著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年的时间专心致志地工作。这一漫长的旅程为我们打开了一扇大门，开启了丰富的新问题和令人兴奋的探索之路。这项工作的一个非常重要的发现是存在一个一般的Hopf代数结构，它同时包括上面提到的组合Hopf代数，以及单李代数(量子群)的包络代数的变形。这种联系还没有被深入探讨，但它肯定值得深入探讨。因此，这个项目的一个重要目标将是在组合Hopf代数世界和抽象Hopf代数世界之间架起桥梁。后者(Andruskiewitsch、Schneider和其他人)最近取得了一些重要进展，这对这一努力应该是互惠互利的。这项工作的核心是乔亚尔的物种概念。它为将范畴思想应用于代数组合学提供了正确的框架。Fock函子理论的第一个阶段是在PI与Mahajan的工作中发展起来的，它将物种中Hopf么半群的设置与分次Hopf代数的设置联系起来。我们将探索这一理论的延伸。代数组合数学的具体应用将是这个项目的一个主要焦点。在过去的二十年里，由罗塔、斯坦利和其他许多人开创的利用代数技术来研究具体组合问题已经取得了坚实的基础。这个项目通过利用范畴理论中的概念和想法来朝着这个方向发展。如果运用得当，直截了当的想法可能会非常有力和清晰。相反，这个项目中研究的具体问题将成为代数和范畴理论新发展的动力和指南。国际和平研究所将向研究生介绍这一研究领域，并在欧洲、拉丁美洲和印度进行国际联系和合作。

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.