Group and ring-like structures in Category Theory and applications to Algebraic Combinatorics

该项目涉及代数组合学，范畴理论和霍普夫代数理论的研究，特别是后者与李理论有关的方面。代数组合学是数学的一个分支，它利用代数技巧来研究有限或可数的离散结构。 范畴论用于形式化抽象数学概念。 建议的研究相当多的使用范畴理论作为一种工具，用于组织各种代数结构相关联的组合对象。PI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。大量的努力致力于一个新的理论，扩大经典霍普夫-李理论的范围，并包括超平面安排和幂等半群作为基础数据的一部分。 这样的扩展是可能的和有意义的是PI目前工作的主要发现。简而言之，该安排的室被视为在自由结合代数中乘以元素序列的不同方式。推广的概念的Hopf代数和李代数，然后制定的山雀的投影映射。经典的情况下恢复限制的情况下，辫子超平面安排。该项目的一个同样重要的组成部分涉及的组合应用理论的霍普夫monoids在物种和相关的代数结构。这是PI与Swapneel Mahajan先前联合工作的延续。一个令人兴奋的最近的发展涉及到环理论的方法，色多项式和推广。建议的研究相当多的使用范畴理论作为一种工具，用于组织各种代数结构相关联的组合对象。具体来说，在Joyal的物种类别中，更高的monoidal结构是突出的。虽然以具体的例子为指导，这种方法导致更抽象的一般性问题，这也将得到解决。他们处理混合分配的法律，单子上的更高monoidal类别，并扩展经典的结果施耐德和Sweedler，除其他议题。

This project involves research in algebraic combinatorics, category theory, and the theory of Hopf algebras, particularly the aspects of the latter related to Lie theory. Algebraic combinatorics is a branch of mathematics which uses algebraic techniques to study finite or countable discrete structures. Category theory is used to formalize concepts of abstract mathematics. The proposed research makes considerable use of category theory as a tool for organizing the various algebraic structures associated to combinatorial objects. The PI will introduce graduate students to this area of research and make international contacts and collaborations in Europe, Latin America, and India.Substantial effort is devoted to a new theory enlarging the scope of classical Hopf-Lie theory and including hyperplane arrangements and idempotent semigroups as part of the fundamental data. That such an extension is possible and meaningful is a main finding of the current work of the PI. Briefly, the chambers of the arrangement are regarded as the different ways of multiplying a sequence of elements in a free associative algebra. Extensions of the notion of Hopf algebra and Lie algebra are then formulated in terms of the projection maps of Tits. The classical case is recovered by restricting to the case of braid hyperplane arrangements. An equally important component of the project involves combinatorial applications of the theory of Hopf monoids in species and related algebraic structures. This is a continuation of previous joint work by the PI with Swapneel Mahajan. An exciting recent development involves a ring-theoretic approach to chromatic polynomials and generalizations. The proposed research makes considerable use of category theory as a tool for organizing the various algebraic structures associated to combinatorial objects. Specifically, higher monoidal structures in Joyal's category of species are prominent. While guided by concrete examples, this approach leads to general questions of a more abstract nature, which will also be addressed. They deal with mixed distributive laws, monads on higher monoidal categories, and extensions of classical results of Schneider and Sweedler, among other topics.