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

Cornell University, Ithaca NY

Investigators

Abstract

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.