Categories, Hopf Algebras, and Algebraic Combinatorics

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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.