Coxeter Bialgebras

This text develops a new theory extending the classical theory of connected graded Hopf algebras to reflection arrangements.

This monograph is aimed at graduate students and researchers in diverse areas of mathematics. It offers a new geometric perspective on the classical theory of connected graded Hopf algebras by extending it to the setting of real reflection arrangements. Discrete geometry, algebra, and combinatorics meet fruitfully at these crossroads.The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras. This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.