Chapter 17. - Structure theory under Fock functors

Summary

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.

The building blocks of the theory are geometric objects associated to a reflection arrangement such as faces, flats, lunes, and their orbits under the action of the Coxeter group. A generalized notion of zeta and Möbius function play a fundamental role in all aspects of the theory, including exp-log correspondences and results such as the Poincarö–Birkhoff–Witt theorem. The Tits algebra and its invariant subalgebra also play key roles.

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.