This monograph is the first to present structural interactions between the theory of Coxeter groups and the theory of Hopf algebras. These interactions occur naturally in various parts of science, notably in algebra, combinatorics, geometry and theoretical physics. The monograph is divided into two parts. The first part (chapters 1-3) introduces basic concepts and properties of Coxeter groups, left regular bands and Hopf algebras. The second part (chapters 4-8) consists mostly of original work by the authors. Firstly, it deals with the descent theory for Coxeter groups. Then it proceeds to constructions of Hopf algebras related to Coxeter groups via certain diagrams of semigroup algebras. The theory is then applied to the Hopf algebra of pairs of permutations, in particular proving its freeness and cofreeness. Finally, cofreeness of the Hopf algebras of (pointed) faces, and of quasi-symmetric functions, is established. It is not just the results but also a unified conceptual treatment of the theory that make this monograph a very valuable one. Moreover, in a number of places further ideas are suggested and new results are announced; more details will appear in a follow-up monograph announced by the authors in the preface.

Reviewer:

jtrl