Coxeter Bialgebras

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

Marcelo Aguiar (Author), Swapneel Mahajan (Author)

9781009243773, Cambridge University Press

Hardback, published 17 November 2022

894 pages
24.1 x 16 x 4.6 cm, 1.59 kg

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.

Introduction
1. Coxeter groups and reflection arrangements
Part I. Coxeter Species: 2. Coxeter species and Coxeter bimonoids
3. Basic theory of Coxeter bimonoids
4. Examples of Coxeter bimonoids
5. Coxeter operads
6. Coxeter Lie monoids
7. Structure theory of Coxeter bimonoids
Part II. Coxeter Spaces: 8. Coxeter spaces and Coxeter bialgebras
9. Basic theory of Coxeter bialgebras
10. Examples of Coxeter bialgebras
11. Coxeter operad algebras
12. Coxeter Lie algebras
13. Structure theory of Coxeter bialgebras
Part III. Fock Functors: 14. Fock functors
15. Coxeter bimonoids and Coxeter bialgebras
16. Adjoints of Fock functors
17. Structure theory under Fock functors
18. Examples of Fock spaces
Appendix A. Category theory
References
List of Notations
List of Tables
List of Figures
List of Summaries
Author Index
Subject Index.

Subject Areas: Combinatorics & graph theory [PBV], Algebra [PBF]