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El. knyga: Coxeter Bialgebras

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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.

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.

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This text develops a new theory extending the classical theory of connected graded Hopf algebras to reflection arrangements.
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.
Marcelo Aguiar is Professor in the Department of Mathematics at Cornell University, Ithaca. Swapneel Mahajan is Associate Professor in the Department of Mathematics at the Indian Institute of Technology, Mumbai.
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