Bimonoids for Hyperplane Arrangements

Develops a new theory, parallel to the classical theory of connected Hopf algebras, including a real hyperplane arrangement.

Marcelo Aguiar (Author), Swapneel Mahajan (Author)

9781108495806, Cambridge University Press

Hardback, published 19 March 2020

824 pages, 59 b/w illus. 7 colour illus. 30 tables 430 exercises
24 x 16 x 4.2 cm, 1.49 kg

The goal of this monograph is to develop Hopf theory in a new setting which features centrally a real hyperplane arrangement. The new theory is parallel to the classical theory of connected Hopf algebras, and relates to it when specialized to the braid arrangement. Joyal's theory of combinatorial species, ideas from Tits' theory of buildings, and Rota's work on incidence algebras inspire and find a common expression in this theory. The authors introduce notions of monoid, comonoid, bimonoid, and Lie monoid relative to a fixed hyperplane arrangement. They also construct universal bimonoids by using generalizations of the classical notions of shuffle and quasishuffle, and establish the Borel–Hopf, Poincaré–Birkhoff–Witt, and Cartier–Milnor–Moore theorems in this setting. This monograph opens a vast new area of research. It will be of interest to students and researchers working in the areas of hyperplane arrangements, semigroup theory, Hopf algebras, algebraic Lie theory, operads, and category theory.

Introduction
Part I. Species and Operads: 1. Hyperplane arrangements
2. Species and bimonoids
3. Bimonads on species
4. Operads
Part II. Basic Theory of Bimonoids: 5. Primitive filtrations and decomposable filtrations
6. Universal constructions
7. Examples of bimonoids
8. Hadamard product
9. Exponential and logarithm
10. Characteristic operations
11. Modules over monoid algebras and bimonoids in species
12. Antipode
Part III. Structure Results for Bimonoids: 13. Loday–Ronco, Leray–Samelson, Borel–Hopf
14. Hoffman–Newman–Radford
15. Freeness under Hadamard products
16. Lie monoids
17. Poincaré–Birkhoff–Witt and Cartier–Milnor–Moore
Appendix A. Linear algebra
Appendix B. Higher monads
Appendix C. Internal hom
Appendix D. Semidirect products
References
Notation index
Author index
Subject index.

Subject Areas: Combinatorics & graph theory [PBV], Groups & group theory [PBG], Algebra [PBF], Discrete mathematics [PBD]