Overview

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.

Full Product Details

Author:   Marcelo Aguiar ,  Swapneel Mahajan
Publisher:   Cambridge University Press
Imprint:   Cambridge University Press
Dimensions:   Width: 16.00cm , Height: 4.20cm , Length: 24.00cm
Weight:   1.490kg
ISBN:  

9781108495806

ISBN 10:   110849580
Pages:   824
Publication Date:   19 March 2020
Audience:   College/higher education ,  Postgraduate, Research & Scholarly
Format:   Hardback
Publisher's Status:   Active
Availability:   Manufactured on demand  
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Table of Contents

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.

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Author Information

Marcelo Aguiar is Professor in the Department of Mathematics at Cornell University, New York. Swapneel Mahajan is Associate Professor in the Department of Mathematics at the Indian Institute of Technology, Bombay.

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