Bimonoids for Hyperplane Arrangements

Marcelo Aguiar, Swapneel Mahajan

• Publisher: Cambridge University Press
• Year: 2020
• Print ISBN: 9781108495806
• eBook ISBN: 9781108852784
• Topic: Mathématiques
• Subtopic: Mathématiques discrètes

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.

Table of contents

• Cover
• Half-title
• Series information
• Title page
• Copyright information
• Contents
• Preface
• Introduction
• Part I. Species and operads
• Part II. Basic theory of bimonoids
• Part III. Structure theory for bimonoids
• Appendices
• References
• List of Notations
• List of Tables
• Author Index
• Subject Index