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  1. Home
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  3. Bimonoids for Hyperplane Arrangements
  • Cited by 4
Publisher:
Cambridge University Press
Online publication date:
February 2020
Print publication year:
2020
Online ISBN:
9781108863117
DOI:
https://doi.org/10.1017/9781108863117
Subjects:
Mathematics (general), Mathematics, Discrete Mathematics Information Theory and Coding, Algebra
Series:
Encyclopedia of Mathematics and its Applications (173)
Information

Book description

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.

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