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Bimonoids for Hyperplane Arrangements (Encyclopedia of Mathematics and Its Applications #173) (Hardcover)


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Bimonoids for Hyperplane Arrangements (Encyclopedia of Mathematics and Its Applications #173) Cover Image


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, Poincar -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.
Product Details
ISBN: 9781108495806
ISBN-10: 110849580X
Publisher: Cambridge University Press
Publication Date: March 19th, 2020
Pages: 824
Language: English
Series: Encyclopedia of Mathematics and Its Applications