Bimonoids for hyperplane arrangements

Bimonoids for hyperplane arrangements

by Aguiar, Marcelo

Type Book
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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, an…

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Title Statements

  • Full title :Bimonoids for hyperplane arrangements / Marcelo Aguiar, Swapneel Mahajan.

Summary

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.

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Edition Statements

  • Publication Date:©2020

Publication Statements

The Physical Object

  • Description:1 online resource (xx, 832 pages)

Edition Identifiers

  • ISBN:9781108863117, 1108863116, 9781108852784, 1108852785
  • oclc:on1143848754
  • LC Classification:QA251.3
  • Dewey Decimal:516/.11

Table of Contents

Part I. Species and Operads: -- Hyperplane arrangements -- Species and bimonoids -- Bimonads on species -- Operads -- Part II. Basic Theory of Bimonoids: -- Primitive filtrations and decomposable filtrations -- Universal constructions -- Examples of bimonoids -- Hadamard product -- Exponential and logarithm -- Characteristic operations -- Modules over monoid algebras and bimonoids in species -- Antipode -- Part III. Structure Results for Bimonoids: -- Loday-Ronco, Leray-Samelson, Borel-Hopf -- Hoffman-Newman-Radford -- Freeness under Hadamard products -- Lie monoids -- Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore.

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Subjects

Algebraic spaces Algèbres d'incidence Espaces algébriques Géométrie plane Geometry, Plane Hyperespace Hyperspace Incidence algebras

Writers

Aguiar, Marcelo Mahajan, Swapneel Arvind

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