Bimonoids for hyperplane arrangements / Marcelo Aguiar, Cornell University, Ithaca, Swapneel Mahajan, Indian Institute of Technology, Bombay.

Format:

Book

Author/Creator:

Aguiar, Marcelo, 1968- author.

Mahajan, Swapneel Arvind, 1974- author.

Series:

Encyclopedia of mathematics and its applications ; v. 173.

Encyclopedia of mathematics and its applications ; 173

Language:

English

Subjects (All):

Hopf algebras.

Incidence algebras.

Algebraic spaces.

Monoids.

Geometry, Plane.

Physical Description:

xx, 832 pages : illustrations ; 24 cm.

Place of Publication:

Cambridge, United Kingdom ; New York, NY, United States : Cambridge University Press, 2020.

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."-- Back cover

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.

Notes:

Includes bibliographical references (pages 763-801) and index.

Other Format:

Online version: Aguiar, Marcelo, 1968- Bimonoids for hyperplane arrangements.

ISBN:

9781108495806

110849580X

OCLC:

1122714430