Bimonoids for Hyperplane Arrangements

Marcelo Aguiar; Swapneel Mahajan

doi:10.1017/9781108863117

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Bastidas, Jose 2021. The polytope algebra of generalized permutahedra. Algebraic Combinatorics, Vol. 4, Issue. 5, p. 909.

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Ardizzoni, Alessandro Goyvaerts, Isar and Menini, Claudia 2023. Pre-rigid monoidal categories. Quaestiones Mathematicae, Vol. 46, Issue. 8, p. 1503.

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Aguiar, Marcelo and Ardila, Federico 2023. Hopf Monoids and Generalized Permutahedra. Memoirs of the American Mathematical Society, Vol. 289, Issue. 1437,

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Qadri, Qamar Raza Lai, Xueshuang Zhao, Wei Zhang, Zhenyang Zhao, Qingbo Ma, Peipei Pan, Yuchun and Wang, Qishan 2024. Exploring the Interplay between the Hologenome and Complex Traits in Bovine and Porcine Animals Using Genome-Wide Association Analysis. International Journal of Molecular Sciences, Vol. 25, Issue. 11, p. 6234.

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Forcey, Stefan 2025. Counting Plane Arrangements via Oriented Matroids. Mathematics Magazine, p. 1.

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Marcelo Aguiar, Cornell University, Ithaca, Swapneel Mahajan, Indian Institute of Technology, Mumbai

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)

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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, 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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Bimonoids for Hyperplane Arrangements

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