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Bimonoids for Hyperplane Arrangements

Part of Encyclopedia of Mathematics and its Applications

  • Date Published: April 2020
  • availability: Available
  • format: Hardback
  • isbn: 9781108495806


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About the Authors
  • 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.

    • The first book on the subject; readers will learn the theory first-hand from its original creators
    • Includes carefully designed chapters, with effective use of tables, diagrams, pictures, and exercises, making the book accessible to a wide audience
    • Touches many different areas of mathematics with minimum prerequisites, so readers can choose entry points depending on their background and interest
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    Product details

    • Date Published: April 2020
    • format: Hardback
    • isbn: 9781108495806
    • length: 824 pages
    • dimensions: 240 x 160 x 42 mm
    • weight: 1.49kg
    • contains: 59 b/w illus. 7 colour illus. 30 tables 430 exercises
    • availability: Available
  • Table of Contents

    Part I. Species and Operads:
    1. Hyperplane arrangements
    2. Species and bimonoids
    3. Bimonads on species
    4. Operads
    Part II. Basic Theory of Bimonoids:
    5. Primitive filtrations and decomposable filtrations
    6. Universal constructions
    7. Examples of bimonoids
    8. Hadamard product
    9. Exponential and logarithm
    10. Characteristic operations
    11. Modules over monoid algebras and bimonoids in species
    12. Antipode
    Part III. Structure Results for Bimonoids:
    13. Loday–Ronco, Leray–Samelson, Borel–Hopf
    14. Hoffman–Newman–Radford
    15. Freeness under Hadamard products
    16. Lie monoids
    17. Poincaré–Birkhoff–Witt and Cartier–Milnor–Moore
    Appendix A. Linear algebra
    Appendix B. Higher monads
    Appendix C. Internal hom
    Appendix D. Semidirect products
    Notation index
    Author index
    Subject index.

  • Authors

    Marcelo Aguiar, Cornell University, Ithaca
    Marcelo Aguiar is Professor in the Department of Mathematics at Cornell University, New York.

    Swapneel Mahajan, Indian Institute of Technology, Mumbai
    Swapneel Mahajan is Associate Professor in the Department of Mathematics at the Indian Institute of Technology, Bombay.

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