Bimonoids for Hyperplane Arrangements
Bimonoids for Hyperplane Arrangements
AUD$330.00
inc GSTThe 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
Product details
March 2020Hardback
9781108495806
824 pages
240 × 160 × 42 mm
1.49kg
59 b/w illus. 7 colour illus. 30 tables 430 exercises
Available
Table of Contents
- Introduction
- 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
- References
- Notation index
- Author index
- Subject index.
