Ebook: Bimonoids for Hyperplane Arrangements [slightly glitchy PDF]

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 common ground in
this theory.
The authors introduce notions of monoid, comonoid, bimonoid, and Lie monoid
relative to a fixed hyperplane arrangement. Faces, flats, and lunes of the arrangement
provide the building blocks for these concepts. They also construct universal
bimonoids by using generalizations of the classical notions of shuffle and quasishuf-
fle, and establish the Borel–Hopf, Poincaré–Birkhoff–Witt, and Cartier–Milnor–
Moore theorems in this new setting. A key role is played by noncommutative zeta
and Möbius functions which generalize the classical exponential and logarithm, and
by the representation theory of the Tits algebra.
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.