30 research outputs found
Hopf structures on the multiplihedra
We investigate algebraic structures that can be placed on vertices of the
multiplihedra, a family of polytopes originating in the study of higher
categories and homotopy theory. Most compelling among these are two distinct
structures of a Hopf module over the Loday-Ronco Hopf algebra.Comment: 24 pages, 112 .eps file
Hochschild polytopes
The -multiplihedron is a polytope whose faces correspond to
-painted -trees, and whose oriented skeleton is the Hasse diagram of the
rotation lattice on binary -painted -trees. Deleting certain inequalities
from the facet description of the -multiplihedron, we construct the
-Hochschild polytope whose faces correspond to -lighted -shades,
and whose oriented skeleton is the Hasse diagram of the rotation lattice on
unary -lighted -shades. Moreover, there is a natural shadow map from
-painted -trees to -lighted -shades, which turns out to define a
meet semilattice morphism of rotation lattices. In particular, when , our
Hochschild polytope is a deformed permutahedron whose oriented skeleton is the
Hasse diagram of the Hochschild lattice.Comment: 32 pages, 25 figures, 7 tables. Version 2: Minor correction
Review on higher homotopies in the theory of H-spaces
Higher homotopy in the theory of H-spaces started from the works by Sugawara in the 1950th. In this paper we review the development of the theory of H-spaces associated with it. Mainly there are two types of higher homotopies, homotopy associativity and homotopy commutativity. We give explanations of the polytopes used as the parameter spaces of those higher forms
Geometric combinatorial algebras: cyclohedron and simplex
In this paper we report on results of our investigation into the algebraic
structure supported by the combinatorial geometry of the cyclohedron. Our new
graded algebra structures lie between two well known Hopf algebras: the
Malvenuto-Reutenauer algebra of permutations and the Loday-Ronco algebra of
binary trees. Connecting algebra maps arise from a new generalization of the
Tonks projection from the permutohedron to the associahedron, which we discover
via the viewpoint of the graph associahedra of Carr and Devadoss. At the same
time that viewpoint allows exciting geometrical insights into the
multiplicative structure of the algebras involved. Extending the Tonks
projection also reveals a new graded algebra structure on the simplices.
Finally this latter is extended to a new graded Hopf algebra (one-sided) with
basis all the faces of the simplices.Comment: 23 figures, new expanded section about Hopf algebra of simplices,
with journal correction