In this paper, we associate canonically a precyclic mod- ule to a regular multiplier Hopf algebra endowed with a group-like projection and a modular pair in involution satisfying certain con- dition

Consider the coradical filtration of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We show that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980’s by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heap-ordered trees, respectively. We also sho...

Inspired by [3] we introduce the concept of extended Hopf algebra and consider their cyclic cohomology in the spirit of Connes-Moscovici [3, 4, 5]. Extended Hopf algebras are closely related, but different from, Hopf algebroids. Their definition is motivated by attempting to define cyclic cohomology of Hopf algebroids in general. Many of Hopf algebra like structures, including the Connes-Moscov...

The elements of the wide class of quantum universal enveloping algebras are prooved to be Hopf algebras H with spectrum Q(H) in the category of groups. Such quantum algebras are quantum groups for simply connected solvable Lie groups P (H). This provides utilities for a new algorithm of constructing quantum algebras especially useful for nonsemisimple ones. The quantization procedure can be car...

The Hopf algebra of Feynman diagrams, analyzed by A.Connes and D.Kreimer, is considered from the perspective of the theory of effective actions and generalized τ-functions, which describes the action of diffeomorphism and shift groups in the moduli space of coupling constants. These considerations provide additional evidence of the hidden group (integrable) structure behind the standard formali...

In this paper, we define a Hopf algebra structure on the vector space spanned by packed words using a selection-quotient coproduct. We show that this algebra is free on its irreducible packed words. Finally, we give some brief explanations on the Maple codes we have used.

In this paper, we introduce a Hopf algebra, developed by the author and André Henriques, which is usable in the computation of the tmf homology of a space. As an application, we compute the tmf -homology of BΣ3 in a manner analogous to Mahowald and Milgram’s computation of the ko-homology RP∞ in [7].

We study a simple map as a minimal model of excitable cells. The map has two fast variables which mimic the behavior of class I neurons, undergoing a sub-critical Hopf bifurcation. Adding a third slow variable allows the system to present bursts and other interesting biological behaviors. Bifurcation lines which locate the excitability region are obtained for different planes in parameter space.

We introduce here the Hopf algebra structure describing the noncommutative renormalization of a recently introduced translation-invariant model on Moyal space. We define Hochschild one-cocyles B γ + which allows us to write down the combinatorial DysonSchwinger equations for noncommutative quantum field theory. Oneand two-loops examples are explicitly worked out.