Related papers: Hopf algebra techniques to handle dynamical system…
``Bonsai'' Hopf algebras, introduced here, are generalizations of Connes-Kreimer Hopf algebras, which are motivated by Feynman diagrams and renormalization. We show that we can find operad structure on the set of bonsais. We introduce a new…
B-series originated from the work of John Butcher in the 1960s as a tool to analyze numerical integration of differential equations, in particular Runge-Kutta methods. Connections to renormalization theory in perturbative quantum field…
We review the language of differential forms and their applications to Riemannian Geometry with an orientation to General Relativity. Working with the principal algebraic and differential operations on forms, we obtain the structure…
A number of problems in theoretical physics share a common nucleus of combinatoric nature. It is argued here that Hopf algebraic concepts and techiques can be particularly efficient in dealing with such problems. As a first example, a brief…
Since the Connes--Kreimer Hopf algebra was proposed, revisiting present quantum field theory has become meaningful and important from algebraic points. In this paper, the Hopf algebra in the cutting rules is constructed. Its coproduct…
Abstraction is essential for reducing the complexity of systems across diverse fields, yet designing effective abstraction methodology for probabilistic models is inherently challenging due to stochastic behaviors and uncertainties. Current…
Hopf algebraic structures will replace groups and group representations as the leading paradigm in forthcoming times. K-theory, co-homology, entanglement, statistics, representation categories, quantized or twisted structures as well as…
We construct Hopf algebra isomorphisms of discrete multiplier Hopf C*-algebras, and Hopf AF C*-algebras (generalized quantum UHF algebras), from K-theoretical data. Some of the intermediate results are of independent interest, such as a…
We give a universal construction of families of Hopf $P$-algebras for any Hopf operad $P$. As a special case, we recover the Connes-Kreimer Hopf algebra of rooted trees.
The method of regions is an approach for developing asymptotic expansions of Feynman Integrals. We focus on expansions in Euclidean signature, where the method of regions can also be formulated as an expansion by subgraph. We show that for…
This paper is concerned with two generalizations of the Hopf algebra of symmetric functions that have more or less recently appeared. The Hopf algebra of noncommutative symmetric functions and its dual, the Hopf algebra of quasisymmetric…
Bialgebras and Hopf (bi)modules are typical algebraic structures with several interacting operations. Their structural and homological study is therefore quite involved. We develop the machinery of braided systems, tailored for handling…
A detailed expose of the Hopf algebra approach to interconnected input-output systems in nonlinear control theory is presented. The focus is on input-output systems that can been represented in terms of Chen-Fliess functional expansions or…
We study ideals in Hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations. These conditions include the property that the monoid act via partial permutations, that the…
Cyclic cohomology has been recently adapted to the treatment of Hopf symmetry in noncommutative geometry. The resulting theory of characteristic classes for Hopf algebras and their actions on algebras allows to expand the range of…
In this survey, we first review some known results on the representation theory of algebras with triangular decomposition, including the classification of the simple modules. We then discuss a recipe to construct Hopf algebras with…
In this work, the notion of partial representation of a Hopf algebra is introduced and its relationship with partial actions of Hopf algebras is explored. Given a Hopf algebra $H$, one can associate it to a Hopf algebroid $H_{par}$ which…
Using the standard filtration associated with a generalized lifting method, we determine all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradical generates a Hopf subalgebra isomorphic…
Multiple harmonic sums appear in the perturbative computation of various quantities of interest in quantum field theory. In this article we introduce a class of Hopf algebras that describe the structure of such sums, and develop some of…
There is no systematic general procedure by which isomorphism classes of Hopf algebras that are extensions of $\k F$ by ${\k}^G$ can be found. We develop the general procedure for classification of isomorphism classes of Hopf algebras which…