Related papers: Hopf monoids, permutohedral cones, and generalized…
It was recently shown that the renormalization of quantum field theory is organized by the Hopf algebra of decorated rooted trees, whose coproduct identifies the divergences requiring subtraction and whose antipode achieves this. We…
We show that the braided Hochschild cohomology, of an algebra in a suitably algebraic braided monoidal category, admits a graded ring structure under which it is braided commutative. We then give a canonical identification between the usual…
This paper constructs a novel Hopf algebra $\mathsf{cf}(\mathrm{UT}_{\bullet})$ on the class functions of the unipotent upper triangular groups $\mathrm{UT}_{n}(\mathbb{F}_{q})$ over a finite field. This construction is representation…
In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a…
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…
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…
We introduce a noncommutative and noncocommutative Hopf algebra which takes for certain Hopf categories (and therefore braided monoidal bicategories) a similar role as the Grothendieck- Teichmueller group for quasitensor categories. We also…
We give an introductory survey to the use of Hopf algebras in several problems of noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We…
Typed decorated trees are used by Bruned, Hairer and Zambotti to give a description of a renormalisation processon stochastic PDEs. We here study the algebraic structures on these objects: multiple prelie algebrasand related operads…
We report some observations concerning two well-known approaches to construction of quantum groups. Thus, starting from a bialgebra of inhomogeneous type and imposing quadratic, cubic or quartic commutation relations on a subset of its…
We say that a Hopf algebra H is semicocommutative if the right adjoint coaction factorizes through the tensor product of H with the center of H. For instance the commutative and the cocommutative Hopf algebras are semicocommutative. The…
A vector species is a functor from the category of finite sets with bijections to vector spaces; informally, one can view this as a sequence of $S_n$-modules. A Hopf monoid (in the category of vector species) consists of a vector species…
Hopf algebra deformations are merged with a class of Lie systems of Hamiltonian type, the so-called Lie-Hamilton systems, to devise a novel formalism: the Poisson-Hopf algebra deformations of Lie-Hamilton systems. This approach applies to…
A known fundamental Theorem for braided pointed Hopf algebras states that for each coideal subalgebra, that fulfils a few properties, there is an associated quotient coalgebra right module such that the braided Hopf algebra can be…
A combinatorial Hopf algebra is a graded connected Hopf algebra over a field $F$ equipped with a character (multiplicative linear functional) $\zeta:H\to F$. We show that the terminal object in the category of combinatorial Hopf algebras is…
We get new Hopf algebras (HA): 1. A wealth of quotient HA's of the Malvenuto-Reutenauer HA (the Loday-Ronco HA being a special case). They consist of the permutations avoiding an {\it arbitrary} set of permutations without global descents,…
These notes -- originating from a one-semester class by their second author at the University of Minnesota -- survey some of the most important Hopf algebras appearing in combinatorics. After introducing coalgebras, bialgebras and Hopf…
We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in…
The goal of this paper is to show that valuation theory and Hopf theory are compatible on the class of generalized permutahedra. We prove that the Hopf structure $\mathbf{GP}^+$ on these polyhedra descends, modulo the inclusion-exclusion…
In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Fa\`a di Bruno Hopf algebra, the…