Combinatorial Hopf algebras from PROs (original) (raw)
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Commutative combinatorial Hopf algebras
Journal of Algebraic Combinatorics, 2008
We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its non-commutative dual is realized in three different ways, in particular as the Grossman-Larson algebra of heap ordered trees. Extensions to endofunctions, parking functions, set compositions, set partitions, planar binary trees and rooted forests are discussed. Finally, we introduce one-parameter families interpolating between different structures constructed on the same combinatorial objects.
On the structure of cofree Hopf algebras
HAL (Le Centre pour la Communication Scientifique Directe), 2006
We prove an analogue of the Poincaré-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a B∞-algebra. We construct a universal enveloping functor U 2 from B∞-algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra H is of the form U 2(Prim H). We take advantage of the description of the free 2as-algebra in terms of planar trees to unravel the structure of the operad B∞.
On Hopf algebra structures over free operads
Advances in Mathematics, 2006
The operad Lie can be constructed as the operad of primitives PrimAs from the operad As of associative algebras. This is reflected by the theorems of Friedrichs, Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore. We replace the operad As by families of free operads P, which include the operad Mag freely generated by a noncommutative non-associative binary operation and the operad of Stasheff polytopes. We obtain Poincaré-Birkhoff-Witt type theorems and collect information about the operads PrimP, e.g. in terms of characteristic functions.
Polynomial realizations of some combinatorial Hopf algebras
Journal of Noncommutative Geometry, 2014
We construct explicit polynomial realizations of some combinatorial Hopf algebras based on various kinds of trees or forests, and some more general classes of graphs, ranging from the Connes-Kreimer algebra to an algebra of labelled forests isomorphic to the Hopf algebra of parking functions and to a new noncommutative algebra based on endofunctions admitting many interesting subalgebras and quotients.
An antipode formula for the natural Hopf algebra of a set operad
arXiv (Cornell University), 2013
A set-operad is a monoid in the category of combinatorial species with respect to the operation of substitution. From a set-operad, we give here a simple construction of a Hopf algebra that we call the natural Hopf algebra of the operad. We obtain a combinatorial formula for its antipode in terms of Shröder trees, generalizing the Hayman-Schmitt formula for the Faá di Bruno Hopf algebra. From there we derive more readable formulas for specific operads. The classical Lagrange inversion formula is obtained in this way from the set-operad of pointed sets. We also derive antipodes formulas for the natural Hopf algebra corresponding to the operads of connected graphs, the NAP operad, and for its generalization, the setoperad of trees enriched with a monoid. When the set operad is left cancellative, we can construct a family of posets. The natural Hopf algebra is then obtained as an incidence reduced Hopf algebra, by taking a suitable equivalence relation over the intervals of that family of posets. We also present a simple combinatorial construction of an epimorphism from the natural Hopf algebra corresponding to the NAP operad, to the Connes and Kreimer Hopf algebra.
2 Combinatorial Hopf algebras from dual Hopf algebras 2 . 1 Right-sided combinatorial Hopf algebras
2009
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 Faà di Bruno Hopf algebra, the non-commutative version of the charge renormalization Hopf algebra on planar binary trees for quantum electrodynamics, and the non-commutative version of the Pinter renormalization Hopf algebra on any bosonic field. We also describe two general ways to define the associative product in such Hopf algebras, the first one by recursion, and the second one by grafting and shuffling some decorated rooted trees.
On Hopf Algebras and Their Generalizations
Communications in Algebra, 2008
We survey Hopf algebras and their generalizations. In particular, we compare and contrast three well-studied generalizations (quasi-Hopf algebras, weak Hopf algebras, and Hopf algebroids), and two newer ones (Hopf monads and hopfish algebras). Each of these notions was originally introduced for a specific purpose within a particular context; our discussion favors applicability to the theory of dynamical quantum groups. Throughout the note, we provide several definitions and examples in order to make this exposition accessible to readers with differing backgrounds.
Discrete Mathematics, 1998
Hopf algebras play a major rôle in such diverse mathematical areas as algebraic topology, formal group theory, and theoretical physics, and they are achieving prominence in combinatorics through the influence of G.-C. Rota and his school. Our primary purpose in this article is to build on work of W. Schmitt, and establish combinatorial models for several of the Hopf algebras associated with the universal formal group law and the Lazard ring. In so doing, we incorporate and extend certain invariants of simple graphs such as the umbral chromatic polynomial, and R. Stanley's recently introduced symmetric function. Our fundamental combinatorial components are finite set systems, together with a versatile generalization in which they are equipped with a group of automorphisms. Interactions with the Roman-Rota umbral calculus over graded rings of scalars which may contain torsion are a significant feature of our presentation. Whenever f : C * → A * is of degree −1 and f (C 1) contains the identity of A * , we refer to Γ f as a delta operator. We may define the category of coalgebras with delta operator by insisting that the morphisms are coalgebra maps which commute with the delta operators given on source and target respectively. By way of example, consider the graded polynomial algebra A * [x], and the comultiplication, counit, and antipode maps specified by δ(x) = x ⊗ 1 + 1 ⊗ x , ε(x i) = δ i,0 , and S(x) = −x , respectively. These maps invest A * [x] with the structure of a commutative, cocommutative Hopf algebra, which is known as the binomial Hopf algebra over A * (in one variable). The standard basis consists of the powers x n , for n ≥ 0. Note that δ may be rewritten as δ : A * [x] −→ A * [x, y] ,
Cocommutative Hopf Algebras of Permutations and Trees
Journal of Algebraic Combinatorics, 2005
Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing 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. These results follow from the fact that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra.
A class of finite-by-cocommutative Hopf algebras
arXiv (Cornell University), 2023
We present a rich source of Hopf algebras starting from a cofinite central extension of a Noetherian Hopf algebra and a subgroup of the algebraic group of characters of the central Hopf subalgebra. The construction is transparent from a Tannakian perspective. We determine when the new Hopf algebras are co-Frobenius, or cosemisimple, or Noetherian, or regular, or have finite Gelfand-Kirillov dimension. 2020 Mathematics Subject Classification. 16T05; 18M05. Notations. The natural numbers are denoted by N, and N 0 = N ∪ {0}. Given m < n ∈ N 0 , we set I m,n = {i ∈ N 0 : m ≤ i ≤ m} and I n = I 1,n. 'Algebra' means associative unital algebra. The space of algebra homomorphism from a k-algebra A to a k-algebra B is denoted by Alg(A, B). The category of finite-dimensional left R-modules, where R is an algebra, is denoted by rep R. All Hopf algebras are supposed to have bijective antipode. We write M ≤ N to express that M is a subobject of N in a given category.