Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations

We consider the combinatorial Dyson-Schwinger equation X=B⁺(P(X)) in the non-commutative Connes-Kreimer Hopf algebra of planar rooted trees HNCK, where B⁺ is the operator of grafting on a root, and P a formal series. The unique solution X of this equation generates a graded subalgebra AN,P of HNCK. We describe all the formal series P such that AN,P is a Hopf subalgebra. We obtain in this way a 2-parameters family of Hopf subalgebras of HNCK, organized into three isomorphism classes: a first one, restricted to a polynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, isomorphic to the Hopf algebra of quasi-symmetric functions; a last (infinite) one, which gives a non-commutative version of the Faà di Bruno Hopf algebra. By taking the quotient, the last class gives an infinite set of embeddings of the Faà di Bruno algebra into the Connes-Kreimer Hopf algebra of rooted trees. Moreover, we give an embedding of the free Faà di Bruno Hopf algebra on D variables into a Hopf algebra of decorated rooted trees, together with a non-commutative version of this embedding.     

The infinitesimal Hopf algebra and the poset of planar forests

We introduce an infinitesimal Hopf algebra of planar trees, generalising the construction of the non-commutative Connes-Kreimer Hopf algebra. A non-degenerate pairing and a dual basis are defined, and a combinatorial interpretation of the pairing in terms of orders on the vertices of planar forests is given. Moreover, the coproduct and the pairing can also be described with the help of a partial order on the set of planar forests, making it isomorphic to the Tamari poset. As a corollary, the dual basis can be computed with a Möbius inversion.     

带权无穷小双代数*

作为非齐次结合经典Yang-Baxter 方程的代数抽象,带权无穷小双代数在数学和数学物理领域扮演着重要的角色. 本文引入了带权无穷小Hopf模的概念,证明了带权拟三角无穷小单位双代数上的任意模都有一个自然的带权无穷小单位Hopf模结构.利用一种新的方式装饰平面根森林, 并证明根森林的空间,连同它上边的余乘和一组嫁接算子是集合上权为零的自由多重1-余圈无穷小单位双代数. 给出了余乘的一个组合解释.作为应用, 得到了未装饰的平面根森林上的余圈无穷小单位双代数范畴中的初始对象,它也是(非交换)Connes-Kreimer-Hopf代数中的研究对象. 最后,分别从任意带权无穷小双代数和带权交换无穷小双代数导出了两个预李代数,其中第二个构造推广了Novikov 代数上的Gelfand-Dorfman定理.     

On the operad of bigraft algebras

In this paper, we study the notion of a bigraft algebra, generalizing the notions of left and right graft algebras. We construct the free bigraft algebra on one generator in terms of certain planar rooted trees with decorated edges, and therefore describe explicitly the bigraft operad. We then compute its Koszul dual and show that the bigraft operad is Koszul. Moreover, we endow the free bigraft algebra on one generator with a universal Hopf algebra structure and a pairing. Finally, we prove an analogue of the Poincaré–Birkhoff–Witt and Cartier–Milnor–Moore theorems. For this, we define the notion of infinitesimal bigraft bialgebras and we prove the existence of a new good triple of operads.     

Yetter-Drinfeld modules over weak bialgebras

We discuss properties of Yetter-Drinfeld modules over weak bialgebras over commutative rings. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules over a weak Hopf algebra are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. IfH is finitely generated and projective, then we introduce the Drinfeld double using duality results between entwining structures and smash product structures, and show that the category of Yetter-Drinfeld modules is isomorphic to the category of modules over the Drinfeld double. The category of finitely generated projective Yetter-Drinfeld modules over a weak Hopf algebra has duality.     

Comparison of Hopf algebras on trees

Several Hopf algebra structures on vector spaces of trees can be found in the literature (cf. [10], [8], [2]). In this paper, we compare the corresponding notions of trees, the multiplications and comultiplications. The Hopf algebras are connected graded or, equivalently, complete Hopf algebras. The Hopf algebra structure on planar binary trees introduced by Loday and Ronco [10] is noncommutative and not cocommutative. We show that this Hopf algebra is isomorphic to the noncommutative version of the Hopf algebra of Connes and Kreimer [3]. We compute its first Lie algebra structure constants in the sense of [7], and show that there is no cogroup structure compatible with the Hopf algebra on planar binary trees.     

Integrable Renormalization II: The General Case

We extend the results we obtained in an earlier work [1]. The cocommutative case of ladders is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the Birkhoff decomposition of renormalization theory is derived by using the double Rota-Baxter construction, respectively Atkinsons theorem. We also outline the extension to the Hopf algebra of Feynman graphs via decorated rooted trees.submitted 16/03/04, accepted 09/09/04This revised version was published online in May 2005 with correction to the addresses.     

Free and cofree Hopf algebras

We first prove that a graded, connected, free and cofree Hopf algebra is always self-dual. Then, we prove that two graded, connected, free and cofree Hopf algebras are isomorphic if and only if they have the same Poincaré–Hilbert formal series. If the characteristic of the base field is zero, we prove that the Lie algebra of the primitive elements of such an object is free, and we deduce a characterization of the formal series of free and cofree Hopf algebras by a condition of growth of the coefficients. We finally show that two graded, connected, free and cofree Hopf algebras are isomorphic as (nongraded) Hopf algebras if and only if the Lie algebras of their primitive elements have the same number of generators.     

A noncommutative symmetric system over the Grossman-Larson Hopf algebra of labeled rooted trees

In this paper, we construct explicitly a noncommutative symmetric ( CS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the CS system formed by the generating functions of certain noncommutative symmetric functions, we obtain a specialization of noncommutative symmetric functions by labeled rooted trees. Taking the graded duals, we also get a graded Hopf algebra homomorphism from the Connes-Kreimer Hopf algebra of labeled rooted forests to the Hopf algebra of quasi-symmetric functions. A connection of the coefficients of the third generating function of the constructed CS system with the order polynomials of rooted trees is also given and proved.     

The Hopf algebra of diagonal rectangulations

We define and study a combinatorial Hopf algebra dRec with basis elements indexed by diagonal rectangulations of a square. This Hopf algebra provides an intrinsic combinatorial realization of the Hopf algebra tBax of twisted Baxter permutations, which previously had only been described extrinsically as a Hopf subalgebra of the Malvenuto-Reutenauer Hopf algebra of permutations. We describe the natural lattice structure on diagonal rectangulations, analogous to the Tamari lattice on triangulations, and observe that diagonal rectangulations index the vertices of a polytope analogous to the associahedron. We give an explicit bijection between twisted Baxter permutations and the better-known Baxter permutations, and describe the resulting Hopf algebra structure on Baxter permutations.     

Combinatorics of rooted trees and Hopf algebras

We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally associates a multiplicity to each pair of rooted trees. By using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the associated multiplicities.

Now the symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators, as well as the inner product mentioned above, to Kreimer's Hopf algebra. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this leads to the Hopf algebra of Grossman and Larson. We show that the inner product on rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with the graded dual of Kreimer's Hopf algebra, correcting an earlier result of Panaite.

     

Commutative combinatorial Hopf algebras

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 noncommutative 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.     

Zimmermann type cancellation in the free Faà di Bruno algebra

Haiman and Schmitt showed that one can use the antipode SF of the colored Faà di Bruno Hopf algebra F to compute the (compositional) inverse of a multivariable formal power series. It is shown that the antipode SH of an algebraically free analogue H of F may be used to invert non-commutative power series. Whereas F is the incidence Hopf algebra of the colored partitions of finite colored sets, H is the incidence Hopf algebra of the colored interval partitions of finite totally ordered colored sets. Haiman and Schmitt showed that the monomials in the geometric series for SF are labeled by trees. By contrast, the non-commuting monomials of SH are labeled by colored planar trees. The order of the factors in each summand is determined by the breadth first ordering on the vertices of the planar tree. Finally there is a parallel to Haiman and Schmitt's reduced tree formula for the antipode, in which one uses reduced planar trees and the depth first ordering on the vertices. The reduced planar tree formula is proved by recursion, and again by an unusual cancellation technique. The one variable case of H has also been considered by Brouder, Frabetti, and Krattenthaler, who point out its relation to Foissy's free analogue of the Connes-Kreimer Hopf algebra.     

On the Hopf Algebraic Structure of Lie Group Integrators

A commutative but not cocommutative graded Hopf algebra H_N, based on ordered (planar) rooted trees, is studied. This Hopf algebra is a generalization of the Hopf algebraic structure of unordered rooted trees H_C, developed by Butcher in his study of Runge-Kutta methods and later rediscovered by Connes and Moscovici in the context of noncommutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that H_N is naturally obtained from a universal object in a category of noncommutative derivations and, in particular, it forms a foundation for the study of numerical integrators based on noncommutative Lie group actions on a manifold. Recursive and nonrecursive definitions of the coproduct and the antipode are derived. The relationship between H_N and four other Hopf algebras is discussed. The dual of H_N is a Hopf algebra of Grossman and Larson based on ordered rooted trees. The Hopf algebra H_C of Butcher, Connes, and Kreimer is identified as a proper Hopf subalgebra of H_N using the image of a tree symmetrization operator. The Hopf algebraic structure of the shuffle algebra H_Sh is obtained from H_N by a quotient construction. The Hopf algebra H_P of ordered trees by Foissy differs from H_N in the definition of the product (noncommutative concatenation for H_P and shuffle for H_N) and the definitions of the coproduct and the antipode, however, these are related through the tree symmetrization operator.     

A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering

Fourier normal ordering (Unterberger, 2009) [34] is a new algorithm to construct explicit rough paths over arbitrary Hölder-continuous multidimensional paths. We apply in this article the Fourier normal ordering algorithm to the construction of an explicit rough path over multi-dimensional fractional Brownian motion B$B$ with arbitrary Hurst index α$\alpha$ (in particular, for α≤1/4$\alpha \le 1/4$, which was till now an open problem) by regularizing the iterated integrals of the analytic approximation of B$B$ defined in Unterberger (2009) [32]. The regularization procedure is applied to ‘Fourier normal ordered’ iterated integrals obtained by permuting the order of integration so that innermost integrals have highest Fourier modes. The algebraic properties of this rough path are best understood using two Hopf algebras: the Hopf algebra of decorated rooted trees (Connes and Kreimer, 1998) [6] for the multiplicative or Chen property, and the shuffle algebra for the geometric or shuffle property. The rough path lives in Gaussian chaos of integer orders and is shown to have finite moments.     

Categorification of Hopf algebras of rooted trees

We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec ?) whose semiring of functions is (a P-version of) the Connes-Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to ? and collapsing H ₀). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring in the polynomial representation of the free monad on P.