A Hopf Operad of Forests of Binary Trees and Related Finite-Dimensional Algebras

Abstract

Abstract. The structure of a Hopf operad is defined on the vector spaces spanned by forests of leaf-labeled, rooted, binary trees. An explicit formula for the coproduct and its dual product is given, using a poset on forests.Keywords: Hopf operad, binary tree, poset IntroductionThe theme of this paper is the algebraic combinatorics of leaf-labeled rooted binary trees and forests of such trees. We shall endow these objects with several algebraic structures.The main structure is an operad, called the Bessel operad, which is the suspension of an operad defined by a distributive law between the suspended commutative operad and the operad of commutative non-associative algebras (sometimes called Griess algebras). The Bessel operad may be seen as an analog of the Gerstenhaber operad [10], which is the suspension of an operad defined by a distributive law between the suspended commutative operad and the Lie operad. Unlike the Gerstenhaber operad, the Bessel operad has a simple combinatorial basis, given explicitly by forests of leaf-labeled rooted binary trees.The Bessel operad, like the Gerstenhaber operad, is a Hopf operad. More precisely, they are both endowed with a cocommutative coproduct. This gives rise to a family of finitedimensional coalgebras. In the dual vector spaces of the Bessel operad, one gets algebras based on forests of leaf-labeled binary trees.An explicit formula is obtained for the coproduct in these coalgebras of forests (and therefore for their dual products), using a poset structure on the set of forests, which may be of independent interest.After some preliminary material on operads in the first section, the second section is devoted to the definition of a distributive law between the suspended commutative operad and the Griess operad. The suspension of the operad defined by this distributive law is introduced in the next section. The coproduct is defined and shown to be given by an explicit sum in the fourth section. In the last section, the dual algebras are briefly studied.

On a Hopf operad containing the Poisson operad

“…Besides, the parts of the bigraded dimensions corresponding to the Poisson and Bessel suboperads are correct, i.e. match the well-known dimensions of Poisson and the dimensions of Bessel computed in [2].…”

“…This relation was already proposed for the Bessel suboperad in [2]. The algebras underlying R are given by generators and relations which have some resemblance with the Arnold relations and seem to contain the relations satisfied by some simple differential forms.…”

“…The relations (4) and (5) The relation (5) is also known to be a distributive law from Σ Griess • Com to Com •Σ Griess which defines the Bessel operad, see [2].…”

Hopf algebras of planar binary trees: an operated algebra approach

“…Note that the last eight terms in Eq. (5) are precisely corresponding to the eight kinds of admissible cuts in Adm(T ). Thus…”

“…Later, Brouder and Frabetti [3] showed that there exists a noncommutative Hopf algebra on planar binary trees which represents the renormalization group of quantum electrodynamics, and the coaction which describes the renormalization procedure. In the algebraic framework of Chapoton [5] for Bessel operad, a Hopf operad is constructed on the vector spaces spanned by forests of leaf-labeled binary rooted trees. Aguiar and Sottile further studied the structure of the Loday-Ronco Hopf algebra by a new basis in [1], where the product, coproduct and antipode in terms of this basis were also given.…”