On a Hopf operad containing the Poisson operad

Abstract

A new Hopf operad Ram is introduced, which contains both the well-known Poisson operad and the Bessel operad introduced previously by the author. Besides, a structure of cooperad R is introduced on a collection of algebras given by generators and relations which have some similarity with the Arnold relations for the cohomology of the type A hyperplane arrangement. A map from the operad Ram to the dual operad of R is defined which we conjecture to be a isomorphism. AMS Classification 18D50; 16W30Keywords Hopf operad, coalgebra, chain complex IntroductionThe theory of operads has roots in algebraic topology. One well-known way to build algebraic operads is to start from an operad of topological spaces and apply the homology functor. A famous example, due to Cohen [4,5], is given by the little discs operad whose homology is the Gerstenhaber operad. The operads defined in this way inherit more structure from the diagonal of topological spaces: they are in fact Hopf operads. This phenomenon is similar to the existence of a bialgebra structure on the homology of a topological monoid.This article introduces two algebraic objects. The first one is a Hopf operad called the Ramanujan operad and denoted by Ram, which contains both the well-known Poisson operad and the Bessel operad introduced in [2]. The second one is a Hopf cooperad R, which means that the space R(I) associated to a finite set I is an associative algebra and the cocomposition maps are morphisms of algebras.The operad Ram is conjectured to be isomorphic to the linear dual operad R * of the cooperad R. A morphism of operad from Ram to R * is defined, which should give the desired isomorphism.

Gröbner bases for operads

“…Define the bigraded characters of its components, adding up dimensions of the homogeneous part of degree (i, j) multiplied by x i y j−i . The following result conjectured by Chapoton [4] can be deduced from the explicit description of the operad LieGriess above, together with the functional equation for dimensions and characters of an operad and its Koszul dual [13]; details are left to the reader. Corollary 5.…”

Higher structures, quantum groups and genus zero modular operad

“…Each can be represented as the homology ring of the Fulton–MacPherson compactification of the space of configurations of points in . A few other quadratic operads such that components of their dual cooperads are quadratic algebras are studied in . Finally, a vast generalisation and new properties of the picture represented in this article are developed in the paper .…”

A generalization of the Ramanujan polynomials and plane trees

“…The notion of half-mobile trees was introduced by Chapoton [4]. A half-mobile tree on [n] is defined to be a rooted tree with two kinds vertices, called labeled and unlabeled (or white and black, respectively) vertices satisfying the following conditions:…”

“…For example, for the plane tree T in Figure 1 the set of elder vertices is {3, 8, 9, 11, 12, 13} and the set of the improper edges is {(3, 14), (4, 1), (6,5), (10,4), (14, 2), (14, 7)}.…”