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.