Infinitesimal or Cocommutative Dipterous Bialgebras and Good Triples of Operads

The works of Poincaré, Birkhoff, Witt and Cartier, Milnor, Moore on the connected cocommutative Hopf algebras translated in the language of operads means that the triple of operads (Com,As, Lie) endowed with the Hopf compatiblity relation is good. In this paper, we focus on left dipterous (resp. right dipterous) algebras which are associative algebras with an extra left (resp. right) module on themselves and look for good triples were As is replaced by the dipterous operad Dipt. Since the work of Loday and Ronco, the triple of operads (As,Dipt, B∞) endowed with the semi-Hopf compatibility relations is known to be good. In this paper, we prove that the triple of operads (As,Dipt, Grove) endowed with the so-called nonunital semi-infinitesimal compatibility relations is good. For that, explicit constructions of the free dipterous algebra and the free grove-algebra over a K-vector space V are given. These constructions turn out to be related to rooted planar trees and the little an large Schroeder numbers. Many examples of dipterous algebras are given, notably the free L-dipterous algebras (closely related to duplicial/triplicial algebras), whose motivation comes from left Baxter-Rota operators (average operators) and from language theory. We also open this paper on a good triple, related to the Connes-Kreimer Hopf algebra in quantum field theory, (Com,Dipt, P rimCom Dipt) endowed with the Hopf compatibility relations and also present a general theorem giving good triples from entangled dipterous like operads named associative molecules. Notation: In the sequel K is a field and Σn is the group of permutation over n elements. If A is an operad, then the K-vector space of n-ary operations is denoted as usual by A(n). Recall that if A is regular, then A(n) := An⊗KΣn, where An is the K-vector space of n-ary operations without permutations of the entries. We adopt Sweedler notation for binary cooperation ∆ on a K-vector space V and set ∆(x) = x(1) ⊗ x(2). Left dipterous algebras in the sequel will be just abbreviated as dipterous algebras. 1 2000 Mathematics Subject Classification: 16D99, 05E99, 16W30, 17A30, 18D50.