Structure theorems for dendriform and tridendriform algebras
With the help of a method discovered by the same authors in a preceding paper [``Rigidity theorem, freeness of algebras and applications'', Preprint, \url{arXiv:1701.01323}], several rigidity theorems are proved and then used to show the freeness of several well-known combinatorial Hopf algebras as a dendriform or tridendriform algebras. In all these cases, the considered objects are given a coalgebraic and an algebraic structure on two operads, with a certain compatibility between them; it is proved that, under a connectivity condition, such an object is free and cofree on the corresponding operads. In the context of planar binary trees, skew duplicial algebras are defined as a sort of deformation of duplicial algebras. The pairs of operads with a rigidity theorem are duplicial-dendriform, dendriform-dendriform, skew duplicial-dendriform, skew duplicial-duplicial and skew duplicial-skew duplicial. This is used to give a new proof that the Hopf algebras of surjections and of parking functions are free as dendriform algebras. In the context of planar reduced trees, the notion of terplicial algebra is introduced by analogy with skew duplicial algebras. It differs from the notion of triduplicial algebra defined by Novelli and Thibon. The pairs of operads with a rigidity theorem are terplicial-terplicial, terplical-tridendriform and tridendrifom-tridendriform. This is used to show that the Solomon-Tits is a free tridendriform algebra. A combinatorial description of the products and the coproducts induced by these structures on planar binary trees and on reduced trees is given, using paths and cuts. For the entire collection see [Zbl 1437.05003].
scientific article; zbMATH DE number 7219056
scientific article