论文标题
曲面形结构
Tridendriform structures
论文作者
论文摘要
我们首先研究Tridendriform代数的张量产物，以引入Tridendriform Bialgebra的概念。我们将需要这是增强的Tridendriform代数的概念。受J-L工作的启发。 Loday和M. Ronco，我们在还原的树木上建立了自由的Tridendriform代数，并表明它们具有与Tridendriform产品的某些兼容性相关性。这样的对象将称为a（3，1）-dendriform代数。在研究自由（3，1） - 在一个发电机上的树突状双ggebra时，我们以组合方式描述了它的产物和相关。该产品由分支机构洗牌和可允许的切割描述。我们将其与单词上的准剃须刀代数进行了比较。它的分级双重的是N. Bergeron和Al引入的双重tsym，它是通过树木的闪亮分裂描述的。结果，这表明tsym具有（1，3） - 树突状的双gge骨结构。这意味着可以分为三个部分，并具有方便的兼容性。可以将其扩展到（3，1） - bialgebras，而不是任意数量的发电机。最后，我们介绍了（3，2）的概念 - 树突状的双gebra。这是一个HOPF代数，我们可以在其中分为三片，将产品分为两部分，并与HOPF的兼容性分为两部分。我们举了一个示例，说明具有一个发电机的自由（3，1） - 露出的双ggebra构建的代数。我们描述并生成了其代码原始原始素，并通过L. foissy的作品来计算其共同点的基础。我们通过表明（3，2） - 露出的双齿bra的商是Loday-Ronco Bialgebra来结束本文。
We first study tensor products of tridendriform algebras in order to introduce the notion of tridendriform bialgebra. We shall need for this a notion of augmented tridendriform algebras. Inspired by the work of J-L. Loday and M. Ronco, we build free tridendriform algebras over reduced trees and show that they have a coproduct satisfying some compatibilities with the tridendriform products. Such an object will be called a (3, 1)--dendriform algebra. Studying the free (3, 1)--dendriform bialgebra over one generator, we describe its products and coproduct in a combinatorial way. The products are described by branches shuffle and the coproduct by admissible cuts. We compare it with quasi-shuffle algebras over words. Its graded dual is the bialgebra TSym introduced by N. Bergeron and al which is described by the lightening splitting of a tree. As a consequence, this shows that TSym has a (1, 3)--dendriform bialgebra structure. This means that its coproduct can be split in three parts with convenient compatibilities. This can be extended to (3, 1)-bialgebras over an arbitrary number of generators. Finally, we introduce the notion of (3, 2)--dendriform bialgebra. This is a Hopf algebra, where we can split the product in three pieces and the coproduct in two with Hopf compatibilities. We give an example of such an algebra built on the free (3, 1)-dendriform bialgebra with one generator. We describe and generate its codendriform primitives and count its coassociative primitives thanks to L. Foissy's work. We end this paper by showing that a quotient of this (3, 2)-dendriform bialgebra is the Loday-Ronco bialgebra.
