A simplicial complex spliting associativity

Abstract

We introduce a simplicial object $(\{ \Dy^m\}_{m\geq 0}, {\mathbb F}_i, {\mathbb S}_j)$ in the category of non-symmetric algebraic operads, satisfying that $\Dy^0$ is the operad of associative algebras and $\Dy^1$ is J.-L. Loday\rq s operad of dendriform algebras. The dimensions of the operad $\Dy^m$ are given by the Fuss-Catalan numbers. Given a family of partially ordered sets $\mathbf{P}=\{P_n{\}}_{n\ge 1}$ we show that, under certain conditions, the vector space spanned by the set of $m$-simpleces of $\mathbf{P}$ is a $\Dy^m$ algebra. This construction, applied to certain combinatorial Hopf algebras, whose associative product comes from a dendriform structure, provides examples of $\Dy^m$ algebras.

arXiv commentThe manuscript contains some basic constructions of Algebraic structures defined on m-Dyck paths, arxiv:1508.01252; with two sections containing new results

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• The representation of the symmetric group on m-Tamari intervals
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