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arXiv · 1906.02834

A simplicial complex spliting associativity

Daniel López·Louis-François Préville-Ratelle·Marí a Ronco
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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 ${\bold P}=\{P_n\}_{n\geq 1}$ we show that, under certain conditions, the vector space spanned by the set of $m$-simpleces of ${\bold P}$ 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.

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Daniel López, Louis-François Préville-Ratelle, Marí a Ronco. 2019-06-06. A simplicial complex spliting associativity. https://arxiv.org/abs/1906.02834

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