Diagonals on the permutahedra, multiplihedra and associahedra

We construct an explicit diagonal $\Delta_{P}$ on the permutahedra $P.$ Related diagonals on the multiplihedra $J$ and the associahedra $K$ are induced by Tonks’ projection $P\rightarrow K$ \cite{tonks} and its factorization through $J.$ We introduce the notion of a permutahedral set $% \mathcal{Z}$ and lift $\Delta_{P}$ to a diagonal on $\mathcal{Z}$. We show that the double cobar construction $\Omega^{2}C_{\ast}(X)$ is a permutahedral set; consequently $\Delta_{P}$ lifts to a diagonal on $% \Omega^{2}C_{\ast}(X)$. Finally, we apply the diagonal on $K$ to define the tensor product of $A_{\infty}$-(co)algebras in maximal generality.

Keywords

diagonal, permutahedron, multiplihedron, associahedron

2010 Mathematics Subject Classification

Primary 05A18, 05A19, 52B05, 55U05. Secondary 55P35.

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Published 1 January 2004