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# Homology, Homotopy and Applications

## Volume 7 (2005)

### Number 2

### Proceedings of a Special Session of a Joint RSME-AMS Meeting at Sevilla University

### The biderivative and $A_{\infty}$-bialgebras

Pages: 161 – 177

DOI: http://dx.doi.org/10.4310/HHA.2005.v7.n2.a9

#### Authors

#### Abstract

An $A_{\infty}$-bialgebra is a DGM $H$ equipped with structurally compatible operations $\{ \omega^{j,i} : H^{\otimes i} \rightarrow H^{\otimes j} \}$ such that $(H, \omega^{1,i})$ is an $A_{\infty}$-algebra and $(H, \omega^{j,1})$ is an $A_{\infty}$-coalgebra. Structural compatibility is controlled by the biderivative operator $Bd$, defined in terms of two kinds of cup products on certain cochain algebras of permutahedra over the universal PROP $U=\mathrm{End}(TH)$.

#### Keywords

$A_{\infty}$-algebra, $A_{\infty}$-coalgebra, biderivative, Hopf algebra, permutahedron, universal PROP

#### 2010 Mathematics Subject Classification

Primary 55P35, 55P99. Secondary 18D50.