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


Samson Saneblidze (A. Razmadze Mathematical Institute, Georgian Academy of Sciences, Tbilisi, Republic of Georgia)

Ronald Umble (Department of Mathematics, Millersville University of Pennsylvania, Millersville, Penn., U.S.A.)


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 $U=\mathrm{End}(TH)$.


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

2010 Mathematics Subject Classification

Primary 55P35, 55P99. Secondary 18D50.

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