Homology, Homotopy and Applications

Volume 13 (2011)

Number 1

Matrads, biassociahedra, and $A_{\infty}$-bialgebras

Pages: 1 – 57

DOI: https://dx.doi.org/10.4310/HHA.2011.v13.n1.a2


Samson Saneblidze (Department of Geometry and Topology, A. Razmadze Mathematical Institute, Tbilisi, Republic of Georgia)

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


We introduce the notion of a matrad $M=\{M_{n,m}\}$ whose submodules $M_{*,1}$ and $M_{1,*}$ are non-$\Sigma$ operads. We define the free matrad $\mathcal{H}_\infty$ generated by a singleton $\theta^n_m$ in each bidegree $(m,n)$ and realize $\mathcal{H}_\infty$ as the cellular chains on a new family of polytopes $\{KK_{n,m}=KK_{m,n}\}$, called biassociahedra, of which $KK_{n,1}$ is the associahedron $K_n$. We construct the universal enveloping functor from matrads to PROPs and define an $A_\infty$-bialgebra as an algebra over $\mathcal{H}_\infty$.


$A_{\infty}$-bialgebra; operad; matrad; permutahedron; biassociahedron

2010 Mathematics Subject Classification

Primary 55P35, 55Pxx. Secondary 52B05.

Published 12 July 2011