Gerstenhaber-Schack Bialgebras

A Gerstenhaber-Schack (G-S) bialgebra consists of a graded Hopf algebra $H$ together with multilinear operations $\{{\omega }_3^1,{\omega }_2^2,{\omega }_1^3\}\subset \{Ho{m}^{-1}(H^{\otimes m},H^{\otimes n}):m+n=4\},$ whose sum is the degree $-1$ component of a $2$-cocycle in the G-S complex of $H$. A G-S extension of a graded Hopf algebra $H$ is a G-S bialgebra containing $H$. G-S extensions of $H$ are classified up to isomorphism by the degree $-1$ component of the G-S cohomology group $H_{GS}^2(H;H)$. We exhibit a space $X$ and a non-trivial topologically induced G-S bialgebra structure on $H^{\ast }(\mathrm{\Omega }X;{\mathbb{Z}}_2)\mathrm{.}$