Higher Order Extensions of DG Hopf Algebras

The non-vanishing operations $\omega^{n,m}$ in an $A_{\infty}$-bialgebra of order $r\ $have total arity $m+n\leq r$ and satisfy the $A_{\infty}$-bialgebra structure relations for $m+n\leq r+1.$ Quite remarkably, classical deformation theory extends a DG Hopf algebra $\left( H,d,\mu,\Delta\right) $ to an $A_{\infty}$-bialgebra structure of order $4,$ and isomorphism classes of all such extensions are parametrized by the Gerstenhaber-Schack cohomology group $H_{GS}^{2}\left( H;H\right) $. As an application, we extend the loop cohomology of a particular space to a non-trivial $A_{\infty}$-bialgebra of order $4$. ; Comment: 13 pages; 1 figure