Higher Order Extensions of DG Hopf Algebras

The non-vanishing operations ω^n,m in an A_∞-bialgebra of order rhave total arity m+n≤ r and satisfy the A_∞-bialgebra structure relations for m+n≤ r+1. Quite remarkably, classical deformation theory extends a DG Hopf algebra ( H,d,μ,Δ) to an A_∞-bialgebra structure of order 4, and isomorphism classes of all such extensions are parametrized by the Gerstenhaber-Schack cohomology group H_GS^2( H;H). As an application, we extend the loop cohomology of a particular space to a non-trivial A_∞-bialgebra of order 4.