The Deformation Complex For DG Hopf Algebras

Let H be a differential graded Hopf algebra over a field k. This paper gives an explicit construction of a triple cochain complex that defines the Hochschild-Cartier cohomology of H. A certain truncation of this complex is the appropriate setting for deforming H as an H(q)-structure. The direct limit of all such truncations is the appropriate setting for deforming H as a strongly homotopy associative structure. Sign complications are systematically controlled. The connection between rational perturbation theory and the deformation theory of certain free commutative differential graded algebras is clarified.