  The category of rational mixed Hodge-Tate structures is a mixed Tate
category. So thanks to the Tannakian formalism, it is equivalent to the
category of finite dimensional graded comodules over a graded commutative Hopf
algebra H over Q. Since the category has homological dimension 1, the Hopf
algebra H is isomorphic to the commutative graded Hopf algebra provided by the
tensor algebra of the graded vector space given by the direct sum of the groups
C/Q(n) over n>0. However this isomorphism is not natural in any sense, e.g.
does not work in families.
  We give a natural explicit construction of the Hopf algebra H.
  Generalizing this, we define a Hopf dg-algebra related to any dg-algebra R
over a field k, equipped with an invertible line k(1). When R is the sheaf of
algebras given by the holomorphic de Rham complex of a complex manifold X, and
the line is Q(1), the related Hopf dg-algebra describes a dg-model of the
derived category of variations of Hodge-Tate structures on X. Its cobar complex
provides a dg-model for the rational Deligne cohomology of X.
  We consider a variant of our construction which starts from Fontaine's
crystalline / semi-stable period rings and produces graded / dg Hopf algebras,
which we relate to the p-adic Hodge theory.
