The Galois group of the category of mixed Hodge-Tate structures

The category of mixed Hodge-Tate structures over $\mathbb{Q}$ is a mixed Tate category of homological dimension one. By Tannakian formalism, it is equivalent to the category of graded comodules of a commutative graded Hopf algebra. In my recent joint work with A. Goncharov, we give a canonical description $\mathcal{A}_\bullet(\mathbb{C})$ of the Hopf algebra. Such construction can be generalized to $\mathcal{A}_\bullet(R)$ for any dg-algebra $R$ over a field $k$ with a Tate line $k(1)$. As an example, $\mathcal{A}_\bullet(\Omega_X^\bullet)$ gives a dg-model for the derived category of variations of mixed Hodge-Tate structures.