The motivic Galois group for a double zeta value

We consider multiple zeta values, which are periods of mixed Tate motives over $\mathbb Z$. For a given multiple zeta value $\zeta$, there exists a unique minimal motive $M(\zeta)$ such that $\zeta$ is a period of $M(\zeta)$. In general, the motive $M(\zeta)$ is difficult to compute. In this article, we compute the minimal motive $M(a,b)$ associated to a given double zeta value $\zeta(a,b)$. We also compute the motivic Galois group $G(a,b)$ associated to $\zeta(a,b)$ and discuss its dimension. Moreover, we give a period matrix of $M(a,b)$. The period conjecture predicts that the dimension of $G(a,b)$ equals the transcendence degree of the algebra of periods of $M(a,b)$. Hence our results lead to conjectures about algebraic relations between single and double zeta values.