Let $A = \langle a\rangle$ be an infinite cyclic group on one generator. I'm trying to understand the completion $\widehat{\mathbb{Q}}A$ of the group algebra $\mathbb{Q}A$ with respect to the augmentation ideal $I = \ker(a \mapsto 1)$.
Can someone give an explicit (or just helpful) description of this ring?
One direction I have looked at is the following. The group ring $\mathbb{Q}A$ is the ring of Laurent polynomials $\mathbb{Q}[a,a^{-1}]$ and the ideal $I$ is generated by $1-a$ and $1-a^{-1}$. So perhaps the completion $\widehat{\mathbb{Q}[a,a^{-1}]}$ with respect to powers of $I$, is some sort of ring of power series in variables $1-a$ and $1-a^{-1}$...
Ideally I hope to understand the coalgebra structure of $\widehat{\mathbb{Q}}A$ as well, in particular I'd like to determine the Lie algebra of primitive elements (when viewing $\widehat{\mathbb{Q}}A$ as a complete Hopf algebra).