Let $k$ be a perfect field of positive characteristic, let $\alpha $ be the additive group and let $W$ be the Witt vector ring. We compute the group algebra and the group $W$-algebra of $\alpha $ in the sense of affine schemes. The calculation is next extended to the characteristic-free case. We so obtain a $\mathbf W$-algebra with fractional divided powers, where $\mathbf W$ denote the ring of the big Witt vectors.