Smooth affine group schemes over the dual numbers

We provide an equivalence between the category of affine, smooth group schemes over the ring of generalized dual numbers $k[I]$, and the category of extensions of the form $1 \rightarrow $\text{Lie}$(G, I) \rightarrow E \rightarrow G \rightarrow 1$ where G is an affine, smooth group scheme over k. Here k is an arbitrary commutative ring and $k[I]=k\oplus I$ with $I^2=0$. The equivalence is given by Weil restriction, and we provide a quasi-inverse which we call Weil extension. It is compatible with the exact structures and the ${\mathbb{O}}_k$-module stack structures on both categories. Our constructions rely on the use of the group algebra scheme of an affine group scheme; we introduce this object and establish its main properties. As an application, we establish a Dieudonn'e classification for smooth, commutative, unipotent group schemes over $k[I]$.