Let $V$ be a free $\mathbb{Z}$-module of rank $n$, and let $F$ be the functor associating to a ring $R$ the group $Aut_R(V \otimes R)$. If $V$ was finite, say $Z^n$, this functor is representable by the algebra $k[x_{11}, \ldots, x_{nn}, 1/det]$. What about when $V$ has infinite rank? For example, $V = \oplus_{n = 1}^{\infty} \mathbb{Z}$.
Can automorphisms be described by a determinant-like condition on the infinite matrices? Is the functor even representable? If so, what does the Hopf-algebra look like?