Adelic and Rational Grassmannians for finite dimensional algebras

We develop a theory of Wilson’s adelic Grassmannian ${\mathrm{G}\mathrm{r}}^{\mathrm{a}\mathrm{d}}(R)$ and Segal-Wilson’s rational Grasssmannian ${\mathrm{G}\mathrm{r}}^{\mathrm{r}\mathrm{a}\mathrm{t}}(R)$ associated to an arbitrary finite dimensional complex algebra $R$. We provide several equivalent descriptions of the former in terms of the indecomposable projective modules of $R$ and its primitive idempotents, and prove that it classifies the bispectral Darboux transformations of the $R$-valued exponential function. The rational Grasssmannian ${\mathrm{G}\mathrm{r}}^{\mathrm{r}\mathrm{a}\mathrm{t}}(R)$ is defined by using certain free submodules of $R(z)$ and it is proved that it can be alternatively defined via Wilson type conditions imposed in a representation theoretic settings. A canonical embedding ${\mathrm{G}\mathrm{r}}^{\mathrm{a}\mathrm{d}}(R)\hookrightarrow {\mathrm{G}\mathrm{r}}^{\mathrm{r}\mathrm{a}\mathrm{t}}(R)$ is constructed based on a perfect pairing between the $R$-bimodule of quasiexponentials with values in $R$ and the $R$-bimodule $R[z]$.