Construction of simple non-weight sl(2)-modules of arbitrary rank

We study simple non-weight ${\mathfrak{sl}}(2)$-modules which are finitely generated as ${\mathbb C}[z]$-modules. We show that they are described in terms of semilinear endomorphisms and prove that the Smith type induces a stratification on the set of these ${\mathfrak{sl}}(2)$-modules, providing thus new invariants. Moreover, we show that there is a notion of duality for these type of ${\mathfrak{sl}}(2)$-modules. Finally, we show that there are simple non-weight ${\mathfrak{sl}}(2)$-modules of arbitrary rank by constructing a whole new family of them.