An explicit construction of indecomposable vector bundles over an elliptic curve

In the celebrated paper “Vector bundles over an elliptic curve,” M. Atiyah classifies indecomposable vector bundles, namely he provides a bijection \(\alpha_{(r, d)}\) between indecomposable bundles of arbitrary rank \(r\) and degree \(d\) (denoted by \(\mathcal{E}_{(r, d)}\)) and \(\mathcal{E}_{(h, 0)}\) (where \(h = \text{gcd}(d, r)\)). The latter is described explicitly: there is a distinguished element \(F_r \in \mathcal{E}_{(r, 0)}\) such that for any other …