Adelic Loop Groups

We define the adelic sphere \(\mathbb{C}P^1_\mathbb{Q}\), which is the adelic version of the Riemann sphere, and it is a pro-algebraic Riemann surface; topologically, it is the suspension of the adelic one-dimensional solenoid \(A/\mathbb{Q}\) (which in turn is a fiber bundle over the circle, with fiber a Cantor group and it is the Pontrjagin dual of the additive rationals). One can do complex analysis on this object and define holomorphic vector bundles on it. We introduce the adelic loop group of a Lie group \(G\), which is the space of maps from the adele class group \(A/\mathbb{Q}\) to \(G\); we describe their properties and prove Birkhoff’s factorization for these groups. We sketch the proof that the adelic Picard group of holomorphic line bundles over \(\mathbb{C}P^1_\mathbb{Q}\)  is isomorphic to the additive rationals \((\mathbb{Q},+)\), and prove the Birkhoff-Grothendieck splitting theorem for holomorphic bundles of higher rank over \(\mathbb{C}P^1_\mathbb{Q}\), as sums of line bundles.