Connections on \(\mathbb{A}\)-frame bundles (Q2762667)

Let \(\mathbb{A}\) be a topological algebra. The author studies relations between \(\mathbb{A}\)-connections on \(\mathbb{A}\)-bundles \(\ell\) and connections on the frame bundle \(F(\ell)\). Namely, if \(\mathbb{A}\) is a \(Q\)-algebra, then \(\mathbb{A}\)-connections on \(\ell\) correspond bijectively to connections on \(F(\ell)\). If \(\mathbb{A}\) is not a \(Q\)-algebra then \(F(\ell)\) is only a topological principal bundle and \(\mathbb{A}\)-connections on \(\ell\) correspond bijectively to sheaf-theoretic objects which can be called connections on the sheaf of germs of sections on \(F(\ell)\).