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Lie group extensions associated to projective modules of continuous inverse algebras. (English) Zbl 1212.22009
Summary: We call a unital locally convex algebra \(A\) a continuous inverse algebra if its unit group \(A^{\times }\) is open and inversion is a continuous map. For any smooth action of a, possibly infinite-dimensional, connected Lie group \(G\) on a continuous inverse algebra \(A\) by automorphisms and any finitely generated projective right \(A\)-module \(E\), we construct a Lie group extension \(\widehat G\) of \(G\) by the group \(\mathrm {GL}_A(E)\) of automorphisms of the \(A\)-module \(E\). This Lie group extension is a “non-commutative” version of the group \(\mathrm{Aut}({\mathbb V})\) of automorphisms of a vector bundle over a compact manifold \(M\), which arises for \(G = \mathrm{Diff}(M)\), \(A = C^{\infty }(M,{\mathbb C})\) and \(E = \Gamma {\mathbb V}\). We also identify the Lie algebra \(\widehat {\mathfrak g}\) of \(\widehat G\) and explain how it is related to connections of the \(A\)-module \(E\).

MSC:
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
58B34 Noncommutative geometry (à la Connes)
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