Lie group extensions associated to projective modules of continuous inverse algebras

Karl-Hermann Neeb

Address: Mathematics Department Darmstadt University of Technology Schlossgartenstrasse 7, 64289 Darmstadt, Germany


Abstract: 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 $\operatorname{GL}_A(E)$ of automorphisms of the $A$-module $E$. This Lie group extension is a “non-commutative” version of the group $\operatorname{Aut}({\mathbb{V}})$ of automorphism of a vector bundle over a compact manifold $M$, which arises for $G = \operatorname{Diff}(M)$, $A = C^\infty (M,{\mathbb{C}})$ 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 $E$.

AMSclassification: primary 22E65; secondary 58B34.

Keywords: continuous inverse algebra, infinite dimensional Lie group, vector bundle, projective module, semilinear automorphism, covariant derivative, connection.