Article

# Lie group extensions associated to projective modules of continuous inverse algebras

03/2008;
Source: arXiv

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 $\hat G$ of $G$ by the group $\GL_A(E)$ of automorphisms of the $A$-module $E$. This Lie group extension is a non-commutative'' version of the group $\Aut(\V)$ of automorphism of a vector bundle over a compact manifold $M$, which arises for $G = \Diff(M)$, $A = C^\infty(M,\C)$ and $E = \Gamma\V$. We also identify the Lie algebra $\hat\g$ of $\hat G$ and explain how it is related to connections of the $A$-module $E$.

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27 Jan 2013

### Keywords

non-commutative'' version

automorphisms

compact manifold $M$

continuous inverse algebra

continuous inverse algebra $A$

continuous map

inversion

Lie group $G$

Lie group extension

Lie group extension $\hat G$

smooth action

unit group $A^\times$

unital