Lie group extensions associated to projective modules of continuous inverse algebras

TL;DR

This paper constructs Lie group extensions from actions of Lie groups on continuous inverse algebras and projective modules, generalizing automorphism groups of vector bundles to a non-commutative setting.

Contribution

It introduces a new framework for Lie group extensions associated with projective modules over continuous inverse algebras, extending classical geometric concepts.

Findings

Constructed Lie group extensions for smooth actions on continuous inverse algebras.

Identified the Lie algebra of the extension and related it to module connections.

Provided a non-commutative analogue of automorphism groups of vector bundles.

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 $E$.