| Title:
|
Lie group extensions associated to projective modules of continuous inverse algebras (English) |
| Author:
|
Neeb, Karl-Hermann |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
44 |
| Issue:
|
5 |
| Year:
|
2008 |
| Pages:
|
465-489 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $\operatorname{GL}_A(E)$ 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 $E$. (English) |
| Keyword:
|
continuous inverse algebra |
| Keyword:
|
infinite dimensional Lie group |
| Keyword:
|
vector bundle |
| Keyword:
|
projective module |
| Keyword:
|
semilinear automorphism |
| Keyword:
|
covariant derivative |
| Keyword:
|
connection |
| MSC:
|
22E65 |
| MSC:
|
58B34 |
| idZBL:
|
Zbl 1212.22009 |
| idMR:
|
MR2501579 |
| . |
| Date available:
|
2009-01-29T09:16:19Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127115 |
| . |
| Reference:
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