Lie group extensions associated to projective modules of continuous inverse algebras

Karl-Hermann Neeb

Archivum Mathematicum (2008)

  • Volume: 044, Issue: 5, page 465-489
  • ISSN: 0044-8753

Abstract

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We call a unital locally convex algebra A a continuous inverse algebra if its unit group A × 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 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 ∞ ( M , ℂ ) and E = Γ 𝕍 . We also identify the Lie algebra 𝔤 ^ of G ^ and explain how it is related to connections of the A -module E .

How to cite

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Neeb, Karl-Hermann. "Lie group extensions associated to projective modules of continuous inverse algebras." Archivum Mathematicum 044.5 (2008): 465-489. <http://eudml.org/doc/250505>.

@article{Neeb2008,
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 $E$.},
author = {Neeb, Karl-Hermann},
journal = {Archivum Mathematicum},
keywords = {continuous inverse algebra; infinite dimensional Lie group; vector bundle; projective module; semilinear automorphism; covariant derivative; connection; continuous inverse algebra; infinite dimensional Lie group; vector bundle; projective module; semilinear automorphism; covariant derivative; connection},
language = {eng},
number = {5},
pages = {465-489},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Lie group extensions associated to projective modules of continuous inverse algebras},
url = {http://eudml.org/doc/250505},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Neeb, Karl-Hermann
TI - Lie group extensions associated to projective modules of continuous inverse algebras
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 5
SP - 465
EP - 489
AB - 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 $E$.
LA - eng
KW - continuous inverse algebra; infinite dimensional Lie group; vector bundle; projective module; semilinear automorphism; covariant derivative; connection; continuous inverse algebra; infinite dimensional Lie group; vector bundle; projective module; semilinear automorphism; covariant derivative; connection
UR - http://eudml.org/doc/250505
ER -

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