Uniformization of the orbifold of a finite reflection group
Identifieur interne : 000D68 ( Main/Exploration ); précédent : 000D67; suivant : 000D69Uniformization of the orbifold of a finite reflection group
Auteurs : Kyoji Saito [Japon]Source :
- Aspects of Mathematics [ 0179-2156 ] ; 2004.
Abstract
Abstract: Let W be a finite reflection group of a real vector space V. If W is crystallographic, then the quotient space V*//W appears in several contexts in geometry: i) in Lie theory as the quotient space of a simple Lie algebra by the adjoint Lie group action [Ch1,2] and ii) in complex geometry as the base space of the universal unfolding of a simple singularity [Br1]. Having these backgrounds, V*//W carries some distinguished geometric properties and structures, which, fortunately and also amusingly, can be described only in terms of the reflection group regardless whether W is crystallographic or not. We recall two of them: 1. The complexified regular orbit space (V*//W) C reg is a K(π, 1)-space (Brieskorn [Br3], Deligne [De]). In other words, π 1((V*//W) C reg is an Artin group (i.e. a generalized braid group [B-S][De]) and the universal covering space of (V*//W) C reg is contractible (cf. also [Sa]). 2. The quotient space V*//W carries a flat structure (Saito [S3][S6])1. This means roughly that the tangent bundle of V*//W carries a flat metric J together with some additional structures. Nowadays, a flat structure without a primitive form is also called a Frobenius manifold structure with gravitational descendent (Dubrovin [Du], Manin [Ma1,2]).
Url:
DOI: 10.1007/978-3-322-80236-1_11
Affiliations:
Links toward previous steps (curation, corpus...)
- to stream Istex, to step Corpus: 003055
- to stream Istex, to step Curation: 003055
- to stream Istex, to step Checkpoint: 000C87
- to stream Main, to step Merge: 000D79
- to stream Main, to step Curation: 000D68
Le document en format XML
<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Uniformization of the orbifold of a finite reflection group</title><author><name sortKey="Saito, Kyoji" sort="Saito, Kyoji" uniqKey="Saito K" first="Kyoji" last="Saito">Kyoji Saito</name></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:EC6114A7D6ACBBB9E7EE3C4B81EB8DD7065B0CD6</idno><date when="2004" year="2004">2004</date><idno type="doi">10.1007/978-3-322-80236-1_11</idno><idno type="url">https://api.istex.fr/document/EC6114A7D6ACBBB9E7EE3C4B81EB8DD7065B0CD6/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">003055</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">003055</idno><idno type="wicri:Area/Istex/Curation">003055</idno><idno type="wicri:Area/Istex/Checkpoint">000C87</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Checkpoint">000C87</idno><idno type="wicri:doubleKey">0179-2156:2004:Saito K:uniformization:of:the</idno><idno type="wicri:Area/Main/Merge">000D79</idno><idno type="wicri:Area/Main/Curation">000D68</idno><idno type="wicri:Area/Main/Exploration">000D68</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Uniformization of the orbifold of a finite reflection group</title><author><name sortKey="Saito, Kyoji" sort="Saito, Kyoji" uniqKey="Saito K" first="Kyoji" last="Saito">Kyoji Saito</name><affiliation wicri:level="4"><country xml:lang="fr">Japon</country><wicri:regionArea>RIMS, Kyoto University</wicri:regionArea><placeName><settlement type="city">Kyoto</settlement><region type="prefecture">Région du Kansai</region></placeName><orgName type="university">Université de Kyoto</orgName></affiliation></author></analytic><monogr></monogr><series><title level="s">Aspects of Mathematics</title><imprint><date>2004</date></imprint><idno type="ISSN">0179-2156</idno><idno type="ISSN">0179-2156</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0179-2156</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: Let W be a finite reflection group of a real vector space V. If W is crystallographic, then the quotient space V*//W appears in several contexts in geometry: i) in Lie theory as the quotient space of a simple Lie algebra by the adjoint Lie group action [Ch1,2] and ii) in complex geometry as the base space of the universal unfolding of a simple singularity [Br1]. Having these backgrounds, V*//W carries some distinguished geometric properties and structures, which, fortunately and also amusingly, can be described only in terms of the reflection group regardless whether W is crystallographic or not. We recall two of them: 1. The complexified regular orbit space (V*//W) C reg is a K(π, 1)-space (Brieskorn [Br3], Deligne [De]). In other words, π 1((V*//W) C reg is an Artin group (i.e. a generalized braid group [B-S][De]) and the universal covering space of (V*//W) C reg is contractible (cf. also [Sa]). 2. The quotient space V*//W carries a flat structure (Saito [S3][S6])1. This means roughly that the tangent bundle of V*//W carries a flat metric J together with some additional structures. Nowadays, a flat structure without a primitive form is also called a Frobenius manifold structure with gravitational descendent (Dubrovin [Du], Manin [Ma1,2]).</div></front></TEI><affiliations><list><country><li>Japon</li></country><region><li>Région du Kansai</li></region><settlement><li>Kyoto</li></settlement><orgName><li>Université de Kyoto</li></orgName></list><tree><country name="Japon"><region name="Région du Kansai"><name sortKey="Saito, Kyoji" sort="Saito, Kyoji" uniqKey="Saito K" first="Kyoji" last="Saito">Kyoji Saito</name></region></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Wicri/Mathematiques/explor/BourbakiV1/Data/Main/Exploration
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 000D68 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Main/Exploration/biblio.hfd -nk 000D68 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien
|wiki= Wicri/Mathematiques
|area= BourbakiV1
|flux= Main
|étape= Exploration
|type= RBID
|clé= ISTEX:EC6114A7D6ACBBB9E7EE3C4B81EB8DD7065B0CD6
|texte= Uniformization of the orbifold of a finite reflection group
}}
| This area was generated with Dilib version V0.6.33. |  |