Nonholonomic Riemann and Weyl tensors for flag manifolds
Identifieur interne : 000584 ( Istex/Corpus ); précédent : 000583; suivant : 000585Nonholonomic Riemann and Weyl tensors for flag manifolds
Auteurs : P. Ya. Grozman ; D. A. LeitesSource :
- Theoretical and Mathematical Physics [ 0040-5779 ] ; 2007-11-01.
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Abstract
Abstract: On any manifold, any nondegenerate symmetric 2-form (metric) and any nondegenerate skew-symmetric differential form ω can be reduced to a canonical form at any point but not in any neighborhood: the corresponding obstructions are the Riemannian tensor and dω. The obstructions to flatness (to reducibility to a canonical form) are well known for any G-structure, not only for Riemannian or almost symplectic structures. For a manifold with a nonholonomic structure (nonintegrable distribution), the general notions of flatness and obstructions to it, although of huge interest (e.g., in supergravity) were not known until recently, although particular cases have been known for more than a century (e.g., any contact structure is nonholonomically “flat”: it can always be reduced locally to a canonical form). We give a general definition of the nonholonomic analogues of the Riemann tensor and its conformally invariant analogue, the Weyl tensor, in terms of Lie algebra cohomology and quote Premet’s theorems describing these cohomologies. Using Premet’s theorems and the SuperLie package, we calculate the tensors for flag manifolds associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases) and also compute the obstructions to flatness of the G(2)-structure and its nonholonomic superanalogue.
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DOI: 10.1007/s11232-007-0131-z
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The obstructions to flatness (to reducibility to a canonical form) are well known for any G-structure, not only for Riemannian or almost symplectic structures. For a manifold with a nonholonomic structure (nonintegrable distribution), the general notions of flatness and obstructions to it, although of huge interest (e.g., in supergravity) were not known until recently, although particular cases have been known for more than a century (e.g., any contact structure is nonholonomically “flat”: it can always be reduced locally to a canonical form). We give a general definition of the nonholonomic analogues of the Riemann tensor and its conformally invariant analogue, the Weyl tensor, in terms of Lie algebra cohomology and quote Premet’s theorems describing these cohomologies. Using Premet’s theorems and the SuperLie package, we calculate the tensors for flag manifolds associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases) and also compute the obstructions to flatness of the G(2)-structure and its nonholonomic superanalogue.</div></front></TEI><istex><corpusName>springer-journals</corpusName><author><json:item><name>P. Ya. Grozman</name><affiliations><json:string>Equa Simulation AB, Stockholm, Sweden</json:string><json:string>E-mail: pavel@rixetele.com</json:string></affiliations></json:item><json:item><name>D. A. 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The obstructions to flatness (to reducibility to a canonical form) are well known for any G-structure, not only for Riemannian or almost symplectic structures. For a manifold with a nonholonomic structure (nonintegrable distribution), the general notions of flatness and obstructions to it, although of huge interest (e.g., in supergravity) were not known until recently, although particular cases have been known for more than a century (e.g., any contact structure is nonholonomically “flat”: it can always be reduced locally to a canonical form). We give a general definition of the nonholonomic analogues of the Riemann tensor and its conformally invariant analogue, the Weyl tensor, in terms of Lie algebra cohomology and quote Premet’s theorems describing these cohomologies. Using Premet’s theorems and the SuperLie package, we calculate the tensors for flag manifolds associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases) and also compute the obstructions to flatness of the G(2)-structure and its nonholonomic superanalogue.</p></abstract><textClass xml:lang="en"><keywords scheme="keyword"><list><head>Keywords</head><item><term>Lie algebra cohomology</term></item><item><term>Cartan prolongation</term></item><item><term>Riemann tensor</term></item><item><term>nonholonomic manifold</term></item><item><term>flag manifold</term></item><item><term>G 2-structure</term></item></list></keywords></textClass><textClass><keywords scheme="Journal-Subject-Group"><list><label>P</label><label>P19005</label><label>M13003</label><item><term>Physics</term></item><item><term>Mathematical and Computational Physics</term></item><item><term>Applications of 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DisplayOrder="Western"><GivenName>P.</GivenName><GivenName>Ya.</GivenName><FamilyName>Grozman</FamilyName></AuthorName><Contact><Email>pavel@rixetele.com</Email></Contact></Author><Author AffiliationIDS="Aff2 Aff3"><AuthorName DisplayOrder="Western"><GivenName>D.</GivenName><GivenName>A.</GivenName><FamilyName>Leites</FamilyName></AuthorName><Contact><Email>mleites@math.su.se</Email></Contact></Author><Affiliation ID="Aff1"><OrgName>Equa Simulation AB</OrgName><OrgAddress><City>Stockholm</City><Country Code="SE">Sweden</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgName>Max-Planck-Institut fur Mathematik in den Naturwissenschaften</OrgName><OrgAddress><City>Leipzig</City><Country Code="DE">Germany</Country></OrgAddress></Affiliation><Affiliation ID="Aff3"><OrgDivision>Department of Mathematics</OrgDivision><OrgName>University of Stockholm</OrgName><OrgAddress><City>Stockholm</City><Country Code="SE">Sweden</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>On any manifold, any nondegenerate symmetric 2-form (metric) and any nondegenerate skew-symmetric differential form ω can be reduced to a canonical form at any point but not in any neighborhood: the corresponding obstructions are the Riemannian tensor and dω. The obstructions to flatness (to reducibility to a canonical form) are well known for any G-structure, not only for Riemannian or almost symplectic structures. For a manifold with a nonholonomic structure (nonintegrable distribution), the general notions of flatness and obstructions to it, although of huge interest (e.g., in supergravity) were not known until recently, although particular cases have been known for more than a century (e.g., any contact structure is nonholonomically “flat”: it can always be reduced locally to a canonical form). We give a general definition of the nonholonomic analogues of the Riemann tensor and its conformally invariant analogue, the Weyl tensor, in terms of Lie algebra cohomology and quote Premet’s theorems describing these cohomologies. Using Premet’s theorems and the SuperLie package, we calculate the tensors for flag manifolds associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases) and also compute the obstructions to flatness of the G(2)-structure and its nonholonomic superanalogue.</Para></Abstract><KeywordGroup Language="En"><Heading>Keywords</Heading><Keyword>Lie algebra cohomology</Keyword><Keyword>Cartan prolongation</Keyword><Keyword>Riemann tensor</Keyword><Keyword>nonholonomic manifold</Keyword><Keyword>flag manifold</Keyword><Keyword><Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>-structure</Keyword></KeywordGroup><ArticleNote Type="Misc"><SimplePara>__________</SimplePara><SimplePara>Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 186–219, November, 2007.</SimplePara></ArticleNote></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Nonholonomic Riemann and Weyl tensors for flag manifolds</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Nonholonomic Riemann and Weyl tensors for flag manifolds</title></titleInfo><name type="personal" displayLabel="corresp"><namePart type="given">P.</namePart><namePart type="given">Ya.</namePart><namePart type="family">Grozman</namePart><affiliation>Equa Simulation AB, Stockholm, Sweden</affiliation><affiliation>E-mail: pavel@rixetele.com</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">D.</namePart><namePart type="given">A.</namePart><namePart type="family">Leites</namePart><affiliation>Max-Planck-Institut fur Mathematik in den Naturwissenschaften, Leipzig, Germany</affiliation><affiliation>Department of Mathematics, University of Stockholm, Stockholm, Sweden</affiliation><affiliation>E-mail: mleites@math.su.se</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Springer US</publisher><place><placeTerm type="text">Boston</placeTerm></place><dateCreated encoding="w3cdtf">2006-07-06</dateCreated><dateIssued encoding="w3cdtf">2007-11-01</dateIssued><dateIssued encoding="w3cdtf">2007</dateIssued><copyrightDate encoding="w3cdtf">2007</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: On any manifold, any nondegenerate symmetric 2-form (metric) and any nondegenerate skew-symmetric differential form ω can be reduced to a canonical form at any point but not in any neighborhood: the corresponding obstructions are the Riemannian tensor and dω. The obstructions to flatness (to reducibility to a canonical form) are well known for any G-structure, not only for Riemannian or almost symplectic structures. For a manifold with a nonholonomic structure (nonintegrable distribution), the general notions of flatness and obstructions to it, although of huge interest (e.g., in supergravity) were not known until recently, although particular cases have been known for more than a century (e.g., any contact structure is nonholonomically “flat”: it can always be reduced locally to a canonical form). We give a general definition of the nonholonomic analogues of the Riemann tensor and its conformally invariant analogue, the Weyl tensor, in terms of Lie algebra cohomology and quote Premet’s theorems describing these cohomologies. Using Premet’s theorems and the SuperLie package, we calculate the tensors for flag manifolds associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases) and also compute the obstructions to flatness of the G(2)-structure and its nonholonomic superanalogue.</abstract><subject lang="en"><genre>Keywords</genre><topic>Lie algebra cohomology</topic><topic>Cartan prolongation</topic><topic>Riemann tensor</topic><topic>nonholonomic manifold</topic><topic>flag manifold</topic><topic>G 2-structure</topic></subject><relatedItem type="host"><titleInfo><title>Theoretical and Mathematical Physics</title></titleInfo><titleInfo type="abbreviated"><title>Theor Math Phys</title></titleInfo><genre type="journal" displayLabel="Archive Journal" authority="ISTEX" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</genre><originInfo><publisher>Springer</publisher><dateIssued encoding="w3cdtf">2007-11-01</dateIssued><copyrightDate encoding="w3cdtf">2007</copyrightDate></originInfo><subject><genre>Journal-Subject-Group</genre><topic authority="SpringerSubjectCodes" authorityURI="P">Physics</topic><topic authority="SpringerSubjectCodes" authorityURI="P19005">Mathematical and Computational Physics</topic><topic authority="SpringerSubjectCodes" authorityURI="M13003">Applications of Mathematics</topic></subject><identifier type="ISSN">0040-5779</identifier><identifier type="eISSN">1573-9333</identifier><identifier type="JournalID">11232</identifier><identifier type="IssueArticleCount">6</identifier><identifier type="VolumeIssueCount">3</identifier><part><date>2007</date><detail type="volume"><number>153</number><caption>vol.</caption></detail><detail type="issue"><number>2</number><caption>no.</caption></detail><extent unit="pages"><start>1511</start><end>1538</end></extent></part><recordInfo><recordOrigin>Springer Science+Business Media, Inc., 2007</recordOrigin></recordInfo></relatedItem><identifier type="istex">1B7087E7EE757B38E903282CC347F8A7FD91C0C4</identifier><identifier type="ark">ark:/67375/VQC-680DH90F-W</identifier><identifier type="DOI">10.1007/s11232-007-0131-z</identifier><identifier type="ArticleID">131</identifier><identifier type="ArticleID">s11232-007-0131-z</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer Science+Business Media, Inc., 2007</accessCondition><recordInfo><recordContentSource authority="ISTEX" authorityURI="https://loaded-corpus.data.istex.fr" valueURI="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-3XSW68JL-F">springer</recordContentSource><recordOrigin>Springer Science+Business Media, Inc., 2007</recordOrigin></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/document/1B7087E7EE757B38E903282CC347F8A7FD91C0C4/metadata/json</uri></json:item></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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