The Relation Between Lie Groups and Lie Algebras
Identifieur interne : 001960 ( Istex/Corpus ); précédent : 001959; suivant : 001961The Relation Between Lie Groups and Lie Algebras
Auteurs : A. L. OnishchikSource :
- Encyclopaedia of Mathematical Sciences [ 0938-0396 ] ; 1993.
Abstract
Abstract: The basic method of the theory of Lie groups, which makes it possible to obtain deep results with striking simplicity, consists in reducing questions concerning Lie groups to certain problems of linear algebra. This is done by assigning to every Lie group G its “tangent algebra”g, which to a large extent determines the group G, and to every homomorphism f: G → H of Lie groups a homomorphism df: g → h) of their tangent algebras, which to a large extent determines the homomorphism f. In the language of category theory we have a functor from the category of Lie groups into the category of Lie algebras, whose properties are very close to those of an equivalence of categories. In honour of the founder of the theory of Lie groups we will call this functor (following M. M. Postnikov (1982)) the Lie functor.
Url:
DOI: 10.1007/978-3-642-57999-8_3
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This is done by assigning to every Lie group <Emphasis Type="Italic">G</Emphasis> its “tangent algebra”g, which to a large extent determines the group <Emphasis Type="Italic">G</Emphasis>, and to every homomorphism <Emphasis Type="Italic">f</Emphasis>: <Emphasis Type="Italic">G</Emphasis> → <Emphasis Type="Italic">H</Emphasis> of Lie groups a homomorphism <Emphasis Type="Italic">df</Emphasis>: g → h) of their tangent algebras, which to a large extent determines the homomorphism <Emphasis Type="Italic">f</Emphasis>. In the language of category theory we have a functor from the category of Lie groups into the category of Lie algebras, whose properties are very close to those of an equivalence of categories. In honour of the founder of the theory of Lie groups we will call this functor (following M. M. Postnikov (1982)) <Emphasis Type="Italic">the Lie functor.</Emphasis></Para></Abstract></ChapterHeader><NoBody></NoBody></Chapter></Part></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>The Relation Between Lie Groups and Lie Algebras</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>The Relation Between Lie Groups and Lie Algebras</title></titleInfo><name type="personal"><namePart type="given">A.</namePart><namePart type="given">L.</namePart><namePart type="family">Onishchik</namePart><affiliation>Yaroslavl University, Sovetskaya ul. 14, 150000, Yaroslavl, Russia</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" type="research-article" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Springer Berlin Heidelberg</publisher><place><placeTerm type="text">Berlin, Heidelberg</placeTerm></place><dateIssued encoding="w3cdtf">1993</dateIssued><dateIssued encoding="w3cdtf">1993</dateIssued><copyrightDate encoding="w3cdtf">1993</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: The basic method of the theory of Lie groups, which makes it possible to obtain deep results with striking simplicity, consists in reducing questions concerning Lie groups to certain problems of linear algebra. This is done by assigning to every Lie group G its “tangent algebra”g, which to a large extent determines the group G, and to every homomorphism f: G → H of Lie groups a homomorphism df: g → h) of their tangent algebras, which to a large extent determines the homomorphism f. In the language of category theory we have a functor from the category of Lie groups into the category of Lie algebras, whose properties are very close to those of an equivalence of categories. In honour of the founder of the theory of Lie groups we will call this functor (following M. M. Postnikov (1982)) the Lie functor.</abstract><relatedItem type="host"><titleInfo><title>Lie Groups and Lie Algebras I</title><subTitle>Foundations of Lie Theory Lie Transformation Groups</subTitle></titleInfo><name type="personal"><namePart type="given">A.</namePart><namePart type="given">L.</namePart><namePart type="family">Onishchik</namePart><affiliation>Yaroslavl University, Sovetskaya ul. 14, 150000, Yaroslavl, Russia</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><genre type="book-series" displayLabel="Monograph" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0G6R5W5T-Z">book-series</genre><originInfo><publisher>Springer</publisher><copyrightDate encoding="w3cdtf">1993</copyrightDate><issuance>monographic</issuance></originInfo><subject><genre>Book-Subject-Collection</genre><topic authority="SpringerSubjectCodes" authorityURI="SUCO11649">Mathematics and Statistics</topic></subject><subject><genre>Book-Subject-Group</genre><topic authority="SpringerSubjectCodes" authorityURI="SCM">Mathematics</topic><topic authority="SpringerSubjectCodes" authorityURI="SCM11132">Topological Groups, Lie Groups</topic><topic authority="SpringerSubjectCodes" authorityURI="SCM21022">Differential Geometry</topic><topic authority="SpringerSubjectCodes" authorityURI="SCM28027">Manifolds and Cell Complexes (incl. Diff.Topology)</topic><topic authority="SpringerSubjectCodes" authorityURI="SCM28019">Algebraic Topology</topic><topic authority="SpringerSubjectCodes" authorityURI="SCM11019">Algebraic Geometry</topic></subject><identifier type="DOI">10.1007/978-3-642-57999-8</identifier><identifier type="ISBN">978-3-540-61222-3</identifier><identifier type="eISBN">978-3-642-57999-8</identifier><identifier type="ISSN">0938-0396</identifier><identifier type="BookTitleID">22217</identifier><identifier type="BookID">978-3-642-57999-8</identifier><identifier type="BookChapterCount">12</identifier><identifier type="BookVolumeNumber">20</identifier><identifier type="BookSequenceNumber">20</identifier><identifier type="PartChapterCount">5</identifier><part><date>1993</date><detail type="part"><title>I.: Foundations of Lie Theory</title></detail><detail type="volume"><number>20</number><caption>vol.</caption></detail><detail type="chapter"><number>Chapter 2</number></detail><extent unit="pages"><start>29</start><end>59</end></extent></part><recordInfo><recordOrigin>Springer-Verlag Berlin Heidelberg, 1993</recordOrigin></recordInfo></relatedItem><relatedItem type="series"><titleInfo><title>Encyclopaedia of Mathematical Sciences</title></titleInfo><name type="personal"><namePart type="given">R.</namePart><namePart type="given">V.</namePart><namePart type="family">Gamkrelidze</namePart><affiliation>Steklov Mathematical Institute, Russian Academy of Sciences, ul. Vavilova 42, 117966, Moscow, Russia</affiliation><affiliation>Institute for Scientific Information (VINITI), ul. Usievicha 20a, 125219, Moscow, Russia</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">A.</namePart><namePart type="given">A.</namePart><namePart type="family">Agrachev</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">A.</namePart><namePart type="given">A.</namePart><namePart type="family">Gonchar</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">E.</namePart><namePart type="given">F.</namePart><namePart type="family">Mishchenko</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">N.</namePart><namePart type="given">M.</namePart><namePart type="family">Ostianu</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">V.</namePart><namePart type="given">P.</namePart><namePart type="family">Sakharova</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">A.</namePart><namePart type="given">B.</namePart><namePart type="family">Zhishchenko</namePart><role><roleTerm type="text">editor</roleTerm></role></name><originInfo><publisher>Springer</publisher><copyrightDate encoding="w3cdtf">1993</copyrightDate><issuance>serial</issuance></originInfo><identifier type="ISSN">0938-0396</identifier><identifier type="SeriesID">855</identifier><part><detail type="volume"><number>I.</number><caption>vol.</caption></detail><detail type="chapter"><number>Chapter 2</number></detail></part><recordInfo><recordOrigin>Springer-Verlag Berlin Heidelberg, 1993</recordOrigin></recordInfo></relatedItem><identifier type="istex">7C41B9A7C4169522EDA25A1A3867525C8ECB0CA8</identifier><identifier type="ark">ark:/67375/HCB-ZB87JF0Q-G</identifier><identifier type="DOI">10.1007/978-3-642-57999-8_3</identifier><identifier type="ChapterID">3</identifier><identifier type="ChapterID">Chap3</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer-Verlag Berlin Heidelberg, 1993</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-Verlag Berlin Heidelberg, 1993</recordOrigin></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/document/7C41B9A7C4169522EDA25A1A3867525C8ECB0CA8/metadata/json</uri></json:item></metadata><annexes><json:item><extension>txt</extension><original>true</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/7C41B9A7C4169522EDA25A1A3867525C8ECB0CA8/annexes/txt</uri></json:item></annexes></istex></record>Pour manipuler ce document sous Unix (Dilib)
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