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[{"key": "dc.contributor.author", "value": "Korhonen, Ilari", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2014-05-28T11:55:41Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2014-05-28T11:55:41Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2014", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1436078", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/43585", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "<p>Ryhm\u00e4t ovat yksinkertaisina mutta elegantteina algebrallisina rakenteina jo pitk\u00e4\u00e4n olleet keskeinen\r\nosa niin puhdasta kuin sovellettuakin matematiikkaa. Erityisesti ryhm\u00e4t soveltuvat erilaisten\r\n<i>symmetrioiden</i> esitt\u00e4miseen. Ryhmien esitysteoriassa voidaan ryhmien rakennetta koskevia ongelmia\r\npalauttaa <i>lineaarialgebran</i> ongelmiksi, jotka ovat hyvin ratkaistavissa. T\u00e4m\u00e4 tapahtuu\r\nkuvaamalla ryhm\u00e4 <i>homomorfisesti</i> lineaarikuvausten ryhm\u00e4\u00e4n. Osoittautuu my\u00f6s mielenkiintoiseksi\r\ntutkia jo itsess\u00e4\u00e4n lineaarikuvauksista muodostuvien ryhmien ep\u00e4triviaaleja lineaarisia\r\nesityksi\u00e4.</p><p>\r\nErityisen tarkastelun kohteena tutkielmassa on <i>klassinen matriisiryhm\u00e4</i> SO(3), joka siis koostuu\r\navaruuden ℝ<sup>3</sup> rotaatiokuvauksista. Ryhm\u00e4 SO(3) muiden klassisten matriisiryhmien tapaan\r\non erityisesti ns. <i>Lien ryhm\u00e4</i>, ts. <i>sile\u00e4 monisto</i> siten, ett\u00e4 ryhm\u00e4n operaatio ja k\u00e4\u00e4nteisalkion\r\nmuodostaminen ovat vastaavassa mieless\u00e4 <i>sileit\u00e4 kuvauksia</i> (Sophus Lie, 1842-1899). Monistoja\r\npuolestaan voidaan luonnehtia k\u00e4yrien ja pintojen yleistyksiksi. Tarkemmin ilmaistuna monisto\r\non <i>topologinen avaruus</i>, joka on <i>lokaalisti euklidinen</i>, ts. jokaisella pisteell\u00e4 on ymp\u00e4rist\u00f6, joka\r\non <i>homeomorfinen</i> jonkin euklidisen avaruuden ℝ<sup>n</sup> avoimen joukon ts. <i>kartan</i> kanssa. Monisto\r\non <i>sile\u00e4</i>, jos siirtym\u00e4t karttojen v\u00e4lill\u00e4 ovat avaruuden ℝ<sup>n</sup> sileit\u00e4 kuvauksia.\r\n</p><p>\r\nLien ryhm\u00e4t ovat merkillisi\u00e4 monistoja siin\u00e4kin mieless\u00e4, ett\u00e4 niiden geometria on pitk\u00e4lti kuvattavissa\r\nalgebrallisesti ns. <i>Lien algebran</i> avulla. Osoittautuu, ett\u00e4 t\u00e4m\u00e4 Lien ryhm\u00e4\u00e4 <i>vastaava</i>\r\nLien algebra saadaan aina moniston <i>tangenttiavaruudesta</i> neutraalialkiolle. Riitt\u00e4v\u00e4n siistiss\u00e4\r\ntapauksessa koko ryhm\u00e4n geometria m\u00e4\u00e4r\u00e4ytyy pelk\u00e4st\u00e4\u00e4n sit\u00e4 vastaavasta Lien algebrasta.\r\nKuitenkin aina <i>neutraalialkion sis\u00e4lt\u00e4v\u00e4 yhten\u00e4inen komponentti</i> m\u00e4\u00e4r\u00e4ytyy Lien ryhm\u00e4\u00e4 vastaavasta\r\nLien algebrasta.\r\n</p><p>\r\nLien ryhmien esitysteoria eroaa hieman <i>\u00e4\u00e4rellisten</i> ryhmien esitysteoriasta, sill\u00e4 ryhm\u00e4n rakenteen\r\ns\u00e4ilymisen homomorfismissa lis\u00e4ksi vaaditaan moniston <i>sile\u00e4n struktuurin</i> s\u00e4ilymist\u00e4.\r\nLis\u00e4ksi useat Lien ryhm\u00e4t, kuten SO(3), ovat <i>kompakteja</i>. T\u00e4m\u00e4 tarkoittaa puolestaan sit\u00e4,\r\nett\u00e4 k\u00e4ytt\u00f6\u00f6n saadaan my\u00f6s <i>kompaktien topologisten ryhmien</i> esitysteorian tulokset, kuten <i>unitaaristen</i>\r\nesitysten olemassaolo ja <i>Peterin ja Weylin lause</i> (Hermann Weyl, 1885-1955 sek\u00e4\r\nh\u00e4nen oppilaansa Fritz Peter, 1899-1949). N\u00e4m\u00e4 perustuvat pohjimmiltaan ns. <i>Haarin mitan</i>\r\n(Alfr\u00e9d Haar, 1885-1933) olemassaoloon kaikilla <i>lokaalisti kompakteilla topologisilla ryhmill\u00e4</i>,\r\njoita kompaktit ryhm\u00e4t tietysti ovat.\r\n</p><p>\r\nTutkielman p\u00e4\u00e4tuloksena esitet\u00e4\u00e4n ryhm\u00e4n SO(3) redusoitumattomien esitysten konstruktio,\r\nsek\u00e4 todistus sille, ett\u00e4 kaikki muut redusoitumattomat esitykset ovat ekvivalentteja t\u00e4lle konstruktiolle.\r\nKonstruktiossa p\u00e4\u00e4dyt\u00e4\u00e4n ns. <i>palloharmonisiin</i> funktioihin.</p>", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Ilari Korhonen (tiilkorh) on 2014-05-28 11:55:40.387949. Form: Pro gradu -lomake (1 tekij\u00e4) (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake-1-tekijae). JyX data:", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija@noreply.fi) on 2014-05-28T11:55:41Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201405281864.pdf: 3265473 bytes, checksum: 064a8a86cb6feb7dc044ba9883e093fb (MD5)\nlicense.html: 4897 bytes, checksum: d06f79774731a4d290a9af9276dabc30 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2014-05-28T11:55:41Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201405281864.pdf: 3265473 bytes, checksum: 064a8a86cb6feb7dc044ba9883e093fb (MD5)\nlicense.html: 4897 bytes, checksum: d06f79774731a4d290a9af9276dabc30 (MD5)\n Previous issue date: 2014", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (55 sivua)", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Ryhm\u00e4 SO(3) ja sen lineaariset redusoitumattomat esitykset", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201405281864", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2014-05-28T11:55:42Z", "language": null, "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": "fi", "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "lineaarialgebra", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "algebra", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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