Kleinov Erlangenski program i primjena na grupu SL(2,R)

master's thesis

• Author: David Tarandek
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Committee member: Mirko Primc (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Erlangenski program je uvriježeni naziv teksta iz 1872. godine u kojem je Felix Klein izložio projekt objedinjenja geometrije. Klein u središnje mjesto stavlja projektivnu geometriju, a svakoj podgeometriji dodjeljuje pripadnu grupu transformacija pod čijim su djelovanjem istaknuta svojstva invarijantna. Obratno, svako djelovanje neke grupe na odgovarajućem homogenom prostoru ima svoju geometriju kao teoriju invarijanti tog djelovanja. Rad je usredotočen na specijalnu linearnu grupu \(SL(2, \mathbb{R})\). Kroz izbor njenih podgrupa ostvaruje se djelovanje \(SL(2, \mathbb{R})\) na homogenim prostorima Möbiusovim transformacijama. Naposljetku, pokazuje se invarijantnost familije svih cikala pod djelovanjem \(SL(2,\mathbb{R})\).

Abstract (english)

The Erlangen programme is the name given to a 1872. paper in which Felix Klein presented a project of unification of geometry. Klein highlights the status of projective geometry. To each subgeometry a corresponding group of transformations is assigned, keeping certain properties intact. Conversely, each group action on the associated homogeneous space is attributed with its geometry in the form of invariance theory of the given action. In this paper, I focus on the special linear group \(SL(2, \mathbb{R})\). By choosing any of its subgroups, an action of \(SL(2, \mathbb{R})\) on a homogeneous space is induced in the form of Möbius transformations. Lastly, the family of all cycles is proved to be invariant under the action of \(SL(2,\mathbb{R})\).

• Keywords: Erlangenski program; projektivna geometrija; specijalna linearna grupa; Möbiusova transformacija
• Keywords (english): Erlangen programme; projective geometry; special linear group; Möbius transformation
• Language: croatian
• URN:NBN: urn:nbn:hr:217:631788
• Study programme: Title: Pure Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2018-05-03 11:57:58