Konačnodimenzionalne algebre s dijeljenjem

master's thesis

• Author: Petar Dević
• Mentor: Ilja Gogić (mentor)
• Committee member: Ilja Gogić (predsjednik povjerenstva)
• Committee member: Ivan Kupčić (član povjerenstva)
• Committee member: Rudi Mrazović (član povjerenstva)
• Committee member: Vjeran Hari (član povjerenstva)
• Committee member: Matko Milin (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-07-15, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu proučavamo neka osnovna svojstva konačnodimenzionalnih algebri s dijeljenjem. Najprije dokazujemo slavni Frobeniusov teorem, koji kaže da (do na izomorfizam) postoje točno tri realne konačnodimenzionalne algebre s dijeljenjem: Realni brojevi \(\mathbb{R}\), kompleksni brojevi \(\mathbb{C}\) te kvaternioni \(\mathbb{H}\). Nadalje, dokazujemo Wedderburnov strukturalni teorem koji opisuje strukturu konačnodimenzionalnih poluprostih algebri; svaka takva algebra je izomorfna konačnom direktnom produktu matričnih algebri nad algebrama s dijeljenjem. Time se većina pitanja vezana uz konačnodimenzionalne poluproste algebre reducira na analogna pitanja vezana uz konačnodimenzionalne algebre s dijeljenjem.

Abstract (english)

In this thesis we explore some basic properties of finite dimensional division algebras. We first present the proof of the celebrated Frobenius theorem, which states that (up to an isomorphism) there exist precisely three real finite-dimensional division algebras: Real numbers \(\mathbb{R}\), complex numbers \(\mathbb{C}\) and quaternions \(\mathbb{H}\). Next, we present the proof of the Wedderburn’s structure theorem which describes the structure of semisimple finitedimensional algebras; any such algebra is isomorphic to a finite direct product of matrix algebras over some division algebras. In this way most questions concerning semisimple finite-dimensional algebras are reduced to the analogues questions for finite-dimensional division algebras.

• Keywords: konačnodimenzionalne algebre; Frobeniusov teorem; Wedderburnov strukturalni teorem
• Keywords (english): finite dimensional division algebras; Frobenius theorem; Wedderburn’s structure theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:339404
• Study programme: Title: Mathematics and Physics Education; specializations in: Mathematics and Physics Education; Course: Mathematics and Physics Education; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: magistar/magistra edukacije matematike i fizike (magistar/magistra edukacije matematike i fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-09-25 08:56:38