Reprezentacije poluprostih Liejevih algebri

master's thesis

• Author: Karmen Grizelj
• Mentor: Pavle Pandžić (mentor)
• Committee member: Hrvoje Kraljević (predsjednik povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Committee member: Franka Miriam Bruckler (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-09-23, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilj uvodnog dijela ovog rada je bio motivacija za proučavanje kako općenito Liejevih algebri tako i poluprostih Liejevih algebri. U tu svrhu uveden je pojam Liejeve grupe i izneseni su glavni rezultati o vezi Liejeve grupe sa pripadnom Liejevom algebrom. Eksplicitno je konstruiran funktor Lie koji Liejevoj grupi pridružuje Liejevu algebru. Zapravo, taj funktor uspostavlja ekvivalenciju između 1-povezanih Liejevih grupa i konačnodimenzionalnih realnih Liejevih algebri. Dane su definicije svih pojmova važnih za ovaj rad: Liejeve algebre, reprezentacije, derivacije. Uvedene su i određene važne klase ovih objekata poput rješive i poluproste Liejeve algebre, ireducibilne i potpuno reducibilne reprezentacije, unutarnjih derivacija. Nakon uvođenja pojmova dokazani su osnovni rezultati vezani uz Liejeve algebre i reprezentacije, najvažniji medu njima su Engelov teorem, Liejev teorem, Schurova lema i Cartanovi kriteriji rješivosti. Zatim su dana neka važna svojstva poluprostih Liejevih algebri i dokazan je kriterij poluprostote preko Killingove forme. Zatim je uveden pojam Casimirovog operatora reprezentacije i dokazana su njegova osnovna svojstva. Sve navedeno bila je priprema za dokaz fundamentalnog teorema u teoriji reprezentacija i Liejevih algebri, Weylovog teorema, koji kaže da je konačnodimenzionalna reprezentacija poluproste Liejeve algebre potpuno reducibilna. Nakon rezultata iz teorije navedeni su matrični primjeri Liejevih algebri i opisane su reprezentacije od \(\mathfrak{sl}_2(\mathbb{F})\) korištenjem Weylovog teorema.

Abstract (english)

The purpose of the introductory part is the motivation to study Lie algebras and particularly semisimple Lie algebras. Therefore the concept of Lie group was introduced, as well as the main results about the close connection between the Lie group and its Lie algebra. We constructed a functor Lie taking Lie groups to Lie algebras. This functor in fact gives an equivalence between simply connected Lie groups and finite dimensional real Lie algebras. Most important definitions are given: Lie algebra, representation, derivation. Also, some important classes of these objects are introduced, like solvable and semisimple Lie algebras, irreducible and completely reducible representations, inner derivations. Afterwards the main results of the theory of Lie algebras and representations are given; the most important are certainly Engel’s theorem, Lie’s theorem, Schur’s lemma and Cartan’s solvability criteria. The main object in this study are semisimple Lie algebras, therefore some important properties of semisimple Lie algebras are given. Also, we got one crucial semisimplicity criterion using the Killing form. To get to the fundamental theorem of this study the concept of Casimir element is introduced and the main properties are proved. Afterwards we are ready to prove Weyl’s theorem, which states that any finite dimensional representation of a semisimple Lie algebra is completely reducible. After the theoretical part, some examples of Lie algebras are mentioned; these examples belong to matrix algebras. Also, using Weyl’s theorem all representations of \(\mathfrak{sl}_2(\mathbb{F})\) are described.

• Keywords: Liejeve grupe; Liejeve algebre; funktor Lie; reprezentacije; derivacije; Engelov teorem; Liejev teorem; Schurova lema; Cartanovi kriteriji rješivosti; Killingova forma; Casimirov operator reprezentacije; Weylov teorem
• Keywords (english): Lie groups; Lie algebras; functor Lie; representation; derivation; Engel’s theorem; Lie’s theorem; Schur’s lemma; Cartan’s solvability criteria; Killing form; Casimir element; Weyl's theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:255599
• 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: 2019-01-25 13:18:53