Reprezentacije simplektičkih Liejevih algebri

master's thesis

• Author: Josip Grgurić
• Mentor: Pavle Pandžić (mentor)
• Committee member: Pavle Pandžić (predsjednik povjerenstva)
• Committee member: Mirko Primc (član povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Committee member: Vedran Krčadinac (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-09-23, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom smo radu izložili osnovne činjenice o strukturi i reprezentacijama poluprostih Liejevih algebri i dali primjere nekih reprezentacija simplektičke algebre ranga 2. Najprije smo definirali Liejeve grupe, zatim njihove Liejeve algebre i eksponencijalno preslikavanje pomoću kojeg je opravdano proučavanje reprezentacija Liejevih grupa u termininima reprezentacija njihovih Liejevih algebri. Dali smo grubu klasifikaciju Liejevih algebri i dokazali nekoliko ekvivalentnih karakterizacija kompleksnih poluprostih Liejevih algebri. Detaljno je obrađen slučaj algebre \(\mathfrak{sl}_2\mathbb{C}\) i pokazano poslije kako se ona ugrađuje u strukturu proizvoljne kompleksne poluproste algebre \(\mathfrak{g}\). Prezentirali smo između ostalog dokaz egzistencije Cartanove podalgebre, definirali skup korijena i rešetku težina. Uveli smo pojam najveće težine i naposljetku spomenuli fundamentalan rezultat o klasifikaciji ireducibilnih konačnodimenzionalnih reprezentacija od \(\mathfrak{g}\). Zatim smo razvijenu opću teoriju primijenili na konkretan primjer simplektičkih algebri, točnije na algebru \(\mathfrak{sp}_2\mathbb{C}\).

Abstract (english)

In this thesis we exhibited basic facts about the structure and representations of semisimple Lie algebras and gave examples of some representations of the symplectic algebra of rank 2. First we defined Lie groups, followed by their Lie algebras and the exponential mapping. We showed how to use the exponential mapping to study Lie groups representations in terms of representations of their Lie algebras. We then gave a rough classification of Lie algebras and proved several equivalent characterizations of semisimple complex Lie algebras. The case of the algebra \(\mathfrak{sl}_2\mathbb{C}\) was treated in detail and it was shown later how representation theory of \(\mathfrak{sl}_2\mathbb{C}\) incorporates into the structure of an arbitrary complex semisimple algebra \(\mathfrak{g}\). We presented among other things a proof of the existence of Cartan subalgebras and defined the set of roots and the weight lattice. We introduced the notion of the highest weight and finally mentioned the fundamental result on the classification of irreducible finitedimensional representations of \(\mathfrak{g}\). We then applied the general theory to the concrete example of the symplectic algebras, more precisely the algebra \(\mathfrak{sp}_2\mathbb{C}\).

• Keywords: Liejeve grupe; Liejeve algebre; reprezentacije simplektičke algebre ranga 2; Cartanova podalgebra
• Keywords (english): Lie groups; Lie algebras; representations of the symplectic algebra of rank 2; Cartan subalgebra
• Language: croatian
• URN:NBN: urn:nbn:hr:217:302866
• 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 12:59:59