Sistemi korijena i Dynkinovi dijagrami

master's thesis

• Title (english): Root systems and Dynkin diagrams
• Author: Josip Rebić
• Mentor: Slaven Kožić (mentor)
• Committee member: Slaven Kožić (predsjednik povjerenstva)
• Committee member: Zoran Vondraček (član povjerenstva)
• Committee member: Siniša Slijepčević (član povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-07-22, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu promatrali smo sisteme korijena pridružene Liejevim algebrama. Na početku smo definirali što su to sistemi korijena i što je to ortogonalna refleksija pridružena elementima sistema korijena. Nakon toga definiramo i bazu sistema korijena. Ovime i definiranjem ireducibilnih sistema korijena dobili smo važan alat za proučavanje samih Liejevih algebri. Pokazali smo i dva osnovna načina zapisivanja korijena. Jedan od njih je grafički način pomoću Coxeterovih grafova, odnosno Dynkinovih dijagrama. Drugi način je pomoću takozvanih Cartanovih matrica. Nadalje, prelazimo na Weylove grupe koje definiramo kao podgrupe grupe svih automorfizama nekog sistema korijena generirane refleksijama sistema korijena. Sada nam se samo postavlja pitanje možemo li zapisati sve moguće sisteme korijena? Njihovom klasifikacijom i konstrukcijom dobivamo potvrdan odgovor. Kroz klasifikaciju, koja se sastoji od deset koraka, dolazimo do konačnog zaključka da je broj tipova sistema korijena konačan odnosno, drugim riječima, da možemo sve sisteme korijena klasificirati kao tipove A-G. Kroz konstrukciju zaključujemo da za svaki Dynkinov dijagram (ili Cartanovu matricu) tipova A-G postoji ireducibilan sistem korijena koji ima upravo taj dijagram (ili matricu). Naposlijetku, promatramo Liejeve algebre te ilustriramo vezu između njih i sistema korijena uz primjeru specijalne linearne Liejeve algebre \(sl_n\left(\mathbb{C}\right)\).

Abstract (english)

In this thesis we considered the root systems associated with Lie algebras. At the beginning, we defined root systems and orthogonal reflection associated to the elements of the root system. After that, we define the basis of the root system. By doing this and defining the irreducible root systems, we established an important tool for studing Lie algebras. We also showed two basic methods of presentation of root systems. One of them is the graphical with Coxeter graphs and Dynkin diagrams. The other is with the so-called Cartan matrices. Furthermore, we move on to Weyl groups, which we define as the subgroups of the root system automorphism groups, generated by all reflections of the root system. Now the question is, can we write down all the possible root systems? By their classification and construction we get a positive answer. Through the classification which consists of ten steps, we come to the final conclusion that the number of root system types is finite or, in other words, that we can classify all root systems as types A-G. Through construction, we conclude that for each Dynkin diagram (or Cartan matrix) of types A-G there is an irreducible root system that has exactly that diagram (or matrix). Finally, we observe Lie algebras and illustrate the connection between them and the root systems using the example of special linear Lie algebra \(sl_n\left(\mathbb{C}\right)\).

• Keywords: Liejeve algebre; Coxeterovi grafovi; Dynkinovi dijagrami; Cartanove matrice; Weylove grupe
• Keywords (english): Lie algebras; Coxeter graphs; Dynkin diagrams; Cartan matrices; Weyl groups
• Language: croatian
• URN:NBN: urn:nbn:hr:217:683733
• Study programme: Title: Finance and Business 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: 2021-09-16 11:54:41