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master's thesis
Sistemi korijena i Dynkinovi dijagrami
Zagreb: University of Zagreb, Faculty of Science, 2021. urn:nbn:hr:217:683733
Rebić, Josip
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Rebić, J. (2021). Sistemi korijena i Dynkinovi dijagrami (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:683733

Rebić, Josip. "Sistemi korijena i Dynkinovi dijagrami." Master's thesis, University of Zagreb, Faculty of Science, 2021. https://urn.nsk.hr/urn:nbn:hr:217:683733

Rebić, Josip. "Sistemi korijena i Dynkinovi dijagrami." Master's thesis, University of Zagreb, Faculty of Science, 2021. https://urn.nsk.hr/urn:nbn:hr:217:683733

Rebić, J. (2021). 'Sistemi korijena i Dynkinovi dijagrami', Master's thesis, University of Zagreb, Faculty of Science, accessed 23 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:683733

Rebić J. Sistemi korijena i Dynkinovi dijagrami [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2021 [cited 2024 December 23] Available at: https://urn.nsk.hr/urn:nbn:hr:217:683733

J. Rebić, "Sistemi korijena i Dynkinovi dijagrami", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2021. Available at: https://urn.nsk.hr/urn:nbn:hr:217:683733

Metadata
TitleSistemi korijena i Dynkinovi dijagrami
Title (english)Root systems and Dynkin diagrams
AuthorJosip Rebić
MentorSlaven Kožić (mentor)
Committee memberSlaven Kožić (predsjednik povjerenstva)
Committee memberZoran Vondraček (član povjerenstva)
Committee memberSiniša Slijepčević (član povjerenstva)
Committee memberVjekoslav Kovač (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2021-07-22, 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
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:683733
Study programmeTitle: Finance and Business Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
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Created on2021-09-16 11:54:41