University of Zagreb. Faculty of Science. Department of Mathematics.
Abstract
Kao uvodni dio, rad započinje s definicijama osnovnih pojmova s kojima se u matematici često susrećemo, a i važni su nam u nastavku rada. Nakon osnovnih pojmova, napravljen je uvod u teoriju reprezentacija gdje su definirani pojmovi reprezentacije grupe, vjerne reprezentacije, ekvivalentnost reprezentacija, ireducibilne reprezentacije, reducibilne reprezentacije i potpuno reducibilne reprezentacije. Slične definicije koriste se i u slučaju kada je grupa kompaktna. Definirani su i pojmovi vezani uz reprezentacije kompaktnih grupa. Iskazan je Peter-Weylov teorem, te posljedice tog teorema. Prije nego sto je teorem dokazan definirani su pojmovi korišteni u tom dokazu. Dane su definicije Liejevih algebri i Liejevih grupa i navedeni su neki od primjera Liejevih grupa. Dokazano je da svaka kompaktna Liejeva grupa ima vjernu reprezentaciju. U posljednjem poglavlju definiran je karakter ireducibilne reprezentacije i dani su važni rezultati do kojih je Weyl dosao: Weylova integracijska formula, Weylova formula karaktera i Weylova formula dimenzije.As an introductory part, the paper begins with definitions of basic notions that we often meet with in mathematics, and that are important in the sequel. Definitions of basic notions are followed by the introduction into theory of representations, where notions like group representations, faithful representations, equivalence of representations, irreducible representations, reducible representations and completely reducible representations are defined. Similar definitions are used also in the case when the group is compact. Also, the notions related to representations of compact groups are defined. Peter-Weyl theorem, as well as consequences of this theorem, are presented. Before the theorem is proven, the terms used in the proof are defined. The definitions of Lie algebras and Lie groups are given and some of examples of Lie groups are brought up. It is proven that each compact Lie group has a faithful representation. In the final chapter the character of an irreducible representation is defined and the important results that Weyl reached are given: Weyl integration formula, Weyl character and dimension formulae
