LINEAR GROUPS

Diploma je sestavljena iz devetih poglavij. Začetek diplomskega dela vsebuje osnovne pojme in lastnosti matrik, vektorskih prostorov in osnovne lastnosti grup. V naslednjem poglavju je bolj podrobno predstavljena posebna unitarna grupa, kjer opišemo zemljepisne širine, zemljepisne dolžine ter severni in južni pol grupe. Pokažemo tudi, da so konjugirani razredi v unitarni grupi dvodimenzionalne sfere. V poglavju Ortogonalna upodobitev unitarne grupe vpeljemo orbite in pojem vlakna. Tu pokažemo, da je unitarne grupe dvojno pokritje grupe ortogonalne grupe. V nadaljevanju si pogledamo primer nekompaktne grupe. Nato sledi poglavje Enoparametričnih grup, ki so homomorfizmi, ki slikajo iz aditivne grupe v linearno grupo odvedljivih funkcij spremenljivke t ∈ ℝ. Tu omenimo pojem parcialnega odvoda in izrek o inverznih funkcijah. V nadaljevanju se ukvarjamo z Liejevo algebro, ki je prostor vektorjev tangent na G pri identiteti I. S pomočjo pojma gradient in verižnega ulomka podamo potrebne pogoje, da vektor postane tangenta za realno algebrsko množico S. V tem poglavju so definirani pojmi infinitizimalna tangenta, vektor tangent in prostor tangent. Ukvarjamo se z izračunom infinitizimalne spremembe posebne linearne grupe in ortogonalne grupe. Za konec tega poglavja zapišemo definicijo Liejeve algebre bolj abstraktno s pomočjo uporabe operacije komutator. V zadnjem poglavju z naslovom Primeri enostavnih grup navedemo nekaj primerov teh grup ter dokažemo pomemben izrek. The thesis consists from nine chapters. Begining of thesis contains the concepts and properties of matrices, vector spaces and the basic properties of groups, which are often used in thesis. In the next chapter we focus on special unitary group where we present latitude and longitude of unitary group. In this section we show that conjugate classes in unitary groups are two-dimensional spheres. In chapter The ortogonal representation of unitary group are introduced concepts of orbits and fiber. Here we show that unitary group is double covering of ortogonal group. In the continuation we meet special ortogonal group, which is example of noncompact group. Then follows chapter of Oneparametric groups, where is shown that oneparametric groups are homomorphisms from the additive group of real numbers to the general linear group, which are differentiable functions of the variable t ∈ ℝ. Here we mention concept of partial derivatives and the inverse function theorem, which is shown on ortogonal group and special linear group. In the continuation we speak about Lie algebra. Using the concepts of gradient and the chain rule we give necessary conditions for a vector to be tangent to a real algebraic set S. In this chapter we define concepts like infinitizimal tangent, vector tangent and tangent space. We are also dealing with computation of infinitizimal change of and ortogonal groups. In conclusion of this chapter we write the definition od Lie algebra more abstract, where we use the operation bracket. The final chapter, Examples of simple groups, contains some examples of these groups and we prove important theorem.