Reprezentacije poluprostih Liejevih algebri

Grizelj, Karmen (2014) Reprezentacije poluprostih Liejevih algebri. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

The purpose of the introductory part is the motivation to study Lie algebras and particularly semisimple Lie algebras. Therefore the concept of Lie group was introduced, as well as the main results about the close connection between the Lie group and its Lie algebra. We constructed a functor Lie taking Lie groups to Lie algebras. This functor in fact gives an equivalence between simply connected Lie groups and finite dimensional real Lie algebras. Most important definitions are given: Lie algebra, representation, derivation. Also, some important classes of these objects are introduced, like solvable and semisimple Lie algebras, irreducible and completely reducible representations, inner derivations. Afterwards the main results of the theory of Lie algebras and representations are given; the most important are certainly Engel’s theorem, Lie’s theorem, Schur’s lemma and Cartan’s solvability criteria. The main object in this study are semisimple Lie algebras, therefore some important properties of semisimple Lie algebras are given. Also, we got one crucial semisimplicity criterion using the Killing form. To get to the fundamental theorem of this study the concept of Casimir element is introduced and the main properties are proved. Afterwards we are ready to prove Weyl’s theorem, which states that any finite dimensional representation of a semisimple Lie algebra is completely reducible. After the theoretical part, some examples of Lie algebras are mentioned; these examples belong to matrix algebras. Also, using Weyl’s theorem all representations of $\mathfrak{sl}_2(\mathbb{F})$ are described.

Item Type: Thesis (Diploma thesis)
Supervisor: Pandžić, Pavle
Date: 2014
Number of Pages: 47
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 02 Jun 2015 11:43
Last Modified: 02 Jun 2015 11:43
URI: http://digre.pmf.unizg.hr/id/eprint/3946

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