# Reprezentacije simplektičkih Liejevih algebri

Grgurić, Josip (2015) Reprezentacije simplektičkih Liejevih algebri. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

## Abstract

In this thesis we exhibited basic facts about the structure and representations of semisimple Lie algebras and gave examples of some representations of the symplectic algebra of rank 2. First we defined Lie groups, followed by their Lie algebras and the exponential mapping. We showed how to use the exponential mapping to study Lie groups representations in terms of representations of their Lie algebras. We then gave a rough classification of Lie algebras and proved several equivalent characterizations of semisimple complex Lie algebras. The case of the algebra \$\mathfrak{sl}_2\mathbb{C}\$ was treated in detail and it was shown later how representation theory of \$\mathfrak{sl}_2\mathbb{C}\$ incorporates into the structure of an arbitrary complex semisimple algebra \$\mathfrak{g}\$. We presented among other things a proof of the existence of Cartan subalgebras and defined the set of roots and the weight lattice. We introduced the notion of the highest weight and finally mentioned the fundamental result on the classification of irreducible finitedimensional representations of \$\mathfrak{g}\$. We then applied the general theory to the concrete example of the symplectic algebras, more precisely the algebra \$\mathfrak{sp}_2\mathbb{C}\$.

Item Type: Thesis (Diploma thesis) Pandžić, Pavle 2015 50 NATURAL SCIENCES > Mathematics Faculty of Science > Department of Mathematics Iva Prah 12 Oct 2015 12:24 12 Oct 2015 12:24 http://digre.pmf.unizg.hr/id/eprint/4268