Survey on the Irreducible Representations of Lie Algebras and Lie Groups

Abstract

Lie groups and Lie algebras are central objects in differential geometry and physics. Representing Lie groups and algebras as spaces of linear operators is one of the most powerful tools to understand their structures. In our thesis, we introduce the class of matrix Lie groups and algebras with focus on simply connected Lie groups and semisimple Lie algebras. To obtain the irreducible representation for semisimple Lie algebras, we construct the Verma module and obtain a finite-dimensional irreducible quotient space based on the Verma module. The Weyl's character formula, whose consequences include Weyl's dimension formula and Kostant's multiplicity formula, gives informative data to the irreducible representations. Lastly, nested special orthogonal algebras are investigated to obtain a generalization of Weyl's character formula together with a demonstrative application of Weyl's dimension formula and Kostant's multiplicity formula.