Generating functions for the Casimir invariants of simple lie algebras

In this thesis we consider the theory of finite-dimensional, complex, simple Lie algebras, their irreducible representations and Casimir operators. We review the classification of these simple Lie algebras and an overview of constructions of characters of their representations, including Klimyk’s formula for decomposing tensor products of irreducible representations. We describe a new efficient method for evaluating the characters of irreducible representations using Freudenthal’s formula and present the characters of each fundamental representation of the simple Lie algebras. We review Okubo’s formula for calculating the eigenvalues of general degree Casimir operators on irreducible representations. Using Klimyk’s formula and Okubo’s formula, we obtain a new formula for efficiently calculating these eigenvalues. We apply our result to derive closed expressions for the generating functions of these eigenvalues for the first fundamental representations of the classical Lie algebras.

Funder

College of Science and Engineering, University of Galway

Publisher

University of Galway

Rights

CC BY-NC-ND

cbn

Collections

University of Galway Theses (PhD Theses)