Year: 2025
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
DOI: 10.13025/30057
Authors: Flattery, Michael, Tuite, Michael P.
Institution: Ollscoil na Gaillimhe – University of Galway
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7094946945