Casimir eigenvalues for universal Lie algebra

TL;DR

This paper derives polynomial formulas for Casimir eigenvalues in the adjoint representation of simple Lie algebras using Vogel's universal parameters, providing explicit generating functions.

Contribution

It introduces a universal polynomial expression for Casimir eigenvalues in simple Lie algebras based on Vogel's parameters, unifying different definitions.

Findings

Eigenvalues expressed as polynomials in Vogel's parameters

Explicit generating functions for Casimir eigenvalues

Unified approach for different Casimir operator definitions

Abstract

For two different natural definitions of Casimir operators for simple Lie algebras we show that their eigenvalues in the adjoint representation can be expressed polynomially in the universal Vogel's parameters $\alpha, \beta, \gamma$ and give explicit formulae for the generating functions of these eigenvalues.