On characteristic equations, trace identities and Casimir operators of simple Lie algebras

TL;DR

This paper develops two computationally feasible methods to derive new identities and express Casimir operators for simple Lie algebras, enhancing understanding of their structure and invariants.

Contribution

It introduces novel approaches to analyze characteristic equations and Casimir operators, producing new tensorial identities and explicit expressions for non-primitive Casimirs.

Findings

Derived tensorial identities involving structure constants

Obtained trace identities for matrices in key representations

Expressed non-primitive Casimir operators explicitly

Abstract

Two approaches are developed to exploit, for simple complex or compact real Lie algebras g, the information that stems from the characteristic equations of representation matrices and Casimir operators. These approaches are selected so as to be viable not only for `small' Lie algebras and suitable for treatment by computer algebra. A very large body of new results emerges in the forms, a) of identities of a tensorial nature, involving structure constants etc. of g, b) of trace identities for powers of matrices of the adjoint and defining representations of g, c) of expressions of non-primitive Casimir operators of g in terms of primitive ones. The methods are sufficiently tractable to allow not only explicit proof by hand of the non-primitive nature of the quartic Casimir of g2, f4, e6, but also e.g. of that of the tenth order Casimir of f4.