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

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, but also to be suitable for treatment by computer algebra. A very large body of new results emerges in the forms of (a) identities of a tensorial nature, involving structure constants etc. of 𝔤, (b) trace identities for powers of matrices of the adjoint and defining representations of 𝔤, (c) expressions of nonprimitive Casimir operators of 𝔤 in terms of primitive ones. The methods are sufficiently tractable to allow not only explicit proof by hand of the nonprimitive nature of the quartic Casimir of g2, f4, e6, but also, e.g., of that of the tenth order Casimir of f4.