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

Two approaches are developed to exploit, for simple complex or compact real Lie algebras g, the information that stems from the characteristic equatio ns of representation matrices and Casimir operators. These approaches are s elected so as to be viable not only for "small" Lie algebras, but also to b e suitable for treatment by computer algebra. A very large body of new resu lts 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 nonpr imitive Casimir operators of g in terms of primitive ones. The methods are sufficiently tractable to allow not only explicit proof by hand of the nonp rimitive nature of the quartic Casimir of g(2), f(4), e(6), but also, e.g., of that of the tenth order Casimir of f(4). (C) 2000 American Institute of Physics. [S0022-2488(00)01305-0].