A Method for Constructing Invariants of a Lie Group

Abstract

For the adjoint action of a Lie group G (or a subgroup of G) on the Lie algebra Lie (G), we suggest a method for constructing its invariants. The method is easy to implement and may shed light on the algebraical independence of invariants. The main idea is to extend automorphisms of the Cartan subalgebra to automorphisms of the whole Lie algebra Lie (G). The corresponding matrices in the linear space V ∼ = Lie (G) define the Reynolds operator, which “gathers” invariants of the torus T ⊂ G into special polynomials. The condition for a linear combination of polynomials to be G-invariant is equivalent to the existence of a solution for a certain system of linear equations in the coefficients of the combination. As an example, we consider the adjoint action of the Lie group SL(3) (and its subgroup SL(2)) on the Lie algebra sl(3).