A method for construction of Lie group invariants

TL;DR

This paper introduces a straightforward method for constructing invariants under the adjoint action of Lie groups, utilizing automorphisms of Cartan subalgebras and linear systems to generate polynomial invariants, demonstrated on SL(3).

Contribution

It presents a novel, easily implementable approach for deriving Lie group invariants by extending automorphisms and solving linear equations, enhancing understanding of algebraic independence.

Findings

Method effectively constructs invariants for Lie group actions.

Application to SL(3) illustrates the method's practicality.

Provides a systematic way to analyze algebraic independence of invariants.

Abstract

For an adjoint action of a Lie group G (or its subgroup) on Lie algebra Lie(G) we suggest a method for construction of invariants. The method is easy in implementation and may shed the light on algebraical independence of invariants. The main idea is to extent automorphisms of the Cartan subalgebra to automorphisms of the whole Lie algebra Lie(G). Corresponding matrices in a linear space V=Lie(G) define a Reynolds operator "gathering" invariants of torus T in G into special polynomials. A 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 on the coefficients in 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).