A method for construction of Lie group invariants

Yu. G. Palii

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIII, Tome 421 (2014), pp. 138-151

Résumé

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 $V\cong\operatorname{Lie}(G)$ define a Reynolds operator “gathering” invariants of torus $\mathcal T\subset G$ 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 $\operatorname{SL}(3)$ (and its subgroup $\operatorname{SL}(2)$) on the Lie algebra $\mathfrak{sl}(3)$.

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