Rings of invariant polynomials for a class of Lie algebras

Abstract:

Let $G$ be a group and let $\pi :G \to GL(V)$ be a finite-dimensional representation of $G$. Then for $g \in G,\pi (g)$ induces an automorphism of the symmetric algebra $S(V)$ of $V$. We let $I(G,V,\pi )$ be the subring of $S(V)$ consisting of elements invariant under this induced action. If $G$ is a connected complex semisimple Lie group with Lie algebra $L$ and if Ad is the adjoint representation of $G$ on $L$, then Chevalley has shown that $I(G,L,\text {Ad} )$ is generated by a finite set of algebraically independent elements. However, relatively little is known for nonsemisimple Lie groups. In this paper the author exhibits and investigates a class of nonsemisimple Lie groups $G$ with Lie algebra $L$ for which $I(G,L,\text {Ad} )$ is also generated by a finite set of algebraically independent elements.