Triangular actions on \({\mathbb{C}}^ 3\) (Q1102343)

It is proved that any free triangular action of a connected linear algebraic group \(G\) on \({\mathbb{C}}^ 3\) is equivalent to the trivial action of \(G\) on \(G\times {\mathbb{C}}^ n\), \(n=3-\dim(G)\); an analogue result is proved for quasi-algebraic actions. As an application it is proved that if \(\dim (R_ u(G))\leq 3\) and \(H\) is an algebraic subgroup of \(G\) then \(G/H\) is affine iff and only if \(R_ u(H^ 0)\cap M=1\) for each reductive subgroup \(M\) of \(G\).