Consider algebraic stacks of the form $\mathbb{C}^n/G$ where $G$ is a unipotent group satisfying either
- Type 1: $G$ acts on $\mathbb{C}^n$ via affine linear transformations
- Type 2: $G$ acts on $\mathbb{C}^n$ via triangular algebraic transformations (transformations of the form $(x_1,\ldots,x_n) \to (x_1',\ldots,x_n')$ where $x_i'=x_i+p_i(x_1,\ldots,x_{i-1})$ for polynomials $p_i$)
Clearly stacks of Type 1 are also of Type 2. Is the converse true? In other words, for a triangular algebraic quotient $\mathbb{C}^n/G$ can one always construct an equivalent affine linear quotient $\mathbb{C}^{n'}/G'$?
Motivation here is the first example from the paper:
Winkelmann, J. On free holomorphic $\mathbb{C}$-actions on $\mathbb{C}^n$ and homogeneous Stein manifolds. Math. Ann. 286 593–612 (1990)
