Unipotent Actions on Affine Space

Abstract

Algebraic group actions on affine space, C ⁿ, are determined by finite dimensional algebraic subgroups of the full algebraic automorphism group, Aut C ⁿ. This group is anti-isomorphic to the group of algebra automorphisms of $F_{n}= \text{\textbf{C}}[x_{1}, \cdots, x_{n}]$ by identifying the indeterminates x ₁, …, x _n with the standard coordinate functions: σ ∈ Aut C ⁿ defines σ^* ∈ Aut F _n by $\sigma ^{*}f(x)= f(\sigma x)$, where f ∈ F _n and x ∈ C ⁿ. In fact, an automorphism σ is often defined in terms of the polynomials $(\sigma ^{*}x_{1},\cdots, \sigma ^{*}x_{n})$. Two special subgroups of Aut C ⁿ play an important role in determining the structure of the finite dimensional algebraic subgroups. The first is the affine linear group,

$$A_{n}= \left \{ \sigma = (f_{1}, \cdots, f_{n}) \in \text{Aut} \textbf{C}^{n}\vert \text{deg} f_{i}\leq 1 \right \}$$

which is the semi-direct product of the general linear group, GL _n(C), and the abelian group of translations, $T_{n}\cong \textbf{C}^{n}$. The second is the ‘Jonquière’, or ‘triangular’ subgroup

$$B_{n}= \left \{ \sigma = (f_{1}, \cdots, f_{n}) \in \text{Aut} \textbf {C}^{n}\vert f_{i}= c_{i}x_{i}+ h_{i}, c_{i}\in \textbf{C}, h_{i}\in \textbf{C}[x_{i+1}, \cdots, x_{n}] \right \}$$

.