Classical Complex Algebraic Actions and Affine Embeddings — Examples

Multiplication by $t$ and $t^{-1}$ is regular on $T\times X$ and the group identities hold coordinatewise, so this is an action in Classical complex affine algebraic actions and rational modules. Inverse pullback sends $x$ to $t^{-1}x$ and $y$ to $ty$; multiplication gives $t^{b-a}{x}^a{y}^b$. Laurent-monomial independence gives the direct weight decomposition, and degree zero monomials are exactly $(xy{)}^a$, so $A_0=\mathbb{C}[xy]$.

The two coordinate functions span a stable rational submodule generating $A$. Applying F2, evaluation $\mathrm{e}\mathrm{v}(x,y)$ has coordinates $(x,y)$ in the dual basis and dual weights $(1,-1)$. It is the identity isomorphism of affine planes, making the general embedding construction explicit.

Conjugation is additive, so $g\cdot (h\cdot x)=x+\bar{h}+\bar{g}=(g+h)\cdot x$ and $0\cdot x=x$. Each fixed $g$ acts by the polynomial translation $x\mapsto x+\bar{g}$. But the joint action is not a morphism: restriction along $g\mapsto (g,0)$ would make $g\mapsto \bar{g}$ a polynomial on ${\mathbb{A}}^1$. Such a polynomial would equal $g$ on all real numbers, hence be the identity polynomial, but would then send $i$ to $i$ rather than $-i$.

Inverse pullback acts on $W=\mathbb{C}1+\mathbb{C}z$ by $1\mapsto 1$ and $z\mapsto z-\bar{g}$. Thus $W$ is stable, yet its representation has nonregular matrix coefficient $-\bar{g}$, by step 1.1. No larger finite-dimensional stable subspace containing $z$ can be algebraic: it contains every translate $z-\bar{g}$, in particular the translate $z-1$ for the group element $1$, so it also contains the difference $z-(z-1)=1$ and hence the stable $W$, and restriction to $W$ would have regular matrix entries by taking a finite basis extending a basis of $W$. Consequently the coordinate representation is not rational, even though each polynomial's translates lie in its finite-dimensional bounded-degree space. This explicitly violates the rationality conclusion in F2, and refutes the claim.

Inverse pullback gives $g1=1$, $gz=z-g$, and $g{z}^2=z^2-2gz+g^2$. Therefore $W$ is stable with polynomial matrix entries in $g$, and it generates $\mathbb{C}[z]$ because it contains $z$. Dualizing as in F2 gives $g(a,b,c)=(a,b+ga,c+2gb+g^{2}a)$, a linear transformation for fixed $g$ with regular coefficients and the additive group law.

Evaluation is $(1,x,x^2)$, whose image is exactly $\{a=1,c=b^2\}$: each point there is uniquely obtained by $x=b$, a regular inverse. The action in step 1.1 sends it to $(1,x+g,(x+g{)}^2)$, proving equivariance directly and displaying F2's construction.