A note on smooth $SL_2$-surfaces

Working over a field $k$ of characteristic zero, we study the ring $\mathfrak{R}=\mathfrak{D}^{\mathbb{Z}_2}$ where $\mathfrak{D}=k[x_0,x_1,x_2]/(2x_0x_2-x_1^2-1)$ and $\mathbb{Z}_2$ acts by $x_i\to -x_i$. $\mathfrak{D}$ admits an algebraic $SL_2(k)$-action which restricts to $\mathfrak{R}$. Our results include the following. (1) If $k$ is algebraically closed, the smooth $SL_2$-surface $X={\rm Spec}(\mathfrak{R})$ admits an algebraic embedding in $\mathbb{A}_k^4$, and for any such embedding the $SL_2(k)$-action on $X$ does not extend to $\mathbb{A}_k^4$. In addition, there is no algebraic embedding of $X$ in $\mathbb{A}_k^3$. (2) The automorphism group ${\rm Aut}_k(\mathfrak{R})$ acts transitively on the set of irreducible locally nilpotent derivations of $\mathfrak{R}$. (3) Every automorphism of $\mathfrak{R}$ extends to $\mathfrak{D}$, and ${\rm Aut}_k(\mathfrak{R})=PSL_2(k)\ast_HT$ where $T$ is its triangular subgroup. (4) $\mathfrak{R}$ is non-cancellative, i.e., there exists a ring $\mathfrak{S}$ such that $\mathfrak{R}^{[1]}\cong_k\mathfrak{S}^{[1]}$ but $\mathfrak{R}\not\cong_k\mathfrak{S}$. In order to distinguish $\mathfrak{R}$ from $\mathfrak{S}$, we calculate the plinth invariant for $\mathfrak{R}$.