k-Automorphism Group of A₍c,q₎

• The k-automorphism group of A₍c,q₎ is a well-defined algebraic structure that splits as a semidirect product of special automorphisms and a diagonal torus subgroup.
• Special automorphisms, characterized by translations in y that fix the Makar-Limanov invariant, ensure a rigid and precise subgroup classification.
• The torus subgroup, defined by specific scaling conditions on x and y, underpins the diagonal symmetries and aids in classifying isotropy groups for non–locally nilpotent derivations.

The $k$-automorphism group of $A_{c,q}$, where

$A_{c,q} = k[x,y,z]/(c(x)z - q(x,y)),$

with $c(x) \in k[x]$ a polynomial of degree at least $2$ and $q(x,y) \in k[x,y]$ a quasi-monic polynomial in $y$ of degree at least $2$, is a well-structured algebraic group. The most detailed description is given in (Ahouita et al., 8 Oct 2025), which supplies both a canonical decomposition of this group and a classification of isotropy (stabilizer) subgroups for non–locally nilpotent derivations. The structure of $\operatorname{Aut}_k(A_{c,q})$ shares deep connections with classical invariants such as the Makar-Limanov invariant and encodes both “affine-type” automorphisms and a distinguished torus subgroup.

1. Structural Decomposition of $\operatorname{Aut}_k(A_{c,q})$

The automorphism group of $A_{c,q}$ admits a canonical splitting as an inner semidirect product

$$\operatorname{Aut}_k(A_{c,q}) \cong \mathrm{saut}_k(A_{c,q}) \rtimes \mathcal{G}_{c,q},$$

where:

• $\mathrm{saut}_k(A_{c,q})$ is the subgroup of automorphisms that act trivially on the subalgebra $k[\bar{x}]$; these automorphisms fix the Makar-Limanov invariant, which in this setting equals $k[\bar{x}]$ ((Ahouita et al., 8 Oct 2025), Lemma 2.2 and above).
• $\mathcal{G}_{c,q} \subset (k^*)^2$ is the diagonal torus subgroup defined by

$$\mathcal{G}_{c,q} = \left\{ (e, u) \in k^* \times k^* : c(e x) = e^n c(x),\;\; q(e x, u y) = u^d q(x, y) \right\}$$

where $n = \deg c$ and $d = \deg_y q$.

This is realized via the explicit morphism

$$\begin{aligned} \varphi_{e,u} :\quad & x \mapsto e x, \ & y \mapsto u y, \ & z \mapsto e^{-n} u^d z, \end{aligned}$$

as verified directly on the defining relation $c(x)z - q(x,y)$ ((Ahouita et al., 8 Oct 2025), construction of $\varphi$).

2. Special Automorphisms and the Makar-Limanov Invariant

Elements $\tau\in\mathrm{saut}_k(A_{c,q})$ satisfy $\tau(\bar{x}) = \bar{x}$ and typically take the form

$\tau(y) = y + h(x)$

for some $h(x) \in k[x]$ determined by the relation $c(x)z = q(x,y)$ and compatibility with $A_{c,q}$'s defining ideal. Such automorphisms correspond to “translations in $y$ by functions of $x$” that descend from the automorphism group of the Danielewski surface.

The importance of the Makar-Limanov invariant (here $k[\bar{x}]$) is that it distinguishes $\mathrm{saut}_k(A_{c,q})$ as the normal subgroup fixing all variables in $k[\bar{x}]$ ((Ahouita et al., 8 Oct 2025), Section 2).

3. The Torus Subgroup $\mathcal{G}_{c,q}$

The subgroup $\mathcal{G}_{c,q}$ captures “diagonal” symmetries, i.e., scaling actions on $x$ and $y$ (with a compensating scaling on $z$). The defining conditions $c(e x) = e^n c(x)$ and $q(e x, u y) = u^d q(x, y)$ ensure the automorphism descends to $A_{c,q}$; i.e., these are the scalings compatible with the relation $c(x)z = q(x,y)$. This group is always an algebraic subgroup of $(k^*)^2$, typically a torus or a subgroup thereof, its structure depending on $c$ and $q$ ((Ahouita et al., 8 Oct 2025), definition of $\mathcal{G}_{c,q}$).

Such diagonal automorphisms form a subgroup isomorphic to (possibly a sub-torus of) $(k^*)^2$ and are embedded in $\operatorname{Aut}_k(A_{c,q})$ via the map $\varphi$ described above.

4. The Main Splitting Theorem

The composition $$\psi \circ \varphi = \operatorname{id}_{\mathcal{G}_{c,q}}$$ provides a right inverse, yielding a split short exact sequence

$$1 \longrightarrow \mathrm{saut}_k(A_{c,q}) \longrightarrow \operatorname{Aut}_k(A_{c,q}) \longrightarrow \mathcal{G}_{c,q} \longrightarrow 1,$$

and thereby

$$\operatorname{Aut}_k(A_{c,q}) \cong \mathrm{saut}_k(A_{c,q}) \rtimes \mathcal{G}_{c,q}$$

((Ahouita et al., 8 Oct 2025), Theorem 3.1). Thus, every automorphism decomposes uniquely as a special automorphism followed by a diagonal automorphism. This structure provides a complete description of $\operatorname{Aut}_k(A_{c,q})$ as an abstract group.

5. Isotropy Groups for Non–Locally Nilpotent Derivations

Given a $k$–derivation $\delta$ of $A_{c,q}$ which is not locally nilpotent, the isotropy group

$$\operatorname{Aut}_k(A_{c,q}, \delta) = \big\{ \sigma \in \operatorname{Aut}_k(A_{c,q}) \mid \sigma \circ \delta = \delta \circ \sigma \big\}$$

is classified as a linear algebraic group of dimension at most $3$ ((Ahouita et al., 8 Oct 2025), Theorem 4.1).

Two structural possibilities for $\operatorname{Aut}_k(A_{c,q}, \delta)$ are established:

• Either the isotropy is (up to isomorphism) a closed algebraic subgroup of $\mathcal{G}_{c,q}$ (i.e., only diagonal automorphisms commute with $\delta$), in which case it has dimension at most $2$,
• Or $$\operatorname{Aut}_k(A_{c,q}, \delta) \cong \mathbb{G}_a \rtimes H$$ for some closed subgroup $H \leq \mathcal{G}_{c,q}$, where $\mathbb{G}_a$ represents an additive group arising as a one-parameter subgroup of $\mathrm{saut}_k(A_{c,q})$, resulting in overall dimension at most $3$.

In all cases, $\operatorname{Aut}_k(A_{c,q}, \delta)$ is “small”, reflecting rigidity of non–locally nilpotent derivations.

6. Comparison with Related Automorphism Group Frameworks

The decomposition and rigidity found for $\operatorname{Aut}_k(A_{c,q})$ share conceptual similarities with several frameworks:

• In generalized Weyl algebras, affine automorphism groups (including toral and translation components) arise, with a quantum analogue of the Dixmier conjecture (every endomorphism of the simple localization is an automorphism) producing a similarly rigid automorphism group (Tang, 2016).
• For quantum algebras and function fields such as $k_q(x,y)$, automorphism groups are often generated by “elementary” and “torus” automorphisms, with the fixed ring structure and division algebra rigidity offering further analogies (Fryer, 2013).
• In the Weyl algebra context, stabilizer subgroups (Stafford subgroups) of right ideals exhibit strong self-normalizing properties, suggesting deep rigidity of the subgroup structure (Kouakou et al., 2011).

Each of these frameworks supports or is compatible with the decomposition and constraint properties established for $A_{c,q}$.

7. Significance of the Canonical Semidirect Product and Applications

This explicit semidirect product structure provides a unified perspective for the paper of automorphism groups of Danielewski-type algebras. The separation of affine/“special” automorphisms from diagonal torus actions greatly simplifies questions of action on subvarieties, automorphic equivalence, and interplay with derivations. The constraints on isotropy groups for derivations provide a foundation for rigidity results, classification of group actions, and aids in understanding dynamical systems defined via algebraic flows on such varieties.

The results also facilitate the transfer of techniques from the theory of automorphism groups of surfaces, quantum coordinate algebras, and generalized Weyl algebras to the setting of Danielewski-type algebras and affine threefolds.

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Summary Table: Structure of $\operatorname{Aut}_k(A_{c,q})$

Subgroup	Definition/Action	Structure
$\mathrm{saut}_k(A_{c,q})$	Fixes $k[\bar{x}]$, affine “special” automorphisms, e.g., $y \mapsto y + h(x)$	Normal subgroup
$\mathcal{G}_{c,q}$	$(e,u)$ with $c(e x) = e^n c(x)$, $q(e x, u y) = u^d q(x,y)$	Subgroup of $(k^*)^2$
$\operatorname{Aut}_k(A_{c,q})$	Entire automorphism group, all $k$-algebra automorphisms of $A_{c,q}$	Semidirect product $\mathrm{saut}_k \rtimes \mathcal{G}_{c,q}$

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This organizational scheme succinctly encodes the automorphism group’s structure and significance in the context of affine algebraic geometry and invariant theory.