Automorphisms of Weyl-Algebra Automorphism Groups

Definitions and mathematical statement

For each integer $n>1$, the source asks two parallel generation questions for \[ \operatorname{Aut}(\operatorname{Aut}(W_n)) \quad\text{and}\quad \operatorname{Aut}(\operatorname{Sympl}({\mathbb{C}}^{2n})). \] The title identifies a Weyl-algebra context, but the record does not define $W_n$, the symplectic group symbol or the category of outer automorphisms. In the usual interpretation, $W_n$ is the complex Weyl algebra with generators $q_i,p_i$ satisfying $[p_i,q_j]=\delta_{ij}$, $[q_i,q_j]=[p_i,p_j]=0$; $G_1=\operatorname{Aut}_{{\mathbb{C}}\text{-}\mathrm{alg}}(W_n)$. A customary meaning of $G_2=\operatorname{Sympl}({\mathbb{C}}^{2n})$ here is the group of polynomial automorphisms preserving $\omega=\sum_i dq_i\wedge dp_i$, not merely the linear symplectic group. These meanings require confirmation rather than being inferred from the word group alone.

For a group $G$, inner conjugation is $c_g(h)=ghg^{-1}$, and $\operatorname{Inn}(G)=\{c_g:g\in G\}$. For $\sigma\in\operatorname{Aut}({\mathbb{C}})$, coefficientwise application of $\sigma$, fixing the named generators or coordinates, gives the usual induced automorphism $s_\sigma$ of either intended group. In the Weyl case the coefficient map itself is semilinear, but its conjugation action sends complex-linear algebra automorphisms to complex-linear ones; the rational defining relations are preserved. In the polynomial symplectic case the same coefficient operation preserves the displayed rational symplectic form relations.

If "conjugations" means inner conjugations and outer $\operatorname{Aut}$ means abstract group automorphisms, the precise questions are \[ \operatorname{Aut}(G_j)= \left\langle\operatorname{Inn}(G_j),\ s_\sigma\ (\sigma\in\operatorname{Aut}({\mathbb{C}}))\right\rangle \quad(j=1,2), \] for every $n>1$. Generation means every automorphism is a finite composition of those operations. It does not claim unique expressions, a direct product or an isomorphism between the two groups being studied. No assertion for $n=1$ is inserted.

The source does not distinguish abstract, continuous, algebraic or ind-group automorphisms, or specify whether conjugation by a larger ambient normalizer is allowed. Such changes can alter the claim and which field automorphisms are admissible. The literature triage checked 2026-08-21 reports no authoritative complete resolution located, while the verification note reports notation cleanup. The open label remains dated metadata, not an independently checked structural theorem.

Short English statement

For both the Weyl-algebra automorphism group and the intended polynomial symplectic group, ask whether every automorphism of the group is generated by conjugations and coefficient-field automorphisms. The two claims are separate and apply only to $n>1$. The outer-automorphism category and meaning of conjugation are not specified.

Sources

• Local UnsolvedMath registry, permanent ID 30000657, source code catalogue entry, "Automorphisms of Weyl-Algebra Automorphism Groups," including both nested-group questions and verification metadata.
• Affine Algebraic Geometry (2007), Oberwolfach Reports, pp. 5--82, DOI 10.4171/owr/2007/01: https://doi.org/10.4171/owr/2007/01. This is also the sole supplied literature reference.
• Literature triage checked 2026-08-21; local record updated 2026-09-16. No external source has been visited or independently checked here.