Non-surjective local and 2-local automorphisms of simple generalized Witt algebras with non-finitely generated grading group

For a simple generalized Witt algebra $W(A,T,\varphi)$ over a field of characteristic $0$, Chen–Zhao–Zhao (2021) proved that every invertible local or 2-local automorphism is an automorphism, and that invertibility is automatic when the grading group $A$ is finitely generated (covering the classical Witt algebras $W_n$). For non-finitely generated $A$, as happens for the algebra $W_\infty$ of derivations of the Laurent polynomial ring in countably many variables, non-surjective local and 2-local automorphisms exist, so the invertibility hypothesis cannot simply be dropped. The open problem is to complete the description in the infinite-rank setting: classify the local and 2-local automorphisms of $W(A,T,\varphi)$ when $A$ is not finitely generated — in particular the non-surjective ones, since injectivity is automatic — and characterize the structural data $(A,T,\varphi)$ for which every local (resp. 2-local) automorphism is automatically surjective, i.e. the exact conditions under which the invertibility requirement in Theorems 3.3 and 3.4 can be removed.

Background

Let $\mathbb{F}$ be a field of characteristic $0$, $A$ an abelian group, and $T$ an $\mathbb{F}$-vector space. The group algebra $\mathbb{F}A$ has basis elements $t^\alpha$, $\alpha\in A$, with multiplication $t^\alpha t^\beta=t^{\alpha+\beta}$. Given a map $\varphi:T\times A\to\mathbb{F}$ that is $\mathbb{F}$-linear in the first and additive in the second variable, the tensor product $W(A,T,\varphi)=\mathbb{F}A\otimes_{\mathbb{F}}T$ becomes a Lie algebra under $[t^\alpha\partial_\alpha,t^\beta\partial_\beta]=t^{\alpha+\beta}\big(\varphi(\partial_\alpha,\beta)\partial_\beta-\varphi(\partial_\beta,\alpha)\partial_\alpha\big),$ where $t^\alpha\partial$ abbreviates $t^\alpha\otimes\partial$. These are the generalized Witt algebras, defined in characteristic $0$ by Kawamoto (1986) and studied structurally by Djoković and Zhao (1998). The grading $W_\alpha=t^\alpha T$ has $T=W_0$ as a Cartan subalgebra (a nilpotent subalgebra equal to its own normalizer). By Kawamoto's simplicity theorem, $W(A,T,\varphi)$ is simple exactly when $A\neq 0$ and $\varphi$ is nondegenerate, meaning $\varphi(\partial,\alpha)=0$ for all $\alpha$ forces $\partial=0$ and $\varphi(\partial,\alpha)=0$ for all $\partial$ forces $\alpha=0$; such an $A$ is necessarily torsion-free. When $A\cong\mathbb{Z}^n$ and $\dim T=n$ one recovers the classical Witt algebra $W_n=\operatorname{Der}\mathbb{F}[t_1^{\pm1},\dots,t_n^{\pm1}]$, the Lie algebra of vector fields on an $n$-dimensional torus; with $A=\mathbb{Z}^{(\infty)}$ (the direct sum of countably many copies of $\mathbb{Z}$) one obtains $W_\infty=\operatorname{Der}\mathbb{F}[t_1^{\pm1},t_2^{\pm1},\dots]$. Djoković and Zhao showed that every automorphism of a simple $W(A,T,\varphi)$ has the form $\theta(\chi,\sigma,\tau)(t^\alpha\partial)=\chi(\alpha)t^{\sigma(\alpha)}\tau(\partial)$, where $\chi$ is a character of $A$ (a group homomorphism $A\to\mathbb{F}^{\times}$), $\sigma\in\operatorname{Aut}(A)$, and $\tau\in GL(T)$ satisfies $\varphi(\tau(\partial),\sigma(\gamma))=\varphi(\partial,\gamma)$.

A linear map $\Phi$ on a Lie algebra $L$ is a local automorphism (Larson and Sourour, 1990) if for every $x\in L$ there is an automorphism $\theta_x$ of $L$ with $\Phi(x)=\theta_x(x)$; a map $\Phi:L\to L$, not assumed linear, is a 2-local automorphism (Šemrl, 1997) if for all $x,y\in L$ there is an automorphism $\theta_{x,y}$ with $\Phi(x)=\theta_{x,y}(x)$ and $\Phi(y)=\theta_{x,y}(y)$. The general program for such locally defined maps is to decide whether they are automatically automorphisms, i.e. whether automorphisms are determined by their local actions; the derivations analogue (local and 2-local derivations) has been studied for Witt and generalized Witt algebras by Ayupov–Kudaybergenov–Yusupov, Zhao–Chen–Zhao, and Chen–Zhao–Zhao.

Chen–Zhao–Zhao (2021) proved that every invertible local or 2-local automorphism of a simple generalized Witt algebra over any field of characteristic $0$ is an automorphism, and that invertibility is automatic when $A$ is finitely generated, so in particular every local or 2-local automorphism of $W_n$ is an automorphism. Their key tool is that any two elements of $W(A,T,\varphi)$ lie in a subalgebra isomorphic to some $W(\mathbb{Z}^n,V_m,\varphi)$ with finitely generated grading group, where rigidity holds. However, local and 2-local automorphisms are injective by definition yet need not be surjective: on $W_\infty$, the index shift $\Phi(t^\alpha\frac{d}{dt_i})=t^{(0,\alpha)}\frac{d}{dt_{i+1}}$ is simultaneously a local and a 2-local automorphism that is not surjective. Hence the invertibility hypothesis in their Theorems 3.3 and 3.4 is genuinely necessary in general, removable when $A$ is finitely generated, and its removability for the remaining infinite-rank algebras is not determined: neither a description of the non-surjective local/2-local maps nor a criterion distinguishing the infinite-rank algebras on which automatic surjectivity holds is known.

Problem Statement

Let $\mathbb{F}$ be a field of characteristic $0$ and let $W(A,T,\varphi)$ be a simple generalized Witt algebra whose grading group $A$ is not finitely generated (for example $W_\infty=\operatorname{Der}\mathbb{F}[t_1^{\pm1},t_2^{\pm1},\dots]$, where $A=\mathbb{Z}^{(\infty)}$). Since local and 2-local automorphisms are automatically injective, the unresolved issue is exactly surjectivity. The problem, at the source's natural generality over the defining data $(A,T,\varphi)$, has two linked parts:

(i) Classify the local automorphisms and the 2-local automorphisms of $W(A,T,\varphi)$ when $A$ is not finitely generated: determine all maps $\Phi$ (linear for local automorphisms, not assumed linear for 2-local automorphisms) that satisfy the one-point, resp. two-point, matching condition with automorphisms of the form $\theta(\chi,\sigma,\tau)$ yet are not surjective, and describe their structure.

(ii) Give a criterion on the defining data separating the two behaviors: characterize the triples $(A,T,\varphi)$ for which every local (resp. every 2-local) automorphism of $W(A,T,\varphi)$ is automatically surjective — equivalently, the structural hypotheses on $A$, $T$, or $\varphi$ under which the invertibility requirement in Theorems 3.3 and 3.4 of Chen–Zhao–Zhao can be removed, improving on their Corollary 3.6 ($A$ finitely generated).

The two parts form one objective, the complete description of local and 2-local automorphisms of simple generalized Witt algebras; part (i) supplies the non-surjective maps whose existence delimits part (ii), and the criterion of part (ii) is read off from where the classification of part (i) degenerates to automorphisms.

References

1. Yang Chen, Kaiming Zhao, Yueqiang Zhao, 'Local and $2$-local automorphisms of simple generalized Witt algebras', Arkiv för Matematik 59 (2021), no. 1, 1–10. DOI: 10.4310/arkiv.2021.v59.n1.a1; arXiv:2003.13893. https://doi.org/10.4310/arkiv.2021.v59.n1.a1 (preprint: https://arxiv.org/abs/2003.13893)
2. Naoki Kawamoto, 'Generalizations of Witt algebras over a field of characteristic zero', Hiroshima Mathematical Journal 16 (1986), no. 2, 417–426. DOI: 10.32917/hmj/1206130439. https://doi.org/10.32917/hmj/1206130439
3. Dragomir Ž. Djoković and Kaiming Zhao, 'Derivations, isomorphisms, and second cohomology of generalized Witt algebras', Transactions of the American Mathematical Society 350 (1998), no. 2, 643–664. DOI: 10.1090/S0002-9947-98-01786-3. https://doi.org/10.1090/S0002-9947-98-01786-3
4. David R. Larson and Ahmed R. Sourour, 'Local derivations and local automorphisms of $B(X)$', Proceedings of Symposia in Pure Mathematics 51, Part 2 (1990), 187–194. DOI: 10.1090/pspum/051.2/1077437. https://doi.org/10.1090/pspum/051.2/1077437
5. Peter Šemrl, 'Local automorphisms and derivations on $B(H)$', Proceedings of the American Mathematical Society 125 (1997), no. 9, 2677–2680. DOI: 10.1090/s0002-9939-97-04073-2. https://doi.org/10.1090/s0002-9939-97-04073-2
6. Shavkat Ayupov, Karimbergen Kudaybergenov, Baxtiyor Yusupov, '2-Local derivations on generalized Witt algebras', Linear and Multilinear Algebra 69 (2021), no. 16, 3130–3140 (published online 2019). DOI: 10.1080/03081087.2019.1708846. https://doi.org/10.1080/03081087.2019.1708846
7. Yueqiang Zhao, Yang Chen, Kaiming Zhao, '2-Local derivations on Witt algebras', Journal of Algebra and Its Applications 20 (2021), no. 4, 2150068 (published online 2020). DOI: 10.1142/S0219498821500687. https://doi.org/10.1142/S0219498821500687
8. Yang Chen, Kaiming Zhao, Yueqiang Zhao, 'Local derivations on Witt algebras', Linear and Multilinear Algebra 70 (2022), no. 6, 1159–1172 (published online 2020). DOI: 10.1080/03081087.2020.1754750. https://doi.org/10.1080/03081087.2020.1754750
9. Yufeng Yao, Kaiming Zhao, 'Local properties of Jacobson-Witt algebras', Journal of Algebra 586 (2021), 1110–1121. DOI: 10.1016/j.jalgebra.2021.07.025. https://doi.org/10.1016/j.jalgebra.2021.07.025
10. Yulu Chen, Bin Wang, Kaiming Zhao, 'Local and 2-local automorphisms on the N = 2 superconformal algebras', Communications in Algebra 53 (2025), no. 12, 5034–5044. DOI: 10.1080/00927872.2025.2501209. https://doi.org/10.1080/00927872.2025.2501209
11. Qingyan Wu, Shoulan Gao, Dong Liu, Chang Ye, 'Local derivations and local automorphisms on the super Virasoro algebras', Communications in Algebra 52 (2024), no. 6, 2616–2625. DOI: 10.1080/00927872.2024.2302401; arXiv:2302.05837. https://doi.org/10.1080/00927872.2024.2302401
12. Karimbergen Kudaybergenov, Tuuelbay Kurbanbaev, Bakhrom Omirov, 'Local automorphisms of complex solvable Lie algebras of maximal rank', Linear and Multilinear Algebra 72 (2024), no. 13, 2197–2220 (published online 2023). DOI: 10.1080/03081087.2023.2241610. https://doi.org/10.1080/03081087.2023.2241610
13. Abror Khudoyberdiyev, Doston Jumaniyozov, 'Local derivations and automorphisms of nilpotent Lie algebras', Communications in Algebra 53 (2025), no. 5, 1921–1933 (published online 2024). DOI: 10.1080/00927872.2024.2426040. https://doi.org/10.1080/00927872.2024.2426040
14. Ivan Kaygorodov, Abror Khudoyberdiyev, Zarina Shermatova, 'Quasi-derivations of Witt and related algebras', Research in the Mathematical Sciences 13 (2026), no. 1, article 20. DOI: 10.1007/s40687-025-00596-6. https://doi.org/10.1007/s40687-025-00596-6
15. Roman Lubkov, Ilya Shiringovskiy, 'Transposed Poisson structure on the Witt-type algebra $\mathcal{W}(a,-1)$: Derivations, Automorphisms, and Rota–Baxter operators', Communications in Mathematics 34 (2026), no. 1. DOI: 10.46298/cm.18245; arXiv:2606.20584. https://doi.org/10.46298/cm.18245 (preprint: https://arxiv.org/abs/2606.20584)