Automorphism groups of Lie algebras of polynomial vector fields in infinitely many variables (the Bezushchak–Kashuba conjectures)

Over a field $\mathbb{F}$ of characteristic zero, let $W_\infty(\mathbb{F})$ be the union of the Cartan-type Lie algebras $W_n(\mathbb{F})$ of polynomial vector fields in $n$ variables, and let $W_X(\mathbb{F})=\operatorname{Der}\mathbb{F}[X]$ be the Lie algebra of all derivations of the polynomial algebra in countably many variables. While every automorphism of $W_n(\mathbb{F})$ is known (Rudakov, later Bavula and Kraft–Regeta) to be conjugation by a polynomial change of coordinates, and the derivations of $W_\infty(\mathbb{F})$ and $W_X(\mathbb{F})$ were recently described by Bezushchak and Kashuba, the corresponding automorphism question in infinitely many variables is open. Bezushchak and Kashuba pose two conjectures: (1) every automorphism of $W_\infty(\mathbb{F})$ is conjugation by an automorphism of $\mathbb{F}[X]$ from the finiteness subgroup $\operatorname{Aut}_{\mathrm{fin}}\mathbb{F}[X]$, and (2) every automorphism of $W_X(\mathbb{F})$ is conjugation by an arbitrary automorphism of $\mathbb{F}[X]$. The problem is to prove or refute these conjectures.

Background

Let $\mathbb{F}$ be a field of characteristic zero and let $\mathbb{F}[X_n]=\mathbb{F}[x_1,\dots,x_n]$ be the polynomial algebra in $n$ variables. The Lie algebra $W_n(\mathbb{F})=\operatorname{Der}\mathbb{F}[X_n]$ of all derivations (equivalently, polynomial vector fields $\sum_i f_i\,\partial/\partial x_i$) is an infinite-dimensional simple Lie algebra of Cartan type (one of the classical families of simple Lie algebras constructed from polynomial vector fields). Passing to infinitely many variables, set $X=\{x_1,x_2,\dots\}$ and let $\mathbb{F}[X]$ be the polynomial algebra in countably many variables (each polynomial still involves only finitely many variables). Bezushchak and Kashuba consider three nested Lie algebras: $W_\infty(\mathbb{F})=\bigcup_{n\ge1}W_n(\mathbb{F})$, the locally finite union, consisting of vector fields involving only finitely many variables; $W_X(\mathbb{F})=\operatorname{Der}\mathbb{F}[X]$, all derivations, whose elements are sums $\sum_{i\ge1}f_i\,\partial/\partial x_i$ in which a single component polynomial may involve arbitrarily many variables; and an intermediate algebra $W_{\mathrm{fin}}(\mathbb{F})$ of derivations in which each variable occurs in only finitely many component polynomials. Any automorphism $\varphi$ of the commutative algebra $\mathbb{F}[X]$ acts on derivations by conjugation, $d\mapsto\varphi\circ d\circ\varphi^{-1}$, and $\operatorname{Aut}_{\mathrm{fin}}\mathbb{F}[X]$ denotes the subgroup of automorphisms $\varphi$ such that each variable $x_n$ occurs in only finitely many of the polynomials $\varphi(x_i)$ and in only finitely many of the $\varphi^{-1}(x_i)$ — the condition that guarantees this conjugation preserves $W_\infty(\mathbb{F})$. For finitely many variables the structure is classical: Rudakov (1969) showed that every automorphism of $W_n(\mathbb{F})$ is conjugation by an automorphism of $\mathbb{F}[X_n]$, a result re-established and extended by Bavula (2017) for $\operatorname{Der}\mathbb{F}[x_1,\dots,x_n]$ and by Kraft and Regeta (2017) for the Lie algebras of vector fields on affine $n$-space with constant or zero divergence; and Takens (1973) proved that every derivation of $W_n(\mathbb{F})$ is inner (of the form $\operatorname{ad}(a):x\mapsto[x,a]$). For the infinite-variable analogs, Bezushchak and Kashuba (arXiv 2025; Journal of Algebra, volume 710, available online 2026) proved the derivation counterparts: every derivation of $W_\infty(\mathbb{F})$ is the restriction of $\operatorname{ad}(a)$ for some $a\in W_{\mathrm{fin}}(\mathbb{F})$ (so not all derivations of $W_\infty(\mathbb{F})$ are inner), while all derivations of $W_{\mathrm{fin}}(\mathbb{F})$ and of $W_X(\mathbb{F})$ are inner. Relatedly, Đoković and Zhao (1998) had shown that the inner derivations are dense in $W_X$-type algebras in the Tychonoff topology (the topology of pointwise convergence of the coefficients of a derivation relative to the standard basis), and Neeb (2005) proved the analogous statement for the locally simple algebra $\mathfrak{sl}_\infty$ (infinite matrices with finitely many nonzero entries): its derivations are exactly the $\operatorname{ad}(a)$ with $a$ in a larger matrix algebra. What no existing result covers is the automorphism group in the infinite-variable setting: whether the 'change of variables' picture survives, i.e. whether the geometrically evident automorphisms induced from $\operatorname{Aut}\mathbb{F}[X]$ exhaust all Lie-algebra automorphisms of $W_\infty(\mathbb{F})$ and $W_X(\mathbb{F})$. Bezushchak and Kashuba formulate this as two explicitly open conjectures (Section 6, 'Some Conjectures').

Problem Statement

Over a field $\mathbb{F}$ of characteristic zero, prove or refute the two conjectures posed by Bezushchak and Kashuba (2025):

Conjecture 1. Every automorphism of the Lie algebra $W_\infty(\mathbb{F})=\bigcup_{n\ge1}W_n(\mathbb{F})$ is a conjugation $d\mapsto\varphi\circ d\circ\varphi^{-1}$ by an automorphism $\varphi$ of the polynomial algebra $\mathbb{F}[X]$ belonging to the finiteness subgroup $\operatorname{Aut}_{\mathrm{fin}}\mathbb{F}[X]$, i.e. the subgroup of those $\varphi\in\operatorname{Aut}\mathbb{F}[X]$ for which each variable $x_n$ occurs in only finitely many of the polynomials $\varphi(x_i)$ and in only finitely many of the $\varphi^{-1}(x_i)$, $i\ge1$.

Conjecture 2. Every automorphism of the full Lie algebra $W_X(\mathbb{F})=\operatorname{Der}\mathbb{F}[X]$ is a conjugation by some automorphism $\varphi\in\operatorname{Aut}\mathbb{F}[X]$.

The two conjectures share one objective — extending Rudakov's finite-variable automorphism theorem to the infinite-variable setting — and are paired in Bezushchak–Kashuba; Conjecture 1 concerns the locally finite algebra $W_\infty(\mathbb{F})$ where the inducing automorphism must satisfy a finiteness condition, while Conjecture 2 concerns the full derivation algebra, where no finiteness condition is imposed.

References

1. Oksana Bezushchak, Iryna Kashuba, Derivations of Lie Algebras of Vector Fields in Infinitely Many Variables, arXiv:2507.04541 (2025), DOI 10.48550/arXiv.2507.04541, https://arxiv.org/abs/2507.04541
2. Oksana Bezushchak, Iryna Kashuba, Derivations of Lie algebras of vector fields in infinitely many variables, Journal of Algebra 710 (2027), 164–185, available online 2026, DOI 10.1016/j.jalgebra.2026.06.015, https://doi.org/10.1016/j.jalgebra.2026.06.015
3. A. N. Rudakov, Groups of automorphisms of infinite-dimensional simple Lie algebras, Mathematics of the USSR-Izvestiya 3 (1969), 707–722, DOI 10.1070/im1969v003n04abeh000798, https://doi.org/10.1070/im1969v003n04abeh000798
4. V. V. Bavula, The group of automorphisms of the Lie algebra of derivations of a polynomial algebra, Journal of Algebra and its Applications 16 (2017) 1750088, DOI 10.1142/S0219498817500888, https://arxiv.org/abs/1304.3836
5. Hanspeter Kraft, Andriy Regeta, Automorphisms of the Lie algebra of vector fields on affine n-space, Journal of the European Mathematical Society 19 (2017) 1577–1588, DOI 10.4171/jems/700, https://arxiv.org/abs/1402.5012
6. D. Ž. Đoković, Kaiming Zhao, Derivations, isomorphisms, and second cohomology of generalized Witt algebras, Transactions of the American Mathematical Society 350 (1998) 643–664, DOI 10.1090/s0002-9947-98-01786-3, https://doi.org/10.1090/s0002-9947-98-01786-3
7. Floris Takens, Derivations of vector fields, Compositio Mathematica 26 (1973) 151–158, http://www.numdam.org/item?id=CM_1973__26_2_151_0
8. K.-H. Neeb, Derivations of Locally Simple Lie Algebras, Journal of Lie Theory 15 (2005) 589–594, DOI 10.5802/jolt.392, https://doi.org/10.5802/jolt.392
9. Pradeep Bisht, Suman Rani, Santanu Tantubay, Automorphism groups and derivation algebras of Hamiltonian Lie algebras, arXiv:2603.11651 (2026), https://arxiv.org/abs/2603.11651