The first Kac–Weisfeiler conjecture for restricted Lie algebras and the index of algebras with isomorphic enveloping algebras

The first Kac–Weisfeiler conjecture (KW1, 1971) predicts that for a finite-dimensional restricted Lie algebra $\mathfrak g$ over an algebraically closed field of characteristic $p>0$, the maximal dimension $M(\mathfrak g)$ of a simple $\mathfrak g$-module equals $p^{\frac{1}{2}(\dim\mathfrak g-\operatorname{ind}\mathfrak g)}$, where $\operatorname{ind}\mathfrak g$ is the minimal dimension of a coadjoint stabiliser. Topley (2017) disproved the extension of KW1 to arbitrary non-restricted Lie algebras by exhibiting pairs of Lie algebras with isomorphic enveloping algebras but distinct indices, and posed the two questions that define this problem: (i) does KW1 fail for some restricted Lie algebra, and (ii) do there exist restricted Lie algebras with isomorphic enveloping algebras but distinct indices? KW1 is known for Lie algebras of reductive groups, for algebras admitting a linear form with toral stabiliser, and — by Martin–Stewart–Tikaradze–Topley (2019) — for all restricted Lie subalgebras of $\mathfrak{gl}_n$ in sufficiently large characteristic; the general lower bound is due to Premet–Skryabin (1999). No restricted counterexample and no resolution of question (ii) is known in the literature: both questions remain open, with the surviving core concentrated in characteristics small relative to the dimension of the algebra.

Background

Let $k$ be an algebraically closed field of prime characteristic $p$ and let $\mathfrak g$ be a finite-dimensional Lie algebra over $k$. Zassenhaus (1954) proved that the dimensions of simple $\mathfrak g$-modules are uniformly bounded and that the supremum $M(\mathfrak g)$ of these dimensions is a power of $p$. A Lie algebra is called restricted if it carries a $p$-map $x\mapsto x^{[p]}$ satisfying Jacobson's axioms, which are modelled on the matrix power map $X\mapsto X^p$; a subalgebra of $\mathfrak{gl}_n(k)$ closed under taking $p$-th powers of matrices is the basic example. The universal enveloping algebra $U(\mathfrak g)$ is the associative $k$-algebra universal for representations of $\mathfrak g$. For restricted $\mathfrak g$ the elements $x^p-x^{[p]}$ are central in $U(\mathfrak g)$ and generate the $p$-centre; every simple module is annihilated by a unique maximal ideal of the $p$-centre, encoded by a linear form $\chi\in\mathfrak g^*$ called the $p$-character, so the simple modules are exactly the simple modules over the finite-dimensional reduced enveloping algebras $U_\chi(\mathfrak g)$.

The index $\operatorname{ind}\mathfrak g$ is the minimal dimension of a stabiliser $\mathfrak g_\chi=\{x\in\mathfrak g:\chi([x,y])=0\text{ for all }y\in\mathfrak g\}$ of the coadjoint representation of $\mathfrak g$ on $\mathfrak g^*$; for any concrete algebra it is computed by elementary linear algebra, and $\dim\mathfrak g-\operatorname{ind}\mathfrak g$ is always even. In modern notation the first Kac–Weisfeiler conjecture (KW1), stated by Kac and Weisfeiler in 1971, predicts

$M(\mathfrak g)=p^{\frac{1}{2}(\dim\mathfrak g-\operatorname{ind}\mathfrak g)}$

for every restricted Lie algebra $\mathfrak g$. Premet (1995) verified the conjecture for the Lie algebras of connected reductive algebraic groups (with simply connected commutator subgroup, $p$ larger than every coefficient of every positive root, and a nondegenerate invariant trace form when $p=2$). Premet and Skryabin (1999) proved the general lower bound $M(\mathfrak g)\ge p^{\frac{1}{2}(\dim\mathfrak g-\operatorname{ind}\mathfrak g)}$ for every restricted $\mathfrak g$, and showed that KW1 holds whenever some linear form has a stabiliser that is a torus, meaning an abelian subalgebra on which the $p$-map is surjective; hence for restricted algebras the open content of KW1 is exactly the upper bound.

The statement of KW1 makes sense for arbitrary, non-restricted Lie algebras, and it was hoped that it might hold in that generality. Topley (2017) refuted this hope by constructing the first non-restricted counterexamples. His method is a transfer principle between Lie algebras with isomorphic enveloping algebras: both $M(\mathfrak g)$ and $\dim\mathfrak g$ depend only on the isomorphism class of $U(\mathfrak g)$, so if KW1 held for all Lie algebras then $\operatorname{ind}\mathfrak g$ would be determined by $U(\mathfrak g)$; he exhibited non-restricted pairs $\mathfrak g,\mathfrak g'$ with $U(\mathfrak g)\cong U(\mathfrak g')$ and $\operatorname{ind}\mathfrak g\neq\operatorname{ind}\mathfrak g'$, forcing the predicted equality to fail. This ties KW1 to the isomorphism problem for enveloping algebras in positive characteristic (Riley–Usefi 2007): over fields of characteristic zero no pair of non-isomorphic finite-dimensional Lie algebras with isomorphic enveloping algebras is known, while in characteristic $p$ such pairs exist but are far from classified.

Most recently, Martin–Stewart–Tikaradze–Topley (2019) proved KW1 for every restricted Lie subalgebra of $\mathfrak{gl}_n(k)$ whenever $p$ is sufficiently large relative to $n$ — and consequently for the Lie algebras of affine group schemes of finite type in almost all characteristics — by using a first-order version of the Lefschetz principle to transport solvable weak polarisations (isotropic subalgebras of dimension $\frac{1}{2}(\dim\mathfrak g+\operatorname{ind}\mathfrak g)$) from characteristic zero. No result covers the regime where $p$ is small relative to the dimension, and the two questions raised at the end of Topley's paper, reproduced in the problem statement, are the precise formulation of what is missing.

Problem Statement

The problem consists of the two open questions raised at the end of Topley (2017, Proc. Amer. Math. Soc. 145), which together aim to determine the exact status of the first Kac–Weisfeiler conjecture in its original restricted setting:

(i) Does there exist a restricted Lie algebra $\mathfrak g$ over an algebraically closed field of positive characteristic for which the first Kac–Weisfeiler conjecture fails, i.e. for which $M(\mathfrak g)\neq p^{\frac{1}{2}(\dim\mathfrak g-\operatorname{ind}\mathfrak g)}$? Equivalently: prove the upper bound $M(\mathfrak g)\le p^{\frac{1}{2}(\dim\mathfrak g-\operatorname{ind}\mathfrak g)}$ for all restricted Lie algebras, or exhibit a restricted counterexample. The question is unrestricted — any prime characteristic and any dimension are admitted, and by the Premet–Skryabin lower bound a counterexample necessarily satisfies $M(\mathfrak g)>p^{\frac{1}{2}(\dim\mathfrak g-\operatorname{ind}\mathfrak g)}$.

(ii) Do there exist two restricted Lie algebras $\mathfrak g$ and $\mathfrak g'$ over the same algebraically closed field with $U(\mathfrak g)\cong U(\mathfrak g')$ as $k$-algebras but $\operatorname{ind}\mathfrak g\neq\operatorname{ind}\mathfrak g'$?

The two questions share one scientific objective: locating the boundary of validity of KW1 for restricted Lie algebras. They are logically coupled. A positive answer to (ii) implies a positive answer to (i), because $U(\mathfrak g)\cong U(\mathfrak g')$ forces $M(\mathfrak g)=M(\mathfrak g')$ and $\dim\mathfrak g=\dim\mathfrak g'$, so KW1 cannot hold for both algebras when their indices differ. A negative answer to (ii) would show that the index is an isomorphism invariant of the enveloping algebra for restricted Lie algebras, which is supporting evidence for KW1 but does not by itself settle (i). A complete resolution of the problem requires an answer to (i) for the unrestricted class of restricted Lie algebras; question (ii) is a well-defined component of the problem that may also be answered independently.

References

1. Lewis Topley, A non-restricted counterexample to the first Kac–Weisfeiler conjecture, Proceedings of the American Mathematical Society 145 (2017), no. 5, 1937–1942. DOI: 10.1090/proc/13362; arXiv:1605.06033. https://doi.org/10.1090/proc/13362
2. B. Yu. Veĭsfeĭler (Boris Weisfeiler) and V. G. Kac, Irreducible representations of Lie p-algebras, Functional Analysis and Its Applications 5 (1971), no. 2, 111–117 (English translation; Russian original in Funkcional. Anal. i Priložen. 5:2 (1971), 28–36). DOI: 10.1007/BF01076415. https://doi.org/10.1007/BF01076415
3. H. Zassenhaus, The representations of Lie algebras of prime characteristic, Proceedings of the Glasgow Mathematical Association 2 (1954), 1–36. DOI: 10.1017/S2040618500032974. https://doi.org/10.1017/S2040618500032974
4. Alexander Premet, Irreducible representations of Lie algebras of reductive groups and the Kac–Weisfeiler conjecture, Inventiones Mathematicae 121 (1995), 79–117. DOI: 10.1007/BF01884291. https://doi.org/10.1007/BF01884291
5. Alexander Premet and Serge Skryabin, Representations of restricted Lie algebras and families of associative L-algebras, Journal für die reine und angewandte Mathematik 507 (1999), 189–218. DOI: 10.1515/crll.1999.507.189. https://doi.org/10.1515/crll.1999.507.189
6. D. Riley and H. Usefi, The isomorphism problem for universal enveloping algebras of Lie algebras, Algebras and Representation Theory 10 (2007), no. 6, 517–532. DOI: 10.1007/s10468-007-9083-0. https://doi.org/10.1007/s10468-007-9083-0
7. Benjamin Martin, David Stewart and Lewis Topley, with an appendix by Akaki Tikaradze, A proof of the first Kac–Weisfeiler conjecture in large characteristics, Representation Theory 23 (2019), 278–293. DOI: 10.1090/ert/529; arXiv:1810.12632. https://doi.org/10.1090/ert/529
8. Bin Shu, Irreducible modules of modular Lie superalgebras and super version of the first Kac–Weisfeiler conjecture, Canadian Mathematical Bulletin 67 (2024), no. 3, 554–573. DOI: 10.4153/s0008439523000966; arXiv:2312.08666. https://doi.org/10.4153/s0008439523000966
9. Ye Ren, Bin Shu, Fanlei Yang and An Zhang, Modular representations of strange classical Lie superalgebras and the first super Kac–Weisfeiler conjecture, arXiv:2307.00483 (2023). https://arxiv.org/abs/2307.00483
10. Matthew Chaffe and Lewis Topley, Modular representations of truncated current Lie algebras, arXiv:2311.08208 (2023, revised 2024). https://arxiv.org/abs/2311.08208
11. Xabier García-Martínez, Non-isomorphic restricted Lie algebras with isomorphic restricted enveloping algebras, arXiv:2606.17282 (2026). https://arxiv.org/abs/2606.17282
12. Akaki Tikaradze, A generalization of Veldkamp's theorem for a class of Lie algebras, Journal of Algebra 579 (2021), 64–72. DOI: 10.1016/j.jalgebra.2021.03.018; arXiv:1907.03031. https://doi.org/10.1016/j.jalgebra.2021.03.018
13. Benjamin Martin, David I. Stewart and Lewis Topley, Smoothness of stabilisers in generic characteristic, Algebra & Number Theory 20 (2026), 1159–1184. DOI: 10.2140/ant.2026.20.1159; arXiv:1810.12628. https://doi.org/10.2140/ant.2026.20.1159