Martin, Stewart and Topley proved the first Kac–Weisfeiler conjecture (KW1) — that the maximal dimension M(g)M(\mathfrak g) of a simple module over a restricted Lie algebra g\mathfrak g equals p(dim⁡g−ind⁡g)/2p^{(\dim\mathfrak g-\operatorname{ind}\mathfrak g)/2} — for every restricted Lie subalgebra of gld(k)\mathfrak{gl}_d(k) over an algebraically closed field of characteristic p>p0(d)p>p_0(d), and deduced it for Lie algebras of group schemes in almost all characteristics. Both statements rest on a qualitative model-theoretic transfer (a first-order Lefschetz principle), so the threshold p0(d)p_0(d) exists but is not effectively computable from the proof, and the method is confined to algebras admitting a faithful restricted representation of bounded dimension. This problem records the two linked open targets the paper itself left: (1) give an explicit, effectively computable bound p0(d)p_0(d) that makes the large-characteristic theorem effective; (2) settle the Frobenius-characterisation conjecture for arbitrary restricted Lie algebras — g\mathfrak g has index 00 if and only if the reduced enveloping algebras Uχ(g)U_\chi(\mathfrak g) are simple for all χ\chi in some non-empty open subset of g∗\mathfrak g^*. Both targets remain open.

Background

A restricted Lie algebra (or Lie pp-algebra) over an algebraically closed field kk of prime characteristic pp is a finite-dimensional Lie algebra g\mathfrak g equipped with a pp-mapping x↦x[p]x\mapsto x^{[p]} satisfying Jacobson's axioms; the basic example is a Lie subalgebra of the full matrix algebra gln(k)\mathfrak{gl}_n(k) closed under ordinary matrix pp-th powers. For each linear form χ∈g∗\chi\in\mathfrak g^* one forms the reduced enveloping algebra Uχ(g)U_\chi(\mathfrak g), the quotient of the universal enveloping algebra by the relations xp−x[p]−χ(x)=0x^p-x^{[p]}-\chi(x)=0 for all x∈gx\in\mathfrak g; it has dimension pdim⁡gp^{\dim\mathfrak g}, and every simple g\mathfrak g-module is a module over exactly one such Uχ(g)U_\chi(\mathfrak g) (its pp-character). Write M(g)M(\mathfrak g) for the maximal dimension of a simple g\mathfrak g-module, and define the index ind⁡(g)=min⁡χ∈g∗dim⁡gχ\operatorname{ind}(\mathfrak g)=\min_{\chi\in\mathfrak g^*}\dim\mathfrak g_\chi, where gχ\mathfrak g_\chi is the stabiliser of χ\chi for the coadjoint action. In 1971 Kac and Weisfeiler conjectured that M(g)=p(dim⁡g−ind⁡g)/2M(\mathfrak g)=p^{(\dim\mathfrak g-\operatorname{ind}\mathfrak g)/2} for every restricted Lie algebra (KW1). Premet and Skryabin proved the lower bound M(g)≥p(dim⁡g−ind⁡g)/2M(\mathfrak g)\ge p^{(\dim\mathfrak g-\operatorname{ind}\mathfrak g)/2} in full generality and established KW1 for algebras admitting a linear form whose coadjoint stabiliser is a torus (a Lie algebra that coincides with its own span of pp-th powers); KW1 was previously known for restricted completely solvable Lie algebras (all adjoint maps have a triangular form), and later for centralisers of nilpotent matrices in gln\mathfrak{gl}_n and for truncated current Lie algebras over reductive Lie algebras (algebras of the form h⊗k[t]/(tn)\mathfrak h\otimes k[t]/(t^n) with the bracket taken factorwise). For non-restricted Lie algebras the equality genuinely fails, with explicit counterexamples depending on the arithmetic of pp.

Martin, Stewart and Topley proved that for each dd there is a prime threshold p0(d)p_0(d) such that KW1 holds for every restricted Lie subalgebra of gld(k)\mathfrak{gl}_d(k) whenever kk is algebraically closed of characteristic p>p0(d)p>p_0(d), and consequently for the Lie algebra of any fixed affine group scheme of finite type (the scheme-theoretic generalisation of an algebraic group, allowing infinitesimal group laws) in almost all characteristics. Their proof finds weak polarisations — solvable subalgebras s⊆g\mathfrak s\subseteq\mathfrak g of dimension 12(dim⁡g+ind⁡g)\tfrac12(\dim\mathfrak g+\operatorname{ind}\mathfrak g) on which χ\chi vanishes on brackets — over fields of characteristic 00, then transfers their existence to large characteristic through a first-order version of the Lefschetz principle (the model-theoretic fact that a first-order sentence in the language of rings true of all algebraically closed fields of characteristic 00 holds over algebraically closed fields of all sufficiently large characteristics), quantifying over all Lie algebras of bounded dimension simultaneously. This transfer is qualitative — an ultraproduct/compactness argument in model theory — so it yields no numerical value of p0(d)p_0(d); the authors state that an explicit bound is deferred to future work. A 2026 companion paper by the same authors developed Gröbner-basis methods (algorithmic rewriting systems for polynomial ideals) and complexity machinery to make an analogous smoothness theorem first-order, but its thresholds remain existential as well.

The same paper records a structural conjecture linking index and simplicity: call g\mathfrak g Frobenius if ind⁡(g)=0\operatorname{ind}(\mathfrak g)=0. The conjecture states that g\mathfrak g is Frobenius if and only if there exists a non-empty open subset O⊆g∗\mathcal O\subseteq\mathfrak g^* such that Uχ(g)U_\chi(\mathfrak g) is simple for every χ∈O\chi\in\mathcal O. Premet and Skryabin had proved the forward implication in general. For the converse, if some Uχ(g)U_\chi(\mathfrak g) is simple then g\mathfrak g has a simple module of the maximal possible dimension dim⁡Uχ(g)=p(dim⁡g)/2\sqrt{\dim U_\chi(\mathfrak g)}=p^{(\dim\mathfrak g)/2}, so M(g)≥p(dim⁡g)/2M(\mathfrak g)\ge p^{(\dim\mathfrak g)/2}, which is compatible with the KW1 formula M(g)=p(dim⁡g−ind⁡g)/2M(\mathfrak g)=p^{(\dim\mathfrak g-\operatorname{ind}\mathfrak g)/2} only when ind⁡(g)=0\operatorname{ind}(\mathfrak g)=0; hence the converse follows from KW1 for g\mathfrak g. The large-characteristic theorem therefore settles the conjecture for restricted subalgebras of gld(k)\mathfrak{gl}_d(k) with p>p0(d)p>p_0(d), and the open part of the conjecture is exactly the KW1 upper bound for restricted Lie algebras beyond those admitting a faithful restricted representation of bounded dimension in large characteristic. Neighbouring work has developed super-analogues of KW1 for modular Lie superalgebras, but the two targets recorded here — effectivity of the threshold and the characterisation for arbitrary restricted Lie algebras — remain untouched.

Problem Statement

Two linked targets, both of which remove a hypothesis of the large-characteristic KW1 theorem of Martin–Stewart–Topley.

(1) Effective threshold. Give an explicit, effectively computable function d↦p0(d)d\mapsto p_0(d) — a closed formula or an algorithm computing it from dd — together with a proof that for every algebraically closed field kk of prime characteristic p>p0(d)p>p_0(d) and every restricted Lie subalgebra g⊆gld(k)\mathfrak g\subseteq\mathfrak{gl}_d(k), the first Kac–Weisfeiler equality

M(g)=p12(dim⁡g−ind⁡g)M(\mathfrak g)=p^{\tfrac12(\dim\mathfrak g-\operatorname{ind}\mathfrak g)}

holds. The group-scheme version of the theorem (Theorem 1.2 of the paper of Martin–Stewart–Topley) is deduced from this one: by Lemma 3.5 of that paper, the Lie algebra of any fixed affine group scheme of finite type admits a faithful restricted representation of dimension dd independent of the characteristic, so an explicit, effectively computable p0(d)p_0(d) yields an effective threshold for every such group scheme.

(2) Frobenius characterisation for arbitrary restricted Lie algebras. Prove or refute: for every restricted Lie algebra g\mathfrak g over an algebraically closed field of prime characteristic, g\mathfrak g is Frobenius (i.e. ind⁡(g)=0\operatorname{ind}(\mathfrak g)=0) if and only if there exists a non-empty open subset O⊆g∗\mathcal O\subseteq\mathfrak g^* such that Uχ(g)U_\chi(\mathfrak g) is simple for all χ∈O\chi\in\mathcal O. The implication from index zero to generic simplicity is already known in general (Premet–Skryabin); the open direction is the converse for algebras not covered by the known KW1 classes, and any proof of the KW1 upper bound for all restricted Lie algebras would settle it via the argument of Remark 3.4 of that paper. A counterexample is an explicit restricted Lie algebra, given by structure constants and a pp-mapping, on which the equivalence fails.

The common objective is to determine the exact reach of the KW1 conclusion: quantitatively, how large the characteristic must be as an explicit function of the embedding dimension; qualitatively, whether the equivalent geometric simplicity criterion holds for every restricted Lie algebra rather than only those with a faithful restricted representation of bounded dimension in large characteristic. A resolution of the classical KW1 conjecture for all restricted Lie algebras would resolve component (2) and make component (1) immediate.

References

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