The Cherlin–Zilber conjecture for Lie rings predicts that a simple Lie ring definable under a suitable model-theoretic tameness assumption, in sufficiently large characteristic, is a finite-dimensional Lie algebra over a definable algebraically closed field. Invitti (2025) formulated its extension to Lie rings definable in finite-dimensional theories — the 'log-CZ' conjecture — and verified it in characteristic 0: every absolutely simple definable Lie ring of finite dimension and characteristic 0 is, up to a definable finite-index subring and a finite subring, a finite-dimensional Lie algebra over a definable field. In positive characteristic the conjecture is open, even in dimension 3, and the author-identified obstruction is the possible existence of bad Lie rings, i.e., simple connected definable Lie rings admitting a nilpotent Borel subring. This problem asks for a resolution of the positive-characteristic log-CZ conjecture and, as its stated crux, for the existence and structure of bad Lie rings of dimension 3 and 4 and characteristic ≥ 5 — in particular, whether every such ring must be a Lie algebra of finite dimension over a definable perfect field.

Background

A Lie ring is an abelian group (g,+)(\mathfrak{g},+) with a bilinear, antisymmetric bracket [⋅,⋅][\cdot,\cdot] satisfying the Jacobi identity — a Lie algebra without a designated underlying field. Since arbitrary infinite Lie rings are unclassifiable, model theorists study them under tameness assumptions. The classical notion is finite Morley rank: Morley rank is the model-theoretic dimension assigned to definable sets in totally transcendental theories, and a structure has finite Morley rank when every definable set has rank below ω\omega. The Cherlin–Zilber (algebraicity) conjecture predicts that every simple group of finite Morley rank is the group of rational points of an algebraic group over an algebraically closed definable field; its Lie-ring analogue predicts that every definable simple Lie ring of finite Morley rank and sufficiently large characteristic is a finite-dimensional Lie algebra over an algebraically closed definable field. Deloro and Tindzogho Ntsiri proved this Lie-ring conjecture in characteristic 0 for all ranks and in characteristic ≠2,3\neq 2,3 up to rank 4, relying crucially on Macintyre's theorem that a field of finite Morley rank is algebraically closed.

Wagner's finite-dimensional theories form a strictly broader tameness framework: a structure with a dimension function on definable sets satisfying additivity and fibration inequalities, as axiomatized in the theory of dimensional groups and fields. O-minimal theories and superstable theories of finite SUSU-rank are finite-dimensional, so the Lie rings definable in such theories include real Lie algebras and Lie algebras over pseudofinite fields (ultraproducts of finite fields), none of which have finite Morley rank. Two features distinguish this setting. First, definable groups need not have definable connected components, whence the notions of virtually connected (having a definable connected subring of finite index) and of absolutely simple: a definable Lie ring g\mathfrak{g} of finite dimension is absolutely simple if every definable Lie subring of finite index in g\mathfrak{g} has no proper definable infinite ideal of infinite index — the substitute for definable simplicity when connected components are unavailable. Second, definable fields of finite dimension, while always perfect, need not be algebraically closed (a pseudofinite field is a typical example), so Macintyre's theorem and the Cherlin–Macintyre property of fields (the stronger statement that every infinite definable skew-field of finite Morley rank is an algebraically closed field) fail precisely here.

In this framework Invitti (2025) formulated the finite-dimensional version of the Lie-ring Cherlin–Zilber conjecture, the 'log-CZ' conjecture: every absolutely simple definable Lie ring of finite dimension and sufficiently large characteristic is, up to a definable finite-index subring and a finite subring, a finite-dimensional Lie algebra over a definable perfect field. He verified the conjecture in characteristic 0, and classified the definable Lie rings of dimension at most 4 and characteristic >3>3 in the connected and the NIP cases (NIP, 'not the independence property', is the model-theoretic tameness condition requiring that no definable family of sets shatter an infinite parameter set, equivalently that every definable family has finite Vapnik–Chervonenkis dimension); in the connected classification, a non-soluble ring of dimension 3 is either sl2(K)\mathfrak{sl}_2(K) or bad, and a simple ring of dimension 4 is necessarily bad, so the classification is complete exactly modulo the bad case. He also proved that the Fitting and Radical ideals of a finite-dimensional definable Lie ring are definable, nilpotent and soluble respectively. In positive characteristic the conjecture is open even in dimension 3. The identified obstruction is the class of bad Lie rings: for a simple connected definable Lie ring of finite dimension, a Borel is a definable connected proper soluble Lie subring not contained in a strictly larger one, and the ring is bad if it has a nilpotent Borel. In characteristic 0 bad Lie rings exist — so3(R)\mathfrak{so}_3(\mathbb{R}), whose maximal soluble subalgebras are 1-dimensional and hence abelian — yet they are still Lie algebras over the definable field R\mathbb{R}, so badness does not by itself refute the conjecture. In positive characteristic it is unknown whether bad Lie rings exist at all; the partial structure theory shows only that a bad Lie ring of dimension 3 (respectively 4) has additive group isomorphic to K3K^3 (respectively K4K^4) for a definable field KK of dimension 1, leaving the bracket unconstrained. The pseudofinite instances of the question are further tied to the classification of finite simple Lie algebras, which is complete over algebraically closed fields of characteristic p>3p>3 (Premet–Strade) but not in the generality an ultraproduct analysis would require — the obstacle Invitti names when explaining why no Wilson-type description of pseudofinite simple Lie rings (Wilson's theorem describes simple pseudofinite groups as ultraproducts of finite simple groups of bounded Lie rank) is currently accessible.

Problem Statement

Settle the positive-characteristic case of the finite-dimensional Lie-ring Cherlin-Zilber conjecture (the 'log-CZ' conjecture). Let g\mathfrak{g} be an absolutely simple Lie ring definable in a finite-dimensional theory, of finite dimension and of sufficiently large positive characteristic. Prove or refute: there exist a definable Lie subring h\mathfrak{h} of finite index in g\mathfrak{g} and a finite Lie subring h0\mathfrak{h}_0 of h\mathfrak{h} such that h/h0\mathfrak{h}/\mathfrak{h}_0 is a Lie algebra of finite dimension over a definable perfect field KK.

In particular, settle the author-identified crux: do bad Lie rings — simple connected definable Lie rings admitting a nilpotent Borel — of dimension 3 and 4 and characteristic ≥5\geq 5 exist in finite-dimensional theories at all, and is every such bad Lie ring a Lie algebra of finite dimension over a definable perfect field? By the source paper's analysis of the not-bad case, an affirmative answer to the linearity of bad Lie rings in dimensions 3 and 4 establishes the conjecture in those dimensions; the conjecture itself concerns all finite dimensions. The objective of this task is the general all-dimensions conjecture; the dimension 3 and 4 crux is its author-identified essential sub-question, so a complete answer to the crux establishes the conjecture in dimensions 3 and 4, while the general case over all finite dimensions remains required. Both the existence and the structure questions are open in every positive characteristic, including dimension 3.

References

  1. Moreno Invitti: 'Lie rings in finite-dimensional theories', arXiv:2511.06068 (2025), DOI 10.48550/arXiv.2511.06068, https://arxiv.org/abs/2511.06068 (source preprint)
  2. Adrien Deloro, Jules Tindzogho Ntsiri: 'Simple Lie rings of Morley rank 4 (The Spanish Inquisition)', Journal of Algebra 651 (2024), 243-280, DOI 10.1016/j.jalgebra.2024.04.012, https://doi.org/10.1016/j.jalgebra.2024.04.012 (preprint arXiv:2309.12465, https://arxiv.org/abs/2309.12465)
  3. Adrien Deloro, Jules Tindzogho Ntsiri: 'Soluble Lie rings of finite Morley rank', arXiv:2409.07783 (2024), DOI 10.48550/arXiv.2409.07783, https://arxiv.org/abs/2409.07783 (journal version: Model Theory 4(2) (2025), 163-175, DOI 10.2140/mt.2025.4.163, https://doi.org/10.2140/mt.2025.4.163)
  4. Moreno Invitti: 'Endogenies and linearization in the non-virtually connected case', arXiv:2511.07058 (2025), DOI 10.48550/arXiv.2511.07058, https://arxiv.org/abs/2511.07058
  5. Moreno Invitti: 'Almost cohomology of finite-dimensional Lie rings', arXiv:2602.20626 (2026), DOI 10.48550/arXiv.2602.20626, https://arxiv.org/abs/2602.20626
  6. Moreno Invitti: 'Omega-categorical groups and rings of finite dimension', arXiv:2511.07492 (2025), DOI 10.48550/arXiv.2511.07492, https://arxiv.org/abs/2511.07492
  7. Samuel Zamour: 'On the cohomology of finite-dimensional nilpotent groups and Lie rings', Bulletin of the London Mathematical Society 58(6) (2026), DOI 10.1112/blms.70411, https://doi.org/10.1112/blms.70411
  8. Jules Tindzogho Ntsiri, Samuel Zamour: 'Cartan subrings in soluble ranked Lie rings', arXiv:2511.20323 (2025), DOI 10.48550/arXiv.2511.20323, https://arxiv.org/abs/2511.20323
  9. Samuel Zamour: 'On p-Lie algebras of finite Morley rank', arXiv:2605.13700 (2026), DOI 10.48550/arXiv.2605.13700, https://arxiv.org/abs/2605.13700
  10. Katrin Tent: 'From the Cherlin-Zilber Conjecture via sharply 2-transitive groups to the Burnside problem', arXiv:2606.18207 (2026), DOI 10.48550/arXiv.2606.18207, https://arxiv.org/abs/2606.18207
  11. Frank O. Wagner: 'Dimensional groups and fields', The Journal of Symbolic Logic 85(3) (2020), 918-936, DOI 10.1017/jsl.2020.48, https://doi.org/10.1017/jsl.2020.48
  12. John S. Wilson: 'On simple pseudofinite groups', Journal of the London Mathematical Society 51(3) (1995), 471-490, DOI 10.1112/jlms/51.3.471, https://doi.org/10.1112/jlms/51.3.471
  13. Alexander Premet, Helmut Strade: 'Classification of finite dimensional simple Lie algebras in prime characteristics', arXiv:math/0601380 (2006), DOI 10.48550/arXiv.math/0601380, https://arxiv.org/abs/math/0601380