Zelmanov's Conjecture 4.11: is the linearized algebra $J\otimes_F E$ of a quadratic Jordan PI-algebra nil of bounded index in characteristic $p>0$?

In his 2017 paper "Lie algebras and torsion groups with identity", Efim Zelmanov proved that a finitely generated Lie algebra satisfying a polynomial identity, in which every element of the Lie set of the generators is ad-nilpotent, is nilpotent, and deduced that a finitely generated residually-$p$ torsion group whose pro-$p$ completion satisfies a pro-$p$ identity is finite. Inside the positive-characteristic part of the proof he posed Conjecture 4.11: if $J$ is a quadratic Jordan PI-algebra over an (infinite) field of characteristic $p>0$, then the linearized algebra $\tilde J=J\otimes_F E$ — where $E$ is the commutative associative algebra on countably many generators $e_i$ with $e_i^2=0$ — is nil of bounded index, i.e. a single $N$ annihilates every element by its $N$-th Jordan power. The paper proves only a special case, in which $J$ satisfies an additional fully linearized identity. The problem — prove or refute Conjecture 4.11 — is open.

Background

A polynomial identity (PI) for an algebra $A$ over a field $F$ is a nonzero element $f(x_1,\dots,x_r)$ of the appropriate free algebra that vanishes under every substitution of elements of $A$. The classical Kurosh problem (1941) asked whether a finitely generated nil algebra (an algebra in which every element is nilpotent) is nilpotent; the answer is negative in general but positive in the class of PI-algebras, which is one reason PI hypotheses are central in Burnside-type questions. In the source paper, Efim Zelmanov proved (Theorem 1.1) that a Lie algebra $L$ satisfying a polynomial identity and generated by elements $a_1,\dots,a_m$, such that every element of the Lie set $S\langle a_1,\dots,a_m\rangle$ (all iterated commutators in the generators) is ad-nilpotent — the adjoint map $\mathrm{ad}(s):x\mapsto[x,s]$ is a nilpotent operator — must be nilpotent. From this he deduced (Theorems 1.2, 1.3) that a finitely generated residually-$p$ torsion group (a torsion group that separates into finite $p$-groups) whose associated Lie algebra, resp. pro-$p$ completion (the inverse limit of its finite $p$-group quotients), satisfies a polynomial identity, resp. a pro-$p$ identity (an identity between elements of the free pro-$p$ group), is finite — a significant extension of his positive solution of the restricted Burnside problem (the question whether, for a fixed number of generators and a fixed exponent, there are only finitely many finite groups; Zelmanov proved that there are).

The proof passes through Jordan algebras and sandwich arguments. In characteristic $\neq 2,3$, if $s$ is an element of $L$ with $\mathrm{ad}(s)^3=0$, the operation $a\circ b=[a,[s,b]]$ makes a certain quotient of $L$ a Jordan algebra — a commutative algebra satisfying $x\circ y=y\circ x$ and $(x^2\circ y)\circ x=x^2\circ(y\circ x)$ — by a construction of Fernández López, García and Gómez Lozano refining the classical Tits–Kantor–Koecher construction (a standard method that reconstructs a Lie algebra from an associated Jordan algebra). An element $s$ of $L$ is called a sandwich (Kostrikin's term) if $\mathrm{ad}(s)^2=0$ and $\mathrm{ad}(s)\mathrm{ad}(b)\mathrm{ad}(s)=0$ for every $b\in L$ (in characteristic $\neq2$ the first condition implies the second); the source cites Grishkov's theorem that in characteristic zero an arbitrary sandwich generates a locally nilpotent ideal, and sandwiches drive the Kostrikin-type combinatorics behind the restricted Burnside problem.

To handle all positive characteristics, including $p=2,3$ where the linear theory of Jordan algebras degenerates, the paper works with quadratic Jordan algebras in the sense of McCrimmon: a vector space $J$ with a quadratic squaring map $x\mapsto x^2$ and a quadratic representation $Q:J\to\mathrm{End}_F(J)$ satisfying the identities (M1)–(M6) and their partial linearizations (no unit is assumed). Powers are defined by $x^1=x$, $x^{2k}=(x^k)^2$, $x^{2k+1}=x\,Q(x^k)$. The free quadratic Jordan algebra $FJ\langle X\rangle$ admits a canonical homomorphism into the free associative algebra $F\langle X\rangle$ endowed with the special quadratic Jordan structure ($x^2=x\cdot x$ and $Q(x)y=xyx$ in the associative product); the kernel consists of the S-identities, i.e. the identities satisfied by all special quadratic Jordan algebras (those isomorphic to a subalgebra of some $A^{(C)}$). A quadratic Jordan algebra is PI if it satisfies some $f\in FJ\langle X\rangle$ that is not an S-identity. Following the source's convention, $a\in J$ is an absolute zero divisor if $Q(a)=0$ (the standard convention additionally requires $a\neq0$ and $a^2=0$); an algebra with no nonzero absolute zero divisors is nondegenerate, and the smallest ideal with nondegenerate quotient is the McCrimmon radical. The structure theory of McCrimmon and Zel'manov classifies strongly prime quadratic Jordan algebras (nondegenerate prime algebras, i.e. algebras in which two nonzero ideals always meet nontrivially: central localizations of $A^{(C)}$, hermitian parts of algebras with involution, 27-dimensional Albert algebras of an admissible cubic form, and algebras of a nondegenerate quadratic form with a basepoint); it yields Lemma 4.6 of the source: there is a function $d(n)$ such that every nondegenerate prime quadratic Jordan algebra over a field of characteristic $p>0$ satisfying a PI of degree $n$ satisfies the identity $\mathrm{Sym}_{d(n)}(x_1,\dots,x_{d(n)})=0$, the full linearization of $x^{d(n)}$.

The conjecture concerns the linearized algebra $\tilde J=J\otimes_F E$, where $E$ is the associative commutative $F$-algebra presented by countably many generators $e_1,e_2,\dots$ with relations $e_i^2=0$; a basis of $E$ is given by the ordered products $e_{i_1}\cdots e_{i_r}$ with $i_1<\dots<i_r$ and $r\ge1$, and $E$ has no unit element. Tensoring with $E$ is the device by which the paper linearizes its Lie and Jordan structures (via divided polynomials, a generalization of Lie polynomials adapted to $L\otimes_F E$). It does not, however, carry identities of $J$ over to $\tilde J$: the $E$-component of a Jordan word evaluated on $J\otimes_F E$ depends on the shape of the word — a factor occurring in a $Q$-slot contributes its $E$-component squared, since $Q(j\otimes u)(k\otimes v)=Q(j)(k)\otimes uvu$ — so a linear combination of words that vanishes on all of $J$ need not vanish on $\tilde J$; and in characteristic $p$ the usual reduction of an identity to multilinear form, which would otherwise bypass this, is blocked once the degree of the identity reaches $p$. In characteristic $p$ the algebra $E$ is itself nil of bounded index — $u^p=0$ for all $u\in E$ by the Frobenius (freshman's-dream) identity, a computation used in the source — whereas in characteristic $0$ the nilpotency indices of elements of $E$ are unbounded; this is why the conjecture is genuinely a positive-characteristic statement. Proposition 4.10 of the source constructs, for every non-S-identity $f$, a finite absolute zero divisor sequence (a finite sequence $h_1,\dots,h_r$ of homogeneous elements of $FJ\langle X\rangle$ such that, for every algebra $J$ in the variety, the values of $h_r$ on $\tilde J$ are absolute zero divisors of $\tilde J$, and whenever $h_k,\dots,h_r$ vanish identically on $\tilde J$ the values of $h_{k-1}$ on $\tilde J$ are absolute zero divisors of $\tilde J$) in the variety $M(f)$ of quadratic Jordan algebras satisfying $f=0$, and within its proof one finds the special case: if $J\in M(f)$ satisfies $\mathrm{Sym}_d(x_1,\dots,x_d)=0$ identically, then $\tilde J$ is nil of bounded index $\le d$. Conjecture 4.11 removes the extra hypothesis $\mathrm{Sym}_d=0$, a hypothesis that Lemma 4.6 can guarantee only for nondegenerate prime algebras. The conjecture is stated exactly once in the paper, immediately after Proposition 4.10, and the main theorems are proved unconditionally, without it.

Problem Statement

Decide Conjecture 4.11 of Zelmanov (2017), under the standing assumptions of the section in which it is posed.

Let $F$ be an infinite field of prime characteristic $p>0$. Let $E$ be the associative commutative $F$-algebra presented by the countable generating set $e_1,e_2,\dots$ with relations $e_i^2=0$ for all $i\ge1$; a basis of $E$ consists of the ordered products $e_{i_1}\cdots e_{i_r}$ with $i_1<\dots<i_r$ and $r\ge1$, and $E$ has no identity element.

Let $J$ be a quadratic Jordan PI-algebra over $F$ in the sense of the source. Explicitly: $J$ is a vector space over $F$ equipped with a squaring map $x\mapsto x^2$ (a quadratic map) and a quadratic representation $Q:J\to\mathrm{End}_F(J)$ (a quadratic map into the endomorphism algebra), with derived operations $x\circ y=(x+y)^2-x^2-y^2$ and $\{x,y,z\}=y(Q(x+z)-Q(x)-Q(z))$, satisfying McCrimmon's identities

(M1) $\{x,x,y\}=x^2\circ y$; (M2) $(yQ(x))\circ x=(y\circ x)Q(x)$; (M3) $x^2Q(x)=(x^2)^2$; (M4) $x^2Q(y)Q(x)=(yQ(x))^2$; (M5) $Q(x^2)=Q(x)^2$; (M6) $Q(yQ(x))=Q(x)Q(y)Q(x)$;

together with all their partial linearizations (no unit element is assumed). Let $FJ\langle X\rangle$ be the free quadratic Jordan algebra on a countable set $X$ of free generators, and let $\varphi:FJ\langle X\rangle\to F\langle X\rangle^{(C)}$ be the canonical homomorphism into the free associative algebra $F\langle X\rangle$ endowed with the special quadratic Jordan structure ($x^2=x\cdot x$ and $Q(x)y=xyx$ in the associative product); the elements of $\ker\varphi$ are the S-identities, that is, the identities satisfied by all special quadratic Jordan algebras (a quadratic Jordan algebra is special if it is isomorphic to a subalgebra of some $A^{(C)}$ for an associative algebra $A$). The hypothesis on $J$ is that some element $f(x_1,\dots,x_r)\in FJ\langle X\rangle\setminus\ker\varphi$ vanishes identically on $J$.

Let $\tilde J=J\otimes_F E$ as an $F$-vector space, equipped with the quadratic Jordan structure determined on decomposable tensors by

$$(j\otimes u)\circ(k\otimes v)=(j\circ k)\otimes uv,\qquad (j\otimes u)^2=j^2\otimes u^2,\qquad Q(j\otimes u)(k\otimes v)=Q(j)(k)\otimes uvu$$

together with $F$-bilinearity of $\circ$ and $F$-quadraticity of the squaring map and of $Q:\tilde J\to\mathrm{End}_F(\tilde J)$; explicitly, $(\sum_k j_k\otimes u_k)^2=\sum_k j_k^2\otimes u_k^2+\sum_{k<l}(j_k\circ j_l)\otimes u_ku_l$. Jordan powers in $\tilde J$ are defined inductively by $x^1=x$, $x^{2k}=(x^k)^2$, $x^{2k+1}=x\,Q(x^k)$.

Conjecture: $\tilde J$ is nil of bounded index — there exists an integer $N$ such that $x^N=0$ for every element $x\in\tilde J$. (The source does not specify how $N$ may depend on $J$; in the proven special case the bound is controlled by the degree of the identity, so a bound depending only on the identity is a natural but unasked strengthening.)

Decide whether Conjecture 4.11 holds: is $\tilde J$ nil of bounded index for every quadratic Jordan PI-algebra $J$ over every infinite field of characteristic $p>0$, with all primes $p$ including $2$ and $3$, degenerate algebras and algebras with nonzero McCrimmon radical included, and without assuming nondegeneracy, primeness, or speciality of $J$?

One remark delimiting the target: the special case in which $J$ additionally satisfies the fully linearized identity $\mathrm{Sym}_d(x_1,\dots,x_d)=0$ (the full linearization of the power $x^d$) is already proved in the source (Proposition 4.10 with Lemma 4.6) and does not by itself answer the conjecture. The conjecture is stated for infinite ground fields, following the source; an answer valid for arbitrary fields of characteristic $p>0$ would also settle it.

References

1. Efim Zelmanov, "Lie algebras and torsion groups with identity", Journal of Combinatorial Algebra 1 (2017), no. 3, pp. 289–340. DOI: 10.4171/JCA/1-3-2, https://doi.org/10.4171/JCA/1-3-2 (preprint: arXiv:1604.05678, https://arxiv.org/abs/1604.05678). Primary source.
2. Antonio Fernández López, Esther García, Miguel Gómez Lozano, "The Jordan algebras of a Lie algebra", Journal of Algebra 308 (2007), no. 1, pp. 164–177. DOI: 10.1016/j.jalgebra.2006.02.035, https://doi.org/10.1016/j.jalgebra.2006.02.035.
3. Kevin McCrimmon, Ephim Zel'manov, "The structure of strongly prime quadratic Jordan algebras", Advances in Mathematics 69 (1988), no. 2, pp. 133–222. DOI: 10.1016/0001-8708(88)90001-1, https://doi.org/10.1016/0001-8708(88)90001-1.
4. A. I. Kostrikin, "Sandwiches in Lie algebras", Mathematics of the USSR-Sbornik 38 (1981), no. 1, pp. 1–9 (English translation; original: Matematicheskiĭ Sbornik 110(152) (1979), no. 1, pp. 3–12). DOI: 10.1070/SM1981v038n01ABEH001042, https://doi.org/10.1070/SM1981v038n01ABEH001042.
5. Kevin McCrimmon, "A general theory of Jordan rings", Proceedings of the National Academy of Sciences 56 (1966), no. 4, pp. 1072–1079. DOI: 10.1073/pnas.56.4.1072, https://doi.org/10.1073/pnas.56.4.1072.
6. Matej Brešar, Consuelo Martínez, "Waring problems across algebra", preprint (2026), arXiv:2603.07033, https://arxiv.org/abs/2603.07033.
7. Be'eri Greenfeld, "The Quantitative Kurosh Problem", Forum of Mathematics, Sigma 13 (2025), article e42. DOI: 10.1017/fms.2025.1, https://doi.org/10.1017/fms.2025.1.
8. Yuri Bahturin, "Some Thoughts on the Current State of the Theory of Identical Relations in Lie Algebras", in: Polynomial Identities in Algebras, Springer INdAM Series, Springer, Cham (2020), pp. 1–24. DOI: 10.1007/978-3-030-63111-6_1, https://doi.org/10.1007/978-3-030-63111-6_1.
9. Antonio Fernández López, "Jordan structures in Lie algebras", Mathematical Surveys and Monographs 240, American Mathematical Society, Providence, RI (2019). DOI: 10.1090/surv/240, https://doi.org/10.1090/surv/240.
10. Evgeny Khukhro, Pavel Shumyatsky, "On finite groups containing an element whose Engel sink is small", Proceedings of the American Mathematical Society (2026, online first). DOI: 10.1090/proc/17893, https://doi.org/10.1090/proc/17893.
11. Antonio de França, Irina Sviridova, "Köthe's Problem, Kurosh-Levitzky Problem and Graded Rings", Journal of Algebra 602 (2022), pp. 224–246. DOI: 10.1016/j.jalgebra.2022.03.016, https://doi.org/10.1016/j.jalgebra.2022.03.016 (preprint: arXiv:2110.12128, https://arxiv.org/abs/2110.12128, under the variant title "Köthe's Problem, Kurosch-Levitzki Problem and Graded Rings").
12. Matej Brešar, Efim Zelmanov, "Jordan homomorphisms: A survey", preprint (2025), arXiv:2510.16876, https://arxiv.org/abs/2510.16876.