Cohn's conjecture: non-speciality of the Cohn Lie algebra over truncated polynomial algebras for every prime characteristic

For a field $\mathbf{k}$ of prime characteristic $p$ , P. M. Cohn (1963) proposed the Lie algebra $\mathcal{L}_p=\mathrm{Lie}_K(x_1,x_2,x_3 \mid y_3x_3 = y_2x_2 + y_1x_1)$ over the truncated polynomial algebra $K=\mathbf{k}[y_1,y_2,y_3 \mid y_i^p=0]$ and conjectured that it is non-special, i.e., it does not embed into any associative algebra over $K$ and thus fails the Poincaré–Birkhoff–Witt embedding property that holds automatically over fields. Bokut, Chen and Chen (2011), via their Gröbner–Shirshov basis theory for Lie algebras over commutative algebras, confirmed the conjecture for $p=2,3,5$ by showing that the Jacobson–Zassenhaus Lie polynomial $\Lambda_p(y_2x_2, y_1x_1)$ , which vanishes in the universal enveloping algebra, is nonzero in $\mathcal{L}_p$ , and gave a terminating algorithm deciding any fixed $p$ . The general conjecture for all primes (in particular every $p>5$ ) remains open: prove that $\mathcal{L}_p$ is non-special for every prime $p$ , or exhibit a prime $p$ for which $\mathcal{L}_p$ is special.

Background

Over a field, the Poincaré–Birkhoff–Witt (PBW) theorem states that every Lie algebra embeds into its universal enveloping associative algebra, so Lie algebras can always be represented by commutators in an associative algebra. Over a general commutative base ring (or commutative algebra) $K$ this fails: following Cohn, a Lie algebra over $K$ is called special if the canonical map into its universal enveloping associative $K$ -algebra $U_K(L)$ (the associative algebra generated by $L$ with $[u,v]=uv-vu$ ) is injective, and non-special otherwise. The first non-special examples are due to Shirshov (1962) and Cartier (1958), both over commutative algebras over the field $\mathbb{F}_2$ ; Cartier also proved that every Lie algebra over a Dedekind domain (a commutative domain in which every nonzero ideal factors uniquely into prime ideals, e.g., the ring of integers of an algebraic number field) is special, so failure requires coefficients in a genuinely non-domain algebra. Cohn (1963) proved that every Lie algebra over a commutative algebra over a field of characteristic $0$ is special, and proposed a characteristic- $p$ counterexample without proof: let $\mathbf{k}$ be a field of characteristic $p>0$ , let $K=\mathbf{k}[y_1,y_2,y_3 \mid y_i^p=0]$ be the algebra of truncated polynomials (commutative, with each generator nilpotent of index $p$ ), and let $\mathcal{L}_p=\mathrm{Lie}_K(x_1,x_2,x_3 \mid y_3x_3 = y_2x_2 + y_1x_1)$ , the Lie algebra freely generated over $K$ by $x_1,x_2,x_3$ modulo the displayed linear relation. His route uses the Jacobson–Zassenhaus Lie polynomial $\Lambda_p(a,b)$ , defined by the identity $(a+b)^p = a^p + b^p + \Lambda_p(a,b)$ , which is expressible solely through Lie brackets in any associative algebra of characteristic $p$ . Since the coefficients $y_i$ are central and nilpotent, $(y_ix_i)^p = y_i^p x_i^p = 0$ in $U_K(\mathcal{L}_p)$ , so expanding $(y_3x_3)^p=(y_2x_2+y_1x_1)^p$ gives $\Lambda_p(y_2x_2, y_1x_1)=0$ in the enveloping algebra. Cohn conjectured that this element is nevertheless nonzero in $\mathcal{L}_p$ itself, which would make the canonical map non-injective, i.e., $\mathcal{L}_p$ non-special. Bokut, Chen and Chen (2011) developed a Gröbner–Shirshov basis theory (a Composition–Diamond lemma) for Lie algebras over commutative algebras: a Gröbner–Shirshov basis is a set of defining relations all of whose compositions (overlaps) reduce to zero, and then the irreducible words form a linear basis, enabling explicit normal-form computations in algebras given by generators and relations. With this they proved $\mathcal{L}_2$ , $\mathcal{L}_3$ and $\mathcal{L}_5$ non-special: for each such $p$ they computed the truncated Gröbner–Shirshov complement $S_{X^p}$ — the subset of the full complement consisting of relations whose $X$ -degree, i.e., total degree in the Lie generators $x_1,x_2,x_3$ , is at most $p$ — and verified that $\Lambda_p(y_2x_2,y_1x_1)$ has a nonzero normal form. They also presented a finite algorithm — compute $S_{X^p}$ and test membership of the normal form of $\Lambda_p(y_2x_2,y_1x_1)$ in the irreducible set — that decides the question for any fixed prime $p$ . However, the computation grows rapidly with $p$ and yields no uniform argument, and the paper explicitly states that for $p>5$ the problem remains open.

Problem Statement

For a field $\mathbf{k}$ of characteristic $p>0$ , let $K=\mathbf{k}[y_1,y_2,y_3 \mid y_i^p=0]$ be the truncated polynomial algebra and $\mathcal{L}_p=\mathrm{Lie}_K(x_1,x_2,x_3 \mid y_3x_3 = y_2x_2 + y_1x_1)$ the Cohn Lie algebra. Determine whether $\mathcal{L}_p$ is non-special for every prime $p$ , that is, whether the canonical map $\mathcal{L}_p \to U_K(\mathcal{L}_p)$ into its universal enveloping associative $K$ -algebra fails to be injective for every $p$ . Non-speciality is known only for $p=2,3,5$ , so the open core is all primes $p>5$ and, beyond that, a proof uniform in $p$ .

References

L. A. Bokut, Yuqun Chen, Yongshan Chen, “Gröbner–Shirshov bases for Lie algebras over a commutative algebra”, Journal of Algebra 337(1) (2011) 82–102. DOI: 10.1016/j.jalgebra.2011.04.020; arXiv:1006.3217. https://arxiv.org/abs/1006.3217
P. M. Cohn, “A remark on the Birkhoff-Witt theorem”, Journal of the London Mathematical Society s1-38(1) (1963) 197–203. DOI: 10.1112/jlms/s1-38.1.197. https://doi.org/10.1112/jlms/s1-38.1.197
L. A. Bokut, Yuqun Chen, “Gröbner-Shirshov bases and PBW theorems”, preprint arXiv:1502.06472 (submitted November 2014). https://arxiv.org/abs/1502.06472
P. Cartier, “Remarques sur le théorème de Birkhoff-Witt”, Annali della Scuola Normale Superiore di Pisa, Scienze Fisiche e Matematiche (3) 12 (1958) 1–4. zbMATH Zbl 0081.26302; EudML 83207. https://eudml.org/doc/83207
A. I. Shirshov, “Some algorithmic problems for $\varepsilon$ -algebras”, Sibirskii Matematicheskii Zhurnal 3 (1962) 132–137 (in Russian). zbMATH Zbl 0143.25602. https://zbmath.org/?q=an%3A0143.25602
Zerui Zhang, Yuqun Chen, Leonid A. Bokut, “Some algorithmic problems for Poisson algebras”, Journal of Algebra 525 (2019) 562–588. DOI: 10.1016/j.jalgebra.2019.02.001. https://doi.org/10.1016/j.jalgebra.2019.02.001
Sergei O. Ivanov, Fedor Pavutnitskiy, Vladislav Romanovskii, Anatolii Zaikovskii, “On homology of Lie algebras over commutative rings”, Journal of Algebra 586 (2021) 99–139. DOI: 10.1016/j.jalgebra.2021.06.019; arXiv:2010.00369. https://arxiv.org/abs/2010.00369
Yuqun Chen, Yu Li, Qingyan Tang, “Gröbner–Shirshov bases for some Lie algebras”, preprint arXiv:1305.4546 (2013). https://arxiv.org/abs/1305.4546