Algorithmic completeness of the conformal-module Gröbner–Shirshov calculus for finitely presented Poisson algebras: termination, confluence, and the word problem

For a Poisson algebra $P(\mathfrak{g})=\operatorname{gr} U(\mathfrak{g})$ realized, after Kolesnikov and Panasenko, as a conformal module over the universal associative conformal envelope $U(\operatorname{Cur}\mathfrak{g},N=3)$ of the current conformal algebra, a complete confluent rewriting system (Gröbner–Shirshov basis) is known only for the free case. The open problem is to extend this calculus to Poisson algebras presented by an arbitrary finite set of relations $S\subset\operatorname{Pois}(G)$ : characterize when the completion of the fixed rule system with the translated relations terminates in a confluent Gröbner–Shirshov basis — finite modulo the fixed rules, which already contain the infinite scheme (17) — and decide the word problem for finitely presented Poisson algebras within (or beyond) this framework. Later literature (Zhang–Chen–Bokut, 2019–2024) solves the word problem only for special classes of presentations, confirming that the general case remains open.

Background

A Poisson algebra is a commutative associative algebra $P$ over a field $k$ of characteristic zero carrying a Lie bracket $\{\cdot,\cdot\}$ that acts by derivations of the commutative product ( $\{x,yz\}=y\{x,z\}+z\{x,y\}$ ). The central example is $P(L)=\operatorname{gr} U(L)$ , the associated graded of the universal enveloping associative algebra of a Lie algebra $L$ with respect to the standard filtration, the bracket extending the Lie product; for a free Lie algebra $L=\operatorname{Lie}(G)$ this yields the free Poisson algebra $\operatorname{Pois}(G)$ on the set $G$ , and quotienting by the Poisson ideal generated by a set $S\subset\operatorname{Pois}(G)$ gives the finitely presented Poisson algebra $\operatorname{Pois}(G\mid S)$ (for finite $G$ and $S$ ). The equality (word) problem for such a presentation asks to decide whether two given elements of $\operatorname{Pois}(G\mid S)$ coincide.

A Lie conformal algebra is a left module over the polynomial ring $H=k[\partial]$ equipped with a bilinear $\lambda$ -bracket $[x\,_\lambda\,y]\in H[\lambda]\otimes_H L$ satisfying sesqui-linearity, skew-symmetry, and a Jacobi identity; the basic example is the current conformal algebra $\operatorname{Cur}\mathfrak{g}=H\otimes\mathfrak{g}$ over a Lie algebra $\mathfrak{g}$ , with bracket $[a\otimes f(\partial)\,_\lambda\,b\otimes g(\partial)]=(f(-\lambda)g(\partial+\lambda)\otimes [a,b])$ . A representation of a Lie conformal algebra on an $H$ -module $M$ is given by conformal linear operators $\rho(x)$ whose $\lambda$ -products are polynomial in $\lambda$ of bounded degree; the minimal such bound $N$ is the locality level, and by Dong's lemma (the standard locality transfer principle: if a family of conformal operators is pairwise local then all their iterated $\lambda$ -products stay local) the representation extends to the universal associative conformal envelope $U(L,N)$ , a universal object among associative conformal algebras ( $H$ -modules with an associative $\lambda$ -multiplication) generated by $L$ with locality $N$ . Unlike the classical situation, this envelope depends on $N$ and need not be unique.

Kolesnikov and Panasenko (arXiv:2011.07339; J. Algebra Appl. 21 (2022), 2250138) observed that for the Poisson–adjoint action $(a\,_\lambda\,w)=\{a,w\}+\lambda aw$ on $H\otimes P(\mathfrak{g})$ the $\lambda$ -products of two such operators are quadratic in $\lambda$ , so $H\otimes P(\mathfrak{g})$ is a conformal module over $U(\operatorname{Cur}\mathfrak{g},N=3)$ generated by the single element $1$ . A Gröbner–Shirshov (GS) basis is a set of rewriting rules for a (module) presentation such that every element reduces to a unique normal form, so that the irreducible words form a linear basis and the equality problem becomes decidable; the Composition–Diamond lemma states that a rewriting system is a GS basis exactly when all its compositions (overlap and inclusions of leading words) are trivial. Chen and Ni had established a Composition–Diamond lemma for modules over associative conformal algebras directly in conformal language; Kolesnikov and Panasenko instead reduce everything to an ordinary associative algebra $A$ generated by $\partial$ together with coefficient operators $L^a_n$ and $R^a_n$ (the $n$ -th coefficients of the left $\lambda$ -action of $a$ and of its right action), which makes the computation accessible to standard computer algebra packages. Their Theorem 2 gives an explicit finite-plus-scheme rewriting system (rules (7)–(16), taken in part from the Kolesnikov–Kozlov standard basis of the envelope, plus $L^a_0\partial^s 1\to 0$ and the infinite scheme (17)) that is a complete confluent GS basis for the split null extension (the associative algebra $C\oplus M$ in which the product of two elements of the module $M$ is declared zero, so $M$ is a square-zero ideal) $U(\operatorname{Cur}\mathfrak{g},N=3)\oplus(H\otimes P(\mathfrak{g}))$ in the free case $S=\varnothing$ ; Corollary 2 recovers the Poincaré–Birkhoff–Witt-type linear basis (an analogue of the classical PBW monomial basis of an enveloping algebra), and Corollary 3 is a conditional Composition–Diamond lemma: if the elements of $S$ , expressed through non-associative Lyndon–Shirshov words (a lexicographically ordered word basis of the free Lie algebra due to Shirshov), translate into rewriting rules that have no nontrivial compositions with the rules (7) and (17), then those rules together form a GS basis of $H\otimes\operatorname{Pois}(G\mid S)$ .

Independently, Zhang, Chen, and Bokut developed a direct GS-basis theory for Poisson algebras (J. Algebra 525 (2019)) and solved the word problem only for special presentations: a single relation whose leading monomial is of Lie type, and presentations with sufficiently many relations involving the Poisson bracket of generators; they also settled the metabelian variety (preprint arXiv:1907.05953), and in 2024 the same authors settled the word problem for finitely presented metatrivial Poisson algebras, the metabelian and metatrivial classes being varieties of Poisson algebras determined by identities on iterated Poisson brackets. No termination criterion, complexity bounds, or general decision procedure for arbitrary finite $S$ exists on either route, and no undecidability result for the word problem of finitely presented Poisson algebras is known.

Problem Statement

Over a field $k$ of characteristic zero, fix a well-ordered set $G$ and a finite set of relations $S\subset\operatorname{Pois}(G)$ , and translate each element of $S$ into a rewriting rule exactly as in Corollary 3 of Kolesnikov–Panasenko: expressing the element through non-associative Lyndon–Shirshov words in the alphabet $G$ (the lexicographically ordered word basis of the free Lie algebra on $G$ ) makes it a rule in the free left module generated by the single symbol $1$ over the coefficient algebra $A$ of the background, a rule that replaces the leading word by the corresponding linear combination of smaller words. The problem is to complete their conformal-module Gröbner–Shirshov calculus from the free and composition-compatible cases to an algorithmically complete theory for arbitrary presentations; its linked facets are:

(1) Termination and confluence. Characterize the finite sets $S$ for which the rewriting system consisting of the fixed rules (7)–(17) together with the translated $S$ -rules admits a confluent Gröbner–Shirshov basis consisting of the fixed rules plus finitely many additional rules produced by the completion (the fixed system already contains the infinite scheme (17), and the solved free case $S=\varnothing$ of Theorem 2 corresponds to no additional rules); determine also for which $S$ the composition-elimination (completion) procedure of the Composition–Diamond lemma terminates; provide an algorithm that, given $S$ , certifies termination or produces the basis, or prove that no such general certification exists.

(2) The word problem. As the declared application of the calculus, decide the equality problem for finitely presented Poisson algebras $\operatorname{Pois}(G\mid S)$ : either give a terminating decision procedure (within this framework or any other), or prove the problem undecidable. Known partial cases: a single relation with leading monomial of Lie type; presentations with sufficiently many relations involving the Poisson bracket of generators (Zhang–Chen–Bokut 2019); the metabelian and metatrivial varieties (2019, 2024).

Facet (1) feeds facet (2) in the positive direction only: a terminating confluent calculus from (1) yields the decision procedure demanded in (2). The two facets are otherwise independent. An undecidability answer in (2) closes the negative side of the certification question but does not by itself characterize which $S$ admit a terminating confluent completion.

References

P. S. Kolesnikov, A. S. Panasenko, Defining relations and Gröbner–Shirshov bases of Poisson algebras as of conformal modules, arXiv:2011.07339 (2020); published in Journal of Algebra and Its Applications 21(7) (2022), article 2250138, DOI 10.1142/S0219498822501389, https://doi.org/10.1142/S0219498822501389 (arXiv: https://arxiv.org/abs/2011.07339)
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
Leonid A. Bokut, Yuqun Chen, Zerui Zhang, Gröbner–Shirshov basis theory and word problems for metatrivial Poisson algebras, São Paulo Journal of Mathematical Sciences 18 (2024) 569–597, DOI 10.1007/s40863-023-00386-4, https://doi.org/10.1007/s40863-023-00386-4
Zerui Zhang, Yuqun Chen, L. A. Bokut, Word problem for finitely presented metabelian Poisson algebras, arXiv:1907.05953 (2019), https://arxiv.org/abs/1907.05953
Yuqun Chen, Lili Ni, Gröbner–Shirshov bases for associative conformal modules, arXiv:1708.04580 (2017); also in New Trends in Algebras and Combinatorics (World Scientific, 2020) 408–446, DOI 10.1142/9789811215476_0023, https://arxiv.org/abs/1708.04580
P. S. Kolesnikov, R. A. Kozlov, Standard bases for the universal associative conformal envelopes of Kac–Moody conformal algebras, arXiv:2009.12062 (2020); published in Algebras and Representation Theory (2021), DOI 10.1007/s10468-021-10050-0, https://arxiv.org/abs/2009.12062
P. S. Kolesnikov, Universal enveloping Poisson conformal algebras, International Journal of Algebra and Computation 30(5) (2020) 1015–1034, DOI 10.1142/S0218196720500289, https://doi.org/10.1142/S0218196720500289 (arXiv: https://arxiv.org/abs/1902.03001)
R. A. Kozlov, P. S. Kolesnikov, Groebner–Shirshov bases method for vertex algebras, arXiv:2312.01670 (2023), https://arxiv.org/abs/2312.01670; published as The Gröbner–Shirshov bases method for vertex algebras, Journal of Mathematical Sciences 296 (2026) 217–230, DOI 10.1007/s10958-026-08246-0, https://doi.org/10.1007/s10958-026-08246-0
Hassan Alhussein, Pavel Kolesnikov, Viktor Lopatkin, Hochschild cohomology of the universal associative conformal envelope of the Virasoro Lie conformal algebra with coefficients in all finite modules, Communications in Mathematics 33(3) (2025), article 14674, DOI 10.46298/cm.14674, https://doi.org/10.46298/cm.14674 (arXiv: https://arxiv.org/abs/2411.00812)