A Gelfand–Dorfman (GD) algebra carries a Novikov product and a Lie bracket satisfying a mixed identity; the class corresponds to quadratic Lie conformal algebras and Hamiltonian operators. A GD-algebra is special if it embeds into a Poisson algebra with a derivation, and the special GD-algebras form a variety SGD strictly between differential-Poisson-derived algebras and all GD-algebras. Two quartic special identities are known (Kolesnikov–Sartayev–Orazgaliev, 2019), and Kolesnikov–Sartayev (2021/2024) proved they account for all special identities up to degree 5 and computed dim GD(n) for n ≤ 6. The problem asks: (i) whether these two identities, together with the GD defining identities, generate the complete equational theory of SGD in all degrees, or whether an independent special identity of degree ≥ 6 exists; and (ii) for an explicit linear basis of the free GD-algebra, or at least the dimension sequence dim GD(n) beyond the six known values. A 2026 classification paper confirms in print that the completeness question remains open.

Background

A Novikov algebra is a vector space with a bilinear product ∘\circ satisfying right commutativity (a∘b)∘c=(a∘c)∘b(a\circ b)\circ c=(a\circ c)\circ b and the left-symmetry identity (a∘b)∘c−a∘(b∘c)=(b∘a)∘c−b∘(a∘c)(a\circ b)\circ c-a\circ(b\circ c)=(b\circ a)\circ c-b\circ(a\circ c); Novikov algebras arose from the Hamiltonian formalism of partial differential equations. A Gelfand–Dorfman algebra (GD-algebra) is a vector space with two bilinear operations, a Novikov product ∘\circ and a Lie bracket [⋅,⋅][\cdot,\cdot], linked by the mixed identity b∘[a,c]=[a,b∘c]−[c,b∘a]+[b,a]∘c−[b,c]∘ab\circ[a,c]=[a,b\circ c]-[c,b\circ a]+[b,a]\circ c-[b,c]\circ a. These axioms emerged in the study of Hamiltonian operators in the formal calculus of variations, and GD-algebras were later shown to be in one-to-one correspondence with quadratic Lie conformal algebras (Lie conformal algebras generated by a quadratic local family), objects in the theory of vertex operators. The class of GD-algebras is governed by a binary quadratic operad GD\mathop{GD}\nolimits; recall that an operad assigns to each arity nn a space of nn-ary operations encoding the identities of a variety of algebras, and dim⁡GD(n)\dim\mathop{GD}\nolimits(n) is the dimension of the space of multilinear operations of arity nn in the free GD-algebra. A Poisson algebra (P,⋅,{⋅,⋅})(P,\cdot,\{\cdot,\cdot\}) combines an associative commutative product and a Lie bracket with the Leibniz rule; given a linear map dd that is a derivation for both operations, the same space PP with x∘y=x⋅d(y)x\circ y=x\cdot d(y) and [x,y]={x,y}[x,y]=\{x,y\} is a GD-algebra denoted P(d)P^{(d)}. A GD-algebra is special if it embeds into some P(d)P^{(d)}, and Kolesnikov and Sartayev proved the special GD-algebras are closed under homomorphic images, hence form a variety SGD\mathop{SGD}\nolimits. By analogy with special Jordan algebras, a special identity is a polynomial identity holding in all special GD-algebras but not in all GD-algebras; over a field of characteristic zero such identities are determined by their multilinear components of each degree. Two independent quartic special identities were first found in 2019 via the Manin white product of operads (a construction, dual to operadic Koszul duality, that combines two operads into a new one; the operad of special GD-algebras sits inside the Manin white product of the Koszul dual GD!\mathop{GD}\nolimits^{!} with the Poisson operad), and non-special GD-algebras of dimension three exist. The 2021/2024 paper of Kolesnikov and Sartayev, using Gröbner–Shirshov bases (confluent rewriting systems) for Poisson algebras with derivations and Gröbner bases for the shuffle operad (a re-indexing of a symmetric operad that makes composition monomial and enables rewriting computations) governing GD-algebras, proved that every 2-dimensional GD-algebra is special, that all special identities of degree at most 5 are consequences of the two quartic ones, and computed dim⁡GD(n)=1,3,17,140,1524,20699\dim\mathop{GD}\nolimits(n)=1,3,17,140,1524,20699 for n=1,…,6n=1,\dots,6. Related later work clarifies the surrounding landscape: transposed Poisson algebras (Lie algebras with a compatible commutative product, equivalently GD-algebras with commutative Novikov product) satisfy the GD special identities and are conjectured all special, and the analogous basis problem for the free special GD-algebra was solved by Gubarev and Sartayev — but neither the completeness question nor the basis problem for the free GD-algebra itself has been settled. The 2021/2024 paper's technique for special identities is organised around the free GD-algebra: for a linear basis BB of GD⟨X⟩\mathop{GD}\nolimits\langle X\rangle carrying a linear order, the kernel of the natural homomorphism from GD⟨X⟩\mathop{GD}\nolimits\langle X\rangle into its universal Poisson differential envelope is exactly the set of special identities, and the completeness question is approached by comparing the operad SGD\mathop{SGD}\nolimits with the operad defined by the GD identities together with the two quartic identities.

Problem Statement

Let GD\mathop{GD}\nolimits be the variety of Gelfand–Dorfman algebras over a field of characteristic zero (vector spaces with a Novikov product ∘\circ and a Lie bracket [⋅,⋅][\cdot,\cdot] satisfying b∘[a,c]=[a,b∘c]−[c,b∘a]+[b,a]∘c−[b,c]∘ab\circ[a,c]=[a,b\circ c]-[c,b\circ a]+[b,a]\circ c-[b,c]\circ a), and let SGD\mathop{SGD}\nolimits be the variety of special GD-algebras, i.e. those embeddable into a Poisson algebra with a derivation dd via x∘y=x⋅d(y)x\circ y=x\cdot d(y), [x,y]={x,y}[x,y]=\{x,y\}. Consider the two quartic special identities found in Kolesnikov–Sartayev–Orazgaliev (2019), displayed as (13) and (15) in arXiv:2105.13815: [c,a∘d]∘b+([a,c]∘d)∘b=[c,(a∘b)∘d]−[c,a∘b]∘d[c,a\circ d]\circ b+([a,c]\circ d)\circ b=[c,(a\circ b)\circ d]-[c,a\circ b]\circ d 2([a,b]∘c)∘d=[b∘c,a∘d]−[a∘c,b∘d]+([a,b∘c]−[b,a∘c])∘d+([a,b∘d]−[b,a∘d])∘c2([a,b]\circ c)\circ d=[b\circ c,a\circ d]-[a\circ c,b\circ d]+([a,b\circ c]-[b,a\circ c])\circ d+([a,b\circ d]-[b,a\circ d])\circ c The problem has two logically independent parts, both stated as open directions in the source paper and both contributing to one overall objective — a complete structural description of the GD operad and of the boundary between the varieties GD\mathop{GD}\nolimits and SGD\mathop{SGD}\nolimits. A complete resolution requires both parts: a complete answer to (1) alone, or to (2) alone, resolves that part but not the other, and each part is independently refutable:

(1) Do the GD defining identities together with the two displayed quartic identities generate the entire equational theory of SGD\mathop{SGD}\nolimits? Equivalently: does every polynomial identity holding in every GD-algebra of the form P(d)P^{(d)} for a differential Poisson algebra PP follow from the GD identities plus these two? It is known that all special identities of degree ≤5\leq 5 are consequences of the two quartic ones; the question is whether an independent special identity of degree ≥6\geq 6 exists.

(2) Find an explicit (combinatorial, e.g. monomial) linear basis of the free GD-algebra GD⟨X⟩\mathop{GD}\nolimits\langle X\rangle, or at least determine the dimension sequence dim⁡GD(n)\dim\mathop{GD}\nolimits(n) for all nn; currently only dim⁡GD(n)=1,3,17,140,1524,20699\dim\mathop{GD}\nolimits(n)=1,3,17,140,1524,20699 for n=1,…,6n=1,\dots,6 is known (the first five terms coincide with OEIS A322137/A291842, but the sixth term differs, ruling out that model).

The two parts are related but neither implies the other: settling (2) does not by itself settle (1), nor conversely. Answers must address the stated generality (all degrees, all arities); no restriction to particular degrees, dimensions, or subclasses is admissible.

References

  1. P. S. Kolesnikov, B. Sartayev, A. Orazgaliev. Gelfand–Dorfman algebras, derived identities, and the Manin product of operads. Journal of Algebra 539 (2019) 260–284. DOI: 10.1016/j.jalgebra.2019.07.034; arXiv:1903.02238; https://arxiv.org/abs/1903.02238
  2. P. S. Kolesnikov, B. K. Sartayev. On the special identities of Gelfand–Dorfman algebras. Experimental Mathematics 33 (2024), no. 1, 165–174. DOI: 10.1080/10586458.2022.2041134; arXiv:2105.13815; https://arxiv.org/abs/2105.13815
  3. V. Gubarev, B. K. Sartayev. Free special Gelfand–Dorfman algebra. Journal of Algebra and Its Applications 23 (2024), no. 14, 2550005. DOI: 10.1142/S0219498825500057; arXiv:2208.10147; https://arxiv.org/abs/2208.10147
  4. B. K. Sartayev. Some generalizations of the variety of transposed Poisson algebras. Communications in Mathematics 32 (2024), no. 2, art. 11346. DOI: 10.46298/cm.11346; arXiv:2305.12869; https://arxiv.org/abs/2305.12869
  5. Ziyi Zhang, Zeyu Hao, Yining Sun, Liangyun Chen. Gelfand–Dorfman Algebras: Nilpotency, Solvability, Construction and Classification. Preprint (2026). arXiv:2607.09672; https://arxiv.org/abs/2607.09672