Classifying filtered deformations and the completion identification $A^*=G^*$ for the type I and integer-parameter type IV Novikov prototypes of Osborn-Zelmanov

Osborn and Zelmanov (1995) classified the simple infinite-dimensional Novikov algebras over an algebraically closed field of characteristic 0 satisfying a Weisfeiler-type finiteness condition, proving that the associated graded algebra of the induced filtration is one of four explicit prototypes with product $x_i\circ x_j=(\alpha j+\beta)x_{i+j}$, and that for types II and III and for type IV with non-exceptional parameter the completion of the filtered algebra identifies with the completion of its graded model. For type I (the Laurent-polynomial prototype) and for type IV when $\beta$ is an exceptional integer (per the source, a set including $\beta=0,-1,-2,\ldots$), the identification was not established and the filtered deformations were left unclassified: it is unknown which simple filtered Novikov algebras realize these prototypes as associated graded while their completions fail to be topologically isomorphic to the graded completion, and no invariants distinguishing the resulting completions are available. This problem asks to determine the exact exceptional parameter set, to prove or disprove the completion identification $A^*=G^*$ in the remaining cases, to classify the filtered deformations of these prototypes, and to supply distinguishing invariants for the completions. The companion question of classifying the Lie products compatible with the integer-$\beta$ type IV Novikov product was resolved by Xu (2000).

Background

Novikov algebras originated in the study of Hamiltonian operators in the formal calculus of variations, where a Hamiltonian operator is a first-order differential operator on a space of local functionals whose associated Poisson bracket is skew-symmetric and satisfies the Jacobi identity. Gel'fand and Dorfman showed that the algebraic data of a basic class of such operators in the one-spatial-variable setting is what is now called a Novikov algebra: a vector space $A$ with a bilinear product $\circ$ satisfying, for all $x,y,z\in A$, the right-commutativity and left-symmetry identities

$$(x\circ y)\circ z=(x\circ z)\circ y,\qquad (x\circ y)\circ z+y\circ(x\circ z)=x\circ(y\circ z)+(y\circ x)\circ z.$$

Independently, Balinskii and Novikov met the same structure while classifying linear Poisson brackets of hydrodynamic type, that is, Poisson brackets on spaces of fields of one spatial variable that are of first order in the spatial derivatives. A Gel'fand-Dorfman bialgebra is a vector space carrying both a Novikov product $\circ$ and a Lie bracket $[\cdot,\cdot]$ linked by the compatibility identity

$$[w\circ u,v]-[w\circ v,u]+[w,u]\circ v-[w,v]\circ u-w\circ[u,v]=0;$$

these bialgebras are equivalent to quadratic Lie conformal superalgebras (a conformal algebra being a module over the polynomial ring in one formal derivation $\partial$ with a $\lambda$-bracket, and a quadratic one being generated in degree one) and encode Hamiltonian pairs of operators. The commutator $[x,y]^-=x\circ y-y\circ x$ of a Novikov product always yields such a bracket, so one central question of the theory is when it is essentially the only one. Osborn and Zelmanov (1995) classified the graded models of simple infinite-dimensional Novikov algebras under a Weisfeiler-type finiteness condition: $A$ contains a maximal subalgebra $H$ such that $A/H$ has a nonzero finite-dimensional irreducible $H$-submodule (a nonassociative analogue of the filtration technique Weisfeiler introduced for classifying simple infinite-dimensional Lie algebras). The Weisfeiler filtration is constructed from $H$ as follows: with $V$ the irreducible submodule and $K$ the kernel of the representation of $H$ on $V$, one sets $A_0=H$, $A_1=K$, defines $A_{i+1}=\{a\in A_i\mid Va+aV\subseteq A_i\}$ for $i\ge1$, and lets $A_{-i}$ be the linear span of all products of $i$ factors from a preimage of $V$; the resulting chain $\{A_i\}_{i\in\mathbb Z}$ satisfies $A_iA_j\subseteq A_{i+j}$ and defines a linear topology in which the filtration subspaces form the neighborhoods of zero, and $A^*$ denotes the completion (closure) of $A$ in this topology. The associated graded algebra $G=\bigoplus_i G_i$ with $G_i=A_i/A_{i+1}$ is one of four prototypes on a basis $\{x_i\}$ with product $x_i\circ x_j=(\alpha j+\beta)x_{i+j}$: type I ($\alpha=0$, $\beta=1$, $i\in\mathbb Z$, the Laurent-polynomial algebra $\mathbb F[x,x^{-1}]$), type II ($\alpha=\beta=1$, $i\ge-1$, the example of S. Gelfand), type III ($\alpha=1$, $\beta=-1$, $i\le1$, the opposite grading of type II), and type IV ($\alpha=1$, $\beta\neq1$, $i\in\mathbb Z$, realized on a subspace of the group algebra of the additive subgroup $\mathbb Z+\mathbb Z\beta$ of $\mathbb C$ with product induced by a derivation). They proved that for types II and III, and for type IV with $\beta$ outside an exceptional set of integer values, every Lie bracket compatible with $\circ$ is a scalar multiple of the commutator and the completion $A^*$ identifies with the graded completion $G^*$. For type IV with integer $\beta$ they exhibited a compatible Lie bracket that is not a scalar multiple of the commutator; for type I and for type IV at the exceptional integer values of $\beta$ they left the completion identification and the classification of filtered deformations unsettled. Xu (2000) subsequently classified all Lie algebra structures compatible with the type IV Novikov product for every additive subgroup of $\mathbb C$ and every parameter value, closing the Lie-product part of the source question, and Xu (2001) classified the simple Novikov algebras containing an element whose left-multiplication operator is locally finite and whose right-multiplication operator is a constant map, a finiteness hypothesis different from the Weisfeiler condition; neither work treats the completions of the type I and exceptional-integer type IV prototypes.

Problem Statement

Let $\mathbb F$ be an algebraically closed field of characteristic 0, and let $A$ be a simple infinite-dimensional Novikov algebra over $\mathbb F$ satisfying the Weisfeiler condition: $A$ contains a maximal subalgebra $H$ such that $A/H$ has a nonzero finite-dimensional irreducible $H$-submodule. The induced Weisfeiler filtration $\{A_i\}_{i\in\mathbb Z}$ (constructed in the background) defines a linear topology in which the filtration subspaces are the neighborhoods of zero; write $A^*$ for the completion (closure) of $A$ in this topology and $G$ for the associated graded algebra, which by Osborn-Zelmanov is one of four prototypes with product $x_i\circ x_j=(\alpha j+\beta)x_{i+j}$: type I ($\alpha=0$, $\beta=1$, $i\in\mathbb Z$; the Laurent-polynomial prototype $x_i\circ x_j=x_{i+j}$), type II ($\alpha=\beta=1$, $i\ge-1$), type III ($\alpha=1$, $\beta=-1$, $i\le1$), and type IV ($\alpha=1$, $\beta\neq1$, $i\in\mathbb Z$). Let $G^*$ denote the completion of $G$ in the topology whose neighborhoods of zero are the graded tails $\bigoplus_{j\ge i}G_j$, $i\in\mathbb Z$. Osborn-Zelmanov proved $A^*=G^*$ for types II and III and for type IV with $\beta$ outside the exceptional integer values. The unresolved core of the source question is the following. (1) Determine the exact exceptional set of integer values of $\beta$ in type IV for the completion identification. The source record is internally ambiguous: its theorem conclusion is stated for $\beta$ 'not a nonzero integer', which leaves every nonzero integer outside the theorem's scope, while its open-question discussion names the failure values as 'including $\beta=0,-1,-2,\dots$'; the negative integers are outside the theorem's scope under either reading, and the status of $\beta=0$ and of the positive integers must be settled explicitly. (2) For $G$ of type I and for $G$ of type IV with $\beta$ in this exceptional set, decide whether every simple filtered Novikov algebra $A$ with Weisfeiler filtration whose associated graded is isomorphic to $G$ as a graded algebra satisfies $A^*\cong G^*$ as topological Novikov algebras (an algebra isomorphism that is also a homeomorphism); where the identification fails, exhibit explicit counterexamples, that is, explicit filtered algebras with computable structure constants. (3) Classify the filtered deformations of these prototypes: describe all simple Novikov algebras with Weisfeiler filtration whose associated graded is the given prototype, up to isomorphism of filtered Novikov algebras (an algebra isomorphism carrying each filtration subspace onto the corresponding one), and among them separate those whose completion is topologically isomorphic to the graded completion from those whose completion is not. (4) Provide explicit descriptions or computable invariants distinguishing the non-isomorphic completions that arise in this way. The four subgoals (1)-(4) interlock rather than stand alone, since (3) subsumes the classification whose separation step decides (2), and (4) supplies the invariants certifying the separations found in (3). The type I case and the type IV case with exceptional integer $\beta$ (product $x_i\circ x_j=(j+\beta)x_{i+j}$) are the two source-designated unresolved branches; the companion question of classifying the Lie brackets compatible with these Novikov products, also left open by the source for nonzero integer $\beta$, was resolved by Xu (2000) (reference 2), so it is not part of this problem's core.

References

1. J. Marshall Osborn and Efim Zelmanov, Nonassociative algebras related to Hamiltonian operators in the formal calculus of variations, Journal of Pure and Applied Algebra 101 (1995), no. 3, 335-352. DOI: 10.1016/0022-4049(95)00062-2. https://doi.org/10.1016/0022-4049(95)00062-2
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3. Xiaoping Xu, Gel'fand-Dorfman bialgebras, Southeast Asian Bulletin of Mathematics 27 (2003), no. 3, 561-574. arXiv:math/0008223. https://arxiv.org/abs/math/0008223
4. Xiaoping Xu, Classification of simple Novikov algebras and their irreducible modules of characteristic 0, Journal of Algebra 246 (2001), no. 2, 673-707. DOI: 10.1006/jabr.2001.8965. arXiv:math/0008072. https://arxiv.org/abs/math/0008072
5. I. M. Gel'fand and I. Ya. Dorfman, Hamiltonian operators and algebraic structures related to them, Functional Analysis and Its Applications 13 (1979), no. 4, 248-262. DOI: 10.1007/BF01078363. https://doi.org/10.1007/BF01078363
6. A. A. Balinskii and S. P. Novikov, Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras, Soviet Mathematics. Doklady 32 (1985), 228-231; translation of the Russian original in Doklady Akademii Nauk SSSR 283 (1985), 1036-1039. zbMATH Zbl 0606.58018. https://zbmath.org/?q=an%3A0606.58018
7. Yanyong Hong, Chengming Bai, and Li Guo, A bialgebra theory of Gel'fand-Dorfman algebras with applications to Lie conformal bialgebras, Canadian Journal of Mathematics (2025), 1-33. DOI: 10.4153/S0008414X25101181. arXiv:2401.13608. https://doi.org/10.4153/S0008414X25101181
8. A. S. Panasenko, Right representations of Novikov algebras, preprint arXiv:2606.16431 (2026). https://arxiv.org/abs/2606.16431