arXiv subjects

Zafar Normatov

Publications and source records attributed to Zafar Normatov.

8 recordsLinked to original sources

arXiv2026

$δ$-Mock-Novikov Algebras: Structural Theory, Operad and Poisson-Type Constructions

We introduce $δ$-mock-Novikov algebras as a parameter-dependent mock analogue of Novikov algebras. For $δ=1$, the commutator of a mock-Novikov algebra defines a Malcev algebra. Every $δ$-mock-Novikov algebra satisfying $\mathcal{A}^2\subseteq\operatorname{Ann}(\mathcal{A})$ is shown to be differentially special. At the operadic level, the binary quadratic operad governing $δ$-mock-Novikov algebras is quadratically self-dual but not Koszul. We prove that finite-dimensional $δ$-mock-Novikov algebras are nilpotent in characteristic zero and classify those of dimension at most four. Finally, we study the associated Poisson-type structures and establish relations and constructions among them via polarization, depolarization, and tensor products.

math.RA↗

arXiv2026

Poisson $n$-Lie algebras: constructions and the structure of solvable algebras

In this paper, we develop a construction of Poisson $n$-Lie algebras that generalizes the Jacobian $n$-Lie construction. Using the Grassmann--Plücker relations, we derive necessary and sufficient conditions under which the resulting bracket defines a Poisson $n$-Lie algebra. We also prove that suitable quotients of tensor products of Poisson algebras carry natural Poisson $n$-Lie structures. Conversely, we give a tensor-type procedure that associates a Poisson algebra to a given Poisson $n$-Lie algebra. The quotient and converse constructions thus provide two systematic methods for relating Poisson algebras to Poisson $n$-Lie algebras. We further establish analogues of Engel's theorem and Lie's theorem and characterize solvability and nilpotency of Poisson $n$-Lie algebras in terms of their underlying associative and $n$-Lie structures. We introduce hypo-nilpotent ideals and investigate maximal such ideals in finite-dimensional solvable Poisson $n$-Lie algebras. Finally, we prove that the generalized eigenspaces of multiplication operators are ideals.

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arXiv2026

Anti-associative dendriform algebras

The general operadic approach to splitting algebraic operations was developed in \cite{BBGN}. By splitting the product in a given algebraic variety $\mathcal{C}$, notion of $\mathcal{C}$-dendriform algebras was systematically studied in \cite{OPV}. This article aims to study ``anti-associative dendriform algebras", which offer an approach to addressing anti-associativity. These algebras are defined by two operations whose sum is anti-associative. Furthermore, the notion of $\mathcal{O}$-operators on anti-associative algebras is presented as a tool to interpret anti-associative dendriform algebras. Moreover, anti-associative algebras with nondegenerate Connes cocycles admit compatible anti-associative dendriform algebra structures.

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arXiv2026

Mock-pre-Lie bialgebras

In this paper, we systematically develop the theory of mock-pre-Lie bialgebras from multiple perspectives. We introduce the notion of a phase space of a mock-Lie algebra, and show that a mock-Lie algebra admits a phase space if and only if it is sub-adjacent to a mock-pre-Lie algebra. We introduce the notions of Manin triples of mock-pre-Lie algebras and mock-pre-Lie bialgebras, and prove the equivalences between mock-pre-Lie bialgebras, Manin triples of mock-pre-Lie algebras, certain matched pairs of mock-pre-Lie algebras, certain matched pairs of mock-Lie algebras and phase spaces of a mock-Lie algebra, which lays a theoretical foundation for subsequent research. Next, we investigate coboundary mock-pre-Lie bialgebras, and derive an analogue of the classical Yang-Baxter equation. In addition, we introduce two important special classes of mock-pre-Lie bialgebras: quasi-triangular mock-pre-Lie bialgebras and factorizable mock-pre-Lie bialgebras. We show that quasi-triangular mock-pre-Lie bialgebras naturally induce relative Rota-Baxter operators of weight -1. Finally, we provide a new perspective for the study of triangular and factorizable mock-pre-Lie bialgebras by introducing the concept of quadratic Rota-Baxter mock-pre-Lie algebras of arbitrary weight.

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arXiv2025

On the coordinate rings of Calogero-Moser spaces and the invariant commuting variety of a pair of matrices

This paper presents a comprehensive description of the coordinate rings and Poisson brackets associated with the fourth Calogero-Moser space and invariant commuting pairs of matrices of size four. As an application, we compute their respective classes in the Grothendieck ring of the category of complex varieties and we offer some novel insights about the geometry of the Hilbert scheme of points on the affine plane.

math.AG↗

arXiv2024

Compatible anti-pre-Lie algebras

In this paper, we introduce the notion of compatible anti-pre-Lie algebras and study relationship between them and the related structures such as anti-$\mathcal{O}$-operators, commutative $2$-cocycles on compatible Lie algebras. Moreover, we give the classification of $2$-dimensional compatible anti-pre-Lie algebras from the classification of anti-pre-Lie algebras of the same dimension.

math.RA↗

arXiv2024

Classification of four-dimensional anti-dendriform algebras whose associated associative algebra has the center of dimension one

This article is devoted to the classification of anti-dendriform algebras that are associated with associativity. They are characterized as algebras with two operations whose sum is associative. In the paper all four-dimensional complex anti-dendriform algebras associated to four-dimensional associative algebras with one-dimensional center are classified

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arXiv2021

Calogero-Moser spaces and the invariants of two matrices of degree 3

We find a minimal set of generators for the coordinate ring of Calogero-Moser space $\mathcal{C}_3$ and the algebraic relations among them explicitly. We give a new presentation for the algebra of $3\times3$ invariant matrices involving the defining relations of $\mathbb{C}[\mathcal{C}_3]$. We find an explicit description of the commuting variety of $3\times3$ matrices and its orbits under the action of the affine Cremona group.

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