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Ivan Kaygorodov

Publications and source records attributed to Ivan Kaygorodov.

At least 19 recordsLinked to original sources
arXiv2026

The algebraic and geometric classification of $\delta$-Novikov algebras

The notion of $\delta$-Novikov algebras was introduced recently as a generalization of Novikov and bicommutative algebras. It looks like $\delta$-Novikov algebras have a richer structure than Novikov algebras. So, unlike Novikov algebras, they have a $2$-dimensional simple algebra for $\delta=-1.$ The present paper is dedicated to the study of $3$-dimensional $\delta$-Novikov algebras for $\delta \notin \big\{0,1\big\}.$ The algebraic and geometric classifications of complex $3$-dimensional $\delta$-Novikov algebras are given. As a corollary, we prove that there are no simple $3$-dimensional $\delta$-Novikov algebras.

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arXiv2026

Integro-derivation Dzhumadildaev algebras: from the algebra of polynomials

This paper introduces and investigates some properties of algebras constructed from the algebra of polynomials via derivation and integration operators using a process presented by Dzhumadildaev in a previous work. In particular, we discover new classes of infinite-dimensional simple conservative algebras and describe derivations of these algebras of ranks 1 and 2.

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arXiv2026

Unital $3$-dimensional structurable algebras: classification, properties and $\rm{AK}$-construction

This paper is devoted to the classification and studying properties of complex unital $3$-dimensional structurable algebras. We provide a complete list of non-isomorphic classes, identifying five algebras for type $(2, 1)$ and two algebras for type $(1, 2).$ For each obtained algebra, we describe the derivation algebra, the automorphism group, the lattice of subalgebras and ideals, and functional identities of degree $2$. Furthermore, we investigate the Allison-Kantor construction for the classified algebras. We determine the structure of the resulting $\mathbb{Z}$-graded Lie algebras, providing their dimensions and Levi decompositions.

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arXiv2026

The algebraic and geometric classification of noncommutative Jordan superalgebras

The algebraic and geometric classifications of complex $3$-dimensional noncommutative Jordan superalgebras are given. In particular, we obtain the algebraic and geometric classification of $3$-dimensional Kokoris and standard superalgebras, and, due to one-to-one correspondences between suitable superalgebras, we have classifications for generic Poisson-Jordan and generic Poisson superalgebras. As a byproduct, we have the algebraic and geometric classification of the variety of $3$-dimensional anticommutative superalgebras and its principal subvarieties: Lie, Malcev, binary Lie, Tortkara, anticommutative $\mathfrak{CD}$-, $\mathfrak{s}_4$-, anticommutative terminal superalgebras, anticommutative conservative and anticommutative quasi-conservative $\big($rigid$\big)$ superalgebras.

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arXiv2026

The algebraic and geometric classification of right alternative superalgebras

The algebraic and geometric classifications of complex $3$-dimensional right alternative superalgebras are given. As a byproduct, we have the algebraic and geometric classification of the variety of $3$-dimensional $\mathfrak{perm}$, binary $\mathfrak{perm}$, associative, binary associative, $\big(-1,1\big)$-, and binary $\big(-1,1\big)$-superalgebras.

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arXiv2026

The algebraic and geometric classification of commutative post-Lie algebras

We study commutative post-Lie algebras $(${\rm CPA}s$)$ from an algebraic point of view. Firstly, we find some new identities in {\rm CPA}, which shows that the commutative multiplication gives a medial and derived commutative associative algebra. As corollaries, we have that there are no simple nontrivial commutative post-Lie algebras and that perfect Lie and centrless perfect commutative associative algebras do not admit nontrivial {\rm CPA} structures. The identities of depolarized {\rm CPA}s are defined. Based on the obtained identities, we developed a method for the classification of $n$-dimensional {\rm CPA}s and gave the algebraic classification of $3$-dimensional {\rm CPA}. We also developed another method for classifying $n$-dimensional nilpotent {\rm CPA}s from nilpotent {\rm CPA}s of smaller dimension and gave the algebraic classification of $4$-dimensional nilpotent {\rm CPA}s. Based on the obtained results, we present the geometric classifications of complex $3$-dimensional and $4$-dimensional nilpotent {\rm CPA}s.

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arXiv2026

The algebraic and geometric classification of derived Jordan and bicommutative algebras

We developed a new proper method for classifying $n$-dimensional derived Jordan algebras, and apply it to the classification of $3$-dimensional derived Jordan algebras. As a byproduct, we have the algebraic classification of $3$-dimensional metabelian commutative algebras and $3$-dimensional derived commutative associative algebras. After that, we introduced a method of classifying $n$-dimensional bicommutative algebras, based on the classification of $n$-dimensional derived commutative associative algebras, and applied it to the classification of $3$-dimensional bicommutative algebras. The second part of the paper is dedicated to the geometric classification of $3$-dimensional metabelian commutative, derived commutative associative, derived Jordan and bicommutative algebras.

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arXiv2026

$\delta$-Leibniz algebras and related $\delta$-type algebras

This paper introduces and investigates the structure of $\delta$-Leibniz algebras, which serve as a parametric generalization of classical Leibniz algebras defined by a scalar $\delta$. The authors define $\delta$-Lie algebras, $\delta$-Lie dialgebras, and $\delta$-Zinbiel algebras via a standard procedure and study their fundamental properties. Furthermore, the research describes symmetric $\delta$-Leibniz algebras and algebras of $\delta$-biderivation type, establishing their connections with nilalgebras. Finally, these results provide a unified framework for understanding various classes of non-associative algebras through the lens of the $\delta$ parameter.

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arXiv2025

Quasi-derivations of Witt and related algebras

In the present work, we compute quasi-derivations of the Witt algebra and some algebras well-related to the Witt algebra. Namely, we prove that each quasi-derivation of the Witt algebra is a sum of a derivation and a $\frac{1}{2}$-derivation; a similar result is obtained for the Virasoro algebra. A different situation appears for Lie algebras ${\mathcal W}(a,b):$ in the case of $b=-1,$ they do not have interesting examples of quasi-derivations, but the case of $b\neq-1$ provides some new non-trivial examples of quasi-derivations. We also completely describe all quasi-derivations of ${\mathcal W}(a,b).$ As a corollary, we describe the derivations and quasi-derivations of the Novikov-Witt and admissible Novikov-Witt algebras previously constructed by Bai and his co-authors; and $\delta$-derivations and transposed $\delta$-Poisson structures on cited Lie algebras. In particular, we proved that each ${\mathcal W}(a,b)$ admits a nontrivial transposed $\frac 1{1-b}$-Poisson structure.

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arXiv2025

$\delta$-Novikov and $\delta$-Novikov--Poisson algebras

This article considers the structure and properties of $\delta$-Novikov algebras, a generalization of Novikov algebras characterized by a scalar parameter $\delta$. It looks like $\delta$-Novikov algebras have a richer structure than Novikov algebras. So, unlike Novikov algebras, they have non-commutative simple finite-dimensional algebras for $\delta=-1.$ Additionally, we introduce $\delta$-Novikov--Poisson algebras, extending several theorems from the classical Novikov--Poisson algebras. Specifically, we consider the commutator structure $[a, b] = a \circ b - b \circ a$ of $\delta$-Novikov algebras, proving that when $\delta \neq 1$, these algebras are metabelian Lie-admissible. Moreover, we prove that every metabelian Lie algebra can be embedded into a suitable $\delta$-Novikov algebra with respect to the commutator product. We further consider the construction of $\delta$-Poisson and transposed $\delta$-Poisson algebras through $\delta$-derivations on the commutative associative algebras. Finally, we analyze the operad associated with the variety of $\delta$-Novikov algebras, proving that it is not Koszul for any value of $\delta$. This result extends known results for the Novikov operad $(\delta=1)$ and the bicommutative operad $(\delta=0)$.

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arXiv2025

Polynomial invariants for 3-dimensional Leibniz algebras

For each 3-dimensional non-Lie Leibniz algebra over the complex numbers, we describe the algebra of polynomial invariants and determine its group of automorphisms. As a consequence, we establish that any two non-nilpotent 3-dimensional non-Lie Leibniz algebras can be distinguished by the traces of degrees $\leqslant 2$ and by the dimensions of their automorphism groups.

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arXiv2025

Nonassociative algebras of anti-biderivation-type

The main purpose of this paper is to study the class of Jacobi-Jordan-admissible algebras, such that its product is an anti-biderivation of the related Jacobi-Jordan algebra. We called it as $\mathcal A{\rm BD}$-algebras. First, we provide characterizations of algebras in this class. Furthermore, we show that this class of nonassociative algebras includes Jacobi-Jordan algebras, symmetric anti-Leibniz algebras, and anti-${\rm LR}$-algebras. In particular, we proved that anti-${\rm LR}$-algebras under the commutator product give $\mathfrak{s}_4$-algebras, which were recently introduced by Filippov and Dzhumadildaev. In addition, we then study $\mathcal A$flexible ${\mathcal A}{\rm BD}$-algebras. Then, we introduce the post-Jacobi-Jordan structures on Jacobi-Jordan algebras and establish results that each Jacobi-Jordan algebra admits a non-trivial post-Jacobi-Jordan structure. At the end of the paper, we give the algebraic classification of complex $3$-dimensional $\mathcal A{\rm BD}$-algebras.

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arXiv2025

The algebraic and geometric classification of right alternative and semi-alternative algebras

The algebraic and geometric classifications of complex $3$-dimensional right alternative and semi-alternative algebras are given. As corollaries, we have the algebraic and geometric classification of complex $3$-dimensional $\mathfrak{perm}$, binary $\mathfrak{perm}$, associative, $(-1,1)$-, binary $(-1,1)$-, and assosymmetric algebras. In particular, we proved that the first example of non-associative right alternative algebras appears in dimension $3;$ the first example of non-associative assosymmetric algebras appears in dimension $3;$ the first example of non-assosymmetric semi-alternative algebras appears in dimension $4;$ the first example of binary $(-1,1)$-algebras, which is non-$(-1,1)$-, appears in dimension $4;$ the first example of right alternative algebras, which is not binary $(-1,1)$-, appears in dimension $4;$ the first example of binary $\mathfrak{perm}$ non-$\mathfrak{perm}$ algebras appears in dimension $4.$ As a byproduct, we give a more easy answer to problem 2.109 from the Dniester Notebook, previously resolved by Shestakov and Arenas.

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arXiv2024

$\delta$-Poisson and transposed $\delta$-Poisson algebras

We present a comprehensive study of two new Poisson-type algebras. Namely, we are working with $\delta$-Poisson and transposed $\delta$-Poisson algebras. Our research shows that these algebras are related to many interesting identities. In particular, they are related to shift associative algebras, $F$-manifold algebras, algebras of Jordan brackets, etc. We classify simple $\delta$-Poisson and transposed $\delta$-Poisson algebras and found their depolarizations. We study $\delta$-Poisson and mixed-Poisson algebras to be Koszul and self-dual. Bases of the free $\delta$-Poisson and mixed-Poisson algebras generated by a countable set $X$ are constructed.

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arXiv2024

The geometric classification of non-associative algebras: a survey

This is a survey on the geometric classification of different varieties of algebras (nilpotent, nil-, associative, commutative associative, cyclic associative, Jordan, Kokoris, standard, noncommutative Jordan, commutative power-associative, weakly associative, terminal, Lie, Malcev, binary Lie, Tortkara, dual mock Lie, $\mathfrak{CD}$-, commutative $\mathfrak{CD}$-, anticommutative $\mathfrak{CD}$-, symmetric Leibniz, Leibniz, Zinbiel, Novikov, bicommutative, assosymmetric, antiassociative, left-symmetric, right alternative, and right commutative), $n$-ary algebras (Fillipov ($n$-Lie), Lie triple systems and anticommutative ternary), superalgebras (Lie and Jordan), and Poisson-type algebras (Poisson, transposed Poisson, Leibniz-Poisson, generic Poisson, generic Poisson-Jordan, transposed Leibniz-Poisson, Novikov-Poisson, pre-Lie Poisson, commutative pre-Lie, anti-pre-Lie Poisson, pre-Poisson, compatible commutative associative, compatible associative, compatible Novikov, compatible pre-Lie). We also discuss the degeneration level classification.

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arXiv2024

Mutations of $\mathfrak{perm}$ algebras

We describe mutation elements in free $\mathfrak{perm}$ algebras. Moreover, we construct a base of free mutation of free $\mathfrak{perm}$ algebra. Using Cohn's criterion for the specialty of algebras, we show that there is an exceptional homomorphic image of mutation of free $\mathfrak{perm}$ algebras.

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arXiv2024

Shift associative algebras

We present a comprehensive study of algebras satisfying the identity $(xy)z=y(zx),$ named as shift associative algebras. Our research shows that these algebras are related to many interesting identities. In particular, they are related to anti-Poisson-Jordan algebras and algebras of associative type $\sigma$. We study algebras of associative type $\sigma$ to be Koszul and self-dual. A basis of the free shift associative algebra generated by a countable set $X$ was constructed. An analog of Wedderburn-Artin's theorem was established. The algebraic and geometric classifications of complex $4$-dimensional shift associative algebras are given. In particular, we proved that the first non-associative shift associative algebra appears only in dimension $5$.

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