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Ivan Kaygorodov

Publications and source records attributed to Ivan Kaygorodov.

At least 73 records · Page 4Linked to original sources
arXiv2022

Transposed Poisson structures on Witt type algebras

We describe $\frac{1}{2}$-derivations, and hence transposed Poisson algebra structures, on Witt type Lie algebras $V(f)$, where $f:Γ\to\mathbb C$ is non-trivial and $f(0)=0$. More precisely, if $|f(Γ)|\ge 4$, then all the transposed Poisson algebra structures on $V(f)$ are mutations of the group algebra structure $(V(f),\cdot)$ on $V(f)$. If $|f(Γ)|=3$, then we obtain the direct sum of $3$ subspaces of $V(f)$, corresponding to cosets of $Γ_0$ in $Γ$, with multiplications being different mutations of $\cdot$. The case $|f(Γ)|=2$ is more complicated, but also deals with certain mutations of $\cdot$. As a consequence, new Lie algebras that admit non-trivial ${\rm Hom}$-Lie algebra structures are found.

math.RA↗
arXiv2022

Transposed Poisson structures on Block Lie algebras and superalgebras

We describe transposed Poisson algebra structures on Block Lie algebras $\mathcal B(q)$ and Block Lie superalgebras $\mathcal S(q)$, where $q$ is an arbitrary complex number. Specifically, we show that the transposed Poisson structures on $\mathcal B(q)$ are trivial whenever $q\not\in\mathbb Z$, and for each $q\in\mathbb Z$ there is only one (up to an isomorphism) non-trivial transposed Poisson structure on $\mathcal B(q)$. The superalgebra $\mathcal S(q)$ admits only trivial transposed Poisson superalgebra structures for $q\ne 0$ and two non-isomorphic non-trivial transposed Poisson superalgebra structures for $q=0$. As a consequence, new Lie algebras and superalgebras that admit non-trivial ${\rm Hom}$-Lie algebra structures are found.

math.RA↗
arXiv2022

The algebraic and geometric classification of Zinbiel algebras

This paper is devoted to the complete algebraic and geometric classification of complex $5$-dimensional Zinbiel algebras. In particular, we proved that the variety of complex $5$-dimensional Zinbiel algebras has dimension $24$, it is defined by $16$ irreducible components and it has $11$ rigid algebras.

math.RA↗
arXiv2022

The algebraic and geometric classification of nilpotent weakly associative and symmetric Leibniz algebras

This paper is devoted to the complete algebraic and geometric classification of complex $4$-dimensional nilpotent weakly associative, complex $4$-dimensional symmetric Leibniz algebras, and complex $5$-dimensional nilpotent symmetric Leibniz algebras. In particular, we proved that the variety of complex $4$-dimensional symmetric Leibniz algebras has no Vergne--Grunewald--O'Halloran Property (there is an irreducible component formed by only nilpotent algebras), but on the other hand, it has Vergne Property (there are no rigid nilpotent algebras).

math.RA↗
arXiv2022

Symmetric Zinbiel superalgebras

The notion of symmetric Zinbiel superalgebras is introduced. We prove that the nilpotency index of a symmetric Zinbiel superalgebra is not greater than 4 and describe two-generated symmetric Zinbiel algebras and odd generated superalgebras. We discuss identities of mono and binary symmetric Zinbiel and Leibniz algebras. It is proven that each quadratic Zinbiel algebra is 2-step nilpotent. Also, we study double extensions of symmetric Zinbiel algebras.

math.RA↗
arXiv2022

On the Kantor product, II

We describe the Kantor square (and Kantor product) of multiplications, extending the classification proposed in [I. Kaygorodov, On the Kantor product, Journal of Algebra and Its Applications, 16 (2017), 9, 1750167]. Besides, we explicitly describe the Kantor square of some low dimensional algebras and give constructive methods for obtaining new transposed Poisson algebras and Poisson-Novikov algebras; and for classifying Poisson structures and commutative post-Lie structures on a given algebra.

math.RA↗
arXiv2021

$\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras

A relation between $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras was established. Some non-trivial transposed Poisson algebras with a certain Lie algebra (Witt algebra, algebra $\mathcal{W}(a,-1)$, thin Lie algebra and solvable Lie algebra with abelian nilpotent radical) were constructed. In particular, we constructed an example of the transposed Poisson algebra with associative and Lie parts isomorphic to the Laurent polynomials and the Witt algebra. On the other side, it was proven that there are no non-trivial transposed Poisson algebras with Lie algebra part isomorphic to a semisimple finite-dimensional algebra, a simple finite-dimensional superalgebra, the Virasoro algebra, $N=1$ and $N=2$ superconformal algebras, or a semisimple finite-dimensional $n$-Lie algebra.

math.RA↗
arXiv2021

The algebraic classification of nilpotent commutative algebras

This paper is devoted to the complete algebraic classification of complex $5$-dimensional nilpotent commutative algebras. Our method of classification is based on the standard method of classification of central extensions of smaller nilpotent commutative algebras and the recently obtained classification of complex $5$-dimensional nilpotent commutative $\mathfrak{CD}$-algebras.

math.RA↗
arXiv2021

The geometric classification of nilpotent commutative $\mathfrak{CD}$-algebras

We give a geometric classification of complex $5$-dimensional nilpotent commutative $\mathfrak{CD}$-algebras. The corresponding geometric variety has dimension $24$ and decomposes into $10$ irreducible components determined by the Zariski closures of a two-parameter family of algebras, three one-parameter families of algebras, and $6$ rigid algebras.

math.RA↗
arXiv2021

The geometric classification of $2$-step nilpotent algebras and applications

We give a geometric classification of complex $n$-dimensional $2$-step nilpotent (all, commutative and anticommutative) algebras. Namely, has been found the number of irreducible components and their dimensions. As a corollary, we have a geometric classification of complex $5$-dimensional nilpotent associative algebras. In particular, it has been proven that this variety has $14$ irreducible components and $9$ rigid algebras.

math.RA↗
arXiv2021

Local and $2$-local derivations of simple $n$-ary algebras

In the present paper, we prove that every local and $2$-local derivation of the complex finite-dimensional simple Filippov algebra is a derivation. As a corollary we have the description of all local and $2$-local derivations of complex finite-dimensional semisimple Filippov algebras. All local derivations of the ternary Malcev algebra $M_8$ are described. It is the first example of a finite-dimensional simple algebra that admits pure local derivations, i.e. algebra admits a local derivation which is not a derivation.

math.RA↗
arXiv2021

The algebraic and geometric classification of nilpotent right alternative algebras

We present algebraic and geometric classifications of the $4$-dimensional complex nilpotent right alternative algebras. Specifically, we find that, up to isomorphism, there are only $9$ non-isomorphic nontrivial nilpotent right alternative algebras. The corresponding geometric variety has dimension $13$ and it is determined by the Zariski closure of $4$ rigid algebras and one one-parametric family of algebras.

math.RA↗
arXiv2021

The algebraic and geometric classification of nilpotent left-symmetric algebras

This paper is devoted to the complete algebraic and geometric classification of complex $4$-dimensional nilpotent left-symmetric algebras. The corresponding geometric variety has dimension $15$ and decomposes into $3$ irreducible components determined by the Zariski closures of two one-parameter families of algebras and a two-parameter family of algebras (see Theorem B). In particular, there are no rigid $4$-dimensional complex nilpotent left symmetric algebras.

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