arXiv subjects

Yury Popov

Publications and source records attributed to Yury Popov.

18 recordsLinked to original sources

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↗

arXiv2020

Identities of sum of two PI-algebras in the case of positive characteristic

We consider the following question posted by K.I. Beidar and A.V. Mikhalev in 1995 for an associative ring $R=R_1+R_2$: is it true that if the subrings $R_1$ and $R_2$ satisfy polynomial identities, then $R$ also satisfies a polynomial identity? Over a field of positive characteristic we establish new conditions on $R_1$ and $R_2$ that guarantee a positive answer to the question. We find upper and low bounds on the degrees of identities of $R$.

math.RA↗

arXiv2020

Degenerations of Zinbiel and nilpotent Leibniz algebras

We describe degenerations of four-dimensional Zinbiel and four-dimensional nilpotent Leibniz algebras over C. In particular, we describe all irreducible components in the corresponding varieties.

math.RA↗

arXiv2020

Degenerations of Jordan Algebras and ''Marginal'' Algebras

We describe all degenerations of the variety $\mathfrak{Jord}_3$ of Jordan algebras of dimension three over $\mathbb{C}.$ In particular, we describe all irreducible components in $\mathfrak{Jord}_3.$ For every $n$ we define an $n$-dimensional rigid ''marginal'' Jordan algebra of level one. Also, we discuss ''marginal'' algebras in associative, alternative, left alternative, non-commutative Jordan, Leibniz, and anticommutative cases.

math.RA↗

arXiv2020

Jordan algebras admitting derivations with invertible values

The notion of derivation with invertible values as a derivation of ring with unity that only takes multiplicatively invertible or zero values appeared in a paper of Bergen, Herstein and Lanski, in which they determined the structure of associative rings that admit derivations with invertible values. Later, the results of this paper were generalized in many cases, for example, for generalized derivations, associative super-algebras, alternative algebras and many others. The present work is dedicated to description of all Jordan algebras admitting derivations with invertible values.

math.RA↗

arXiv2019

The algebraic and geometric classification of nilpotent terminal algebras

We give algebraic and geometric classifications of $4$-dimensional complex nilpotent terminal algebras. Specifically, we find that, up to isomorphism, there are $41$ one-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, $18$ two-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, $2$ three-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, complemented by $21$ additional isomorphism classes. The corresponding geometric variety has dimension 17 and decomposes into 3 irreducible components determined by the Zariski closures of a one-parameter family of algebras, a two-parameter family of algebras and a three-parameter family of algebras. In particular, there are no rigid $4$-dimensional complex nilpotent terminal algebras.

math.RA↗

arXiv2019

Degenerations of nilpotent associative commutative algebras

We give a complete description of degenerations of complex $5$-dimensional nilpotent associative commutative algebras.

math.RA↗

arXiv2018

The universal conservative superalgebra

We introduce the class of conservative superalgebras, in particular, the superalgebra $\mathcal{U}(V)$ of bilinear operations on a superspace $V.$ Moreover, we show that each conservative superalgebra modulo its maximal Jacobian ideal is embedded into $\mathcal{U}(V)$ for a certain superspace $V.$

math.RA↗

arXiv2018

Representations of simple noncommutative Jordan superalgebras I

In this article we begin the study of representations of simple finite-dimensional noncommutative Jordan superalgebras. In the case of degree $\geq 3$ we show that any finite-dimensional representation is completely reducible and, depending on the superalgebra, quasiassociative or Jordan. Then we study representations of superalgebras $D_t(α,β,γ)$ and $K_3(α, β, γ)$ and prove the Kronecker factorization theorem for superalgebras $D_t(α,β,γ)$. In the last section we use a new approach to study noncommutative Jordan representations of simple Jordan superalgebras.

math.RA↗

arXiv2018

Split Regular $Hom$-Leibniz Color $3$-Algebras

We introduce and describe the class of split regular $Hom$-Leibniz color $3$-algebras as the natural extension of the class of split Lie algebras, split Leibniz algebras, split Lie $3$-algebras, split Lie triple systems, split Leibniz $3$-algebras, and some other algebras. More precisely, we show that any of such split regular $Hom$-Leibniz color $3$-algebras $T$ is of the form ${T}={\mathcal U} +\sum\limits_{j}I_{j}$, with $\mathcal U$ a subspace of the $0$-root space ${T}_0$, and $I_{j}$ an ideal of $T$ satisfying {for} $j\neq k:$ \[[{ T},I_j,I_k]+[I_j,{ T},I_k]+[I_j,I_k,T]=0.\] Moreover, if $T$ is of maximal length, we characterize the simplicity of $T$ in terms of a connectivity property in its set of non-zero roots.

math.RA↗

arXiv2017

The structure of simple noncommutative Jordan superalgebras

In this paper we describe all subalgebras and automorphisms of simple noncommutative Jordan superalgebras $K_3(α,β,γ)$ and $D_t(α,β,γ)$ and compute the derivations of the nontrivial simple finite-dimensional noncommutative Jordan superalgebras.

math.RA↗

arXiv2017

Separating invariants for 2x2 matrices

A minimal separating set is found for the algebra of matrix invariants of several 2x2 matrices over an infinite field of arbitrary characteristic

math.RT↗

arXiv2016

Generalized derivations of multiplicative $n$-ary Hom-$Ω$ color algebras

We generalize the results of Leger and Luks, Zhang R. and Zhang Y.; Chen, Ma, Ni, Niu, Zhou and Fan; Kaygorodov and Popov about generalized derivations of color $n$-ary algebras to the case of $n$-ary Hom-$Ω$ color algebras. Particularly, we prove some properties of generalized derivations of multiplicative $n$-ary Hom-$Ω$ color algebras. Moreover, we prove that the quasiderivation algebra of any multiplicative $n$-ary Hom-$Ω$ color algebra can be embedded into the derivation algebra of a larger multiplicative n-ary Hom-$Ω$ color algebra.

math.RA↗

arXiv2016

A characterization of nilpotent nonassociative algebras by invertible Leibniz-derivations

Moens proved that a finite-dimensional Lie algebra over field of characteristic zero is nilpotent if and only if it has an invertible Leibniz-derivation. In this article we prove the analogous results for finite-dimensional Malcev, Jordan, (-1,1)-, quasiassociative, quasialternative, right alternative and Malcev-admissible noncommutative Jordan algebras over the field of characteristic zero. Also, we describe all Leibniz-derivations of semisimple Jordan, right alternative and Malcev algebras.

math.RA↗

arXiv2016

Degenerations of binary Lie and nilpotent Malcev algebras

We describe degenerations of four-dimensional binary Lie algebras, and five- and six-dimensional nilpotent Malcev algebras over \mathbb{C}. In particular, we describe all irreducible components of these varieties.

math.RA↗

arXiv2015

Generalized derivations of (color) n-ary algebras

We generalize the results of Leger and Luks about generalized derivations of Lie algebras to the case of color $n$-ary $Ω$-algebras. Particularly, we prove some properties of generalized derivations of color $n$-ary algebras; prove that a quasiderivation algebra of a color $n$-ary $Ω$-algebra can be embedded into the derivation algebra of a larger color $n$-ary $Ω$-algebra, and describe (anti)commutative $n$-ary algebras satisfying the condition $QDer = End.$

math.RA↗

arXiv2014

Conservative algebras of 2-dimensional algebras

We describe derivations and subalgebras of codimensions 1 of conservative algebras of small dimensions.

math.RA↗

arXiv2013

Alternative algebras admitting derivations with invertible values and invertible derivations

We descibed all alternative algebras with invertible derivations (the analogue of Bergen-Herstein-Lanski's Theorem) and proved the analogue of Moens's Theorem.

math.RA↗