arXiv subjects
Publications and source records attributed to Ivan Kaygorodov.
We describe all central extensions of all $3$-dimensional non-zero complex Zinbiel algebras. As a corollary, we have a full classification of $4$-dimensional non-trivial complex Zinbiel algebras and a full classification of $5$-dimensional non-trivial complex Zinbiel algebras with $2$-dimensional annihilator, which gives the principal step in the algebraic classification of $5$-dimensional Zinbiel algebras.
We give a geometric classification of $n$-dimensional nilpotent, commutative nilpotent and anticommutative nilpotent algebras. We prove that the corresponding geometric varieties are irreducible, find their dimensions and describe explicit generic families of algebras which define each of these varieties. We show some applications of these results in the study of the length of anticommutative algebras.
We describe all rigid algebras and all irreducible components in the variety of four dimensional Leibniz algebras $\mathfrak{Leib}_4$ over $\mathbb{C}.$ In particular, we prove that the Grunewald--O'Halloran conjecture is not valid and the Vergne conjecture is valid for $\mathfrak{Leib}_4.$
We give the classification of $5$- and $6$-dimensional complex one-generated nilpotent assosymmetric algebras.
We give the complete algebraic classification of all complex 4-dimensional nilpotent algebras. The final list has 234 (parametric families of) isomorphism classes of algebras, 66 of which are new in the literature.
We give a full description of the Poisson structures on the finitary incidence algebra $FI(P,R)$ of an arbitrary poset $P$ over a commutative unital ring $R$.
In this paper we describe central extensions (up to isomorphism) of all complex null-filiform and filiform Zinbiel algebras. It is proven that every non-split central extension of an $n$-dimensional null-filiform Zinbiel algebra is isomorphic to an $(n+1)$-dimensional null-filiform Zinbiel algebra. Moreover, we obtain all pairwise non isomorphic quasi-filiform Zinbiel algebras.
We give an algebraic classification of complex $5$-dimensional one-generated nilpotent terminal algebras.
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 $R$.
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.
We give a geometric classification of complex $4$-dimensional nilpotent $\mathfrak{CD}$-algebras. The corresponding geometric variety has dimension 18 and decomposes into $2$ irreducible components determined by the Zariski closures of a two-parameter family of algebras and a four-parameter family of algebras (see Theorem 2). In particular, there are no rigid $4$-dimensional complex nilpotent $\mathfrak{CD}$-algebras.
We give an algebraic classification of complex $4$-dimensional nilpotent $\mathfrak{CD}$-algebras.
Suppose $\mathfrak{R}$ is a $2$,$3$-torsion free unital alternative ring having an idempotent element $e_1$ $\left(e_2 = 1-e_1\right)$ which satisfies $x \mathfrak{R} \cdot e_i = \{0\} \rightarrow x = 0$ $\left(i = 1,2\right)$. In this paper, we aim to characterize the commuting maps. Let $φ$ be a commuting map of $\mathfrak{R}$ so it is shown that $φ(x) = zx + Ξ(x)$ for all $x \in \mathfrak{R}$, where $z \in \mathcal{Z}(\mathfrak{R})$ and $Ξ$ is an additive map from $\mathfrak{R}$ into $\mathcal{Z}(\mathfrak{R})$. As a consequence a characterization of anti-commuting maps is obtained and we provide as an application, a characterization of commuting maps on von Neumann algebras relative alternative $C^{*}$-algebra with no central summands of type $I_1$.
We give a classification of 5- and 6-dimensional complex one-generated nilpotent Novikov 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.
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.
Let $\Re$ and $\Re'$ unital $2$,$3$-torsion free alternative rings and $φ: \Re \rightarrow \Re'$ be a surjective Lie multiplicative map that preserves idempotents. Assume that $\Re$ has a nontrivial idempotents. Under certain assumptions on $\Re$, we prove that $φ$ is of the form $ψ+ τ$, where $ψ$ is either an isomorphism or the negative of an anti-isomorphism of $\Re$ onto $\Re'$ and $τ$ is an additive mapping of $\Re$ into the centre of $\Re'$ which maps commutators into zero.
We give the algebraic classification of alternative, left alternative, Jordan, bicommutative, left commutative, assosymmetric, Novikov and left symmetric central extensions of null-filiform associative algebras.