arXiv subjects
Publications and source records attributed to Ivan Kaygorodov.
Let $G$ be a finite group, $N(G)$ be the set of conjugacy classes of the group $G$. In the present paper it is proved $G\simeq L$ if $N(G)=N(L)$, where $G$ is a finite group with trivial center and $L$ is a finite simple group of exceptional Lie type or Tits group.
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.
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.
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.
We study the algebra of bilinear multiplications of an $n$-dimensional vector space. In particular, we study the Kantor product of some well-known (associative, Lie, alternative, Novikov and some other) multiplications.
In 1990 Kantor defined the conservative algebra $W(n)$ of all algebras (i.e. bilinear maps) on the $n$-dimensional vector space. If $n>1$, then the algebra $W(n)$ does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). We describe automorphisms, one-sided ideals, and idempotents of $W(2).$ Also similar problems are solved for the algebra $W_2$ of all commutative algebras on the 2-dimensional vector space and for the algebra $S_2$ of all commutative algebras with trace zero multiplication on the 2-dimensional vector space.
We construct a basis of free unital generalized Poisson superalgebras and a basis of free unital superalgebras of Jordan brackets. Also, we prove the analogue of Farkas' Theorem for PI unital generalized Poisson algebras and PI unital algebras of Jordan brackets. Relations beetwen generic Poisson superalgebras and superalgebras of Jordan brackets will be studied.
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 $QDer = End.$
We describe derivations and subalgebras of codimensions 1 of conservative algebras of small dimensions.
We descibed all alternative algebras with invertible derivations (the analogue of Bergen-Herstein-Lanski's Theorem) and proved the analogue of Moens's Theorem.
We described all δ-derivations of semisimple f.-d. structurable algebras over algebraically closed field with characteritic is not equal 2,3,5.
We defined (n+1)-ary derivations of n-ary algebras. We described 3-derivations of simple Malcev algebras and 4-ary derivations of simple ternary Malcev algebra M_8.
We defined (n+1)-ary derivations of $n$-ary algebras. We described (n+1)-derivations of simple and semisimple finite-dimensional Filippov algebras over algebraically closed field zero characteristic. We constructed new examples of non-trivial (n+1)-ary derivations of Filippov algebras.
We defined generalized δ-derivations of algebra A as linear mapping χassociated with usual δ-derivation ϕby the rule χ(xy)=δ(χ(x)y+xϕ(y))=δ(ϕ(x)y+xχ(y)) for any x,y \in A. We described generalized δ-derivations of prime alternative algebras, prime Lie algebras and superalgebras, unital algebras, and semisimple finite-dimensional Jordan superalgebras. In this cases we proved that generalized δ-derivation is a generalized derivation or δ-derivation. After that we described δ-superderivations of superalgebras < >, arising from prime alternative algebras, prime Lie algebras and superalgebras, unital algebras, and semisimple finite-dimensional Jordan superalgebras. In the end, we constructed new examples of non-trivial δ-derivations of Lie algebras.
We defined δ-derivations of n-ary algebras. We described δ-derivations of (n+1)-dimensional n-ary Filippov algebras and simple finite-dimensional Filippov algebras over algebraically closed field zero characteristic, and simple ternary Malcev algebra M_8. We constructed new examples of non-trivial δ-derivations of Filippov algebras and new examples of non-trivial antiderivations of simple Filippov algebras.
We described $δ$-derivations and $δ$-superderivations of simple and semisimple finite-dimensional Jordan superalgebras over algebraic closed fields with characteristic $p\neq2$. We constructed new examples of 1/2-derivations and 1/2-superderivations of simple Zelmanov's superalgebra $V_{1/2}(Z,D).$
We consider generalization of wellknown construction Kantor Double J(Γ, {,}) (KKM Double, Kantor-King-McCrimmon Double), where basic algebra Γ is nonunital algebra. We find necessary and sufficient conditions for a generalized Kantor double to be Jordan. We also describe $δ$-superderivations of a generalized Kantor double whose even part is prime.