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Askar Dzhumadil'daev

Publications and source records attributed to Askar Dzhumadil'daev.

13 recordsLinked to original sources
arXiv2026

The symmetric operation in a free Novikov algebra

We study the symmetrization of the Novikov product. Using the embedding of a free Novikov algebra into a commutative differential algebra over a field of characteristic zero, we first obtain a basis for the subalgebra generated by the free generators with respect to the symmetrized product $a\circ b=ab+ba$. Next, using Euler operators (variational derivatives), we give a criterion for recognizing symmetric elements and show that these elements coincide with null Lagrangians in the corresponding differential realization. We then construct a non-special homomorphic image and show that the class of algebras embeddable into symmetrizations of Novikov algebras does not form a variety. Finally, we determine the module structure of the multilinear components over the symmetric group.

math.RA↗
arXiv2025

On the generalized Poisson and transposed Poisson algebras

We provide the polynomial identities of algebras that are both generalized Poisson algebras and transposed Poisson algebras. We establish defining identities via single operation for generalized Poisson algebras and prove that Ito's theorem holds for generalized Poisson algebras.

math.RA↗
arXiv2022

Symmetry groups of pfaffians of symmetric matrices

We prove that symmetry group of the pfaffian polynomial of a symmetric matrix is a dihedral group. We calculate pfaffians of symmetric matrices with components $(x_i-x_j)^2$ and $\cos(x_i-x_j)$ for $i<j.$

math.CO↗
arXiv2019

On the speciality of Tortkara algebras

A criterion for elements of free Zinbiel algebras to be Lie or Jordan is established. This criterion is used in studying speciality problems of Tortkara algebras. We construct a base of free special Tortkara algebras. Furthermore, we prove analogues of classical Cohn's and Shirshov's theorems for Tortkara algebras.

math.RA↗
arXiv2018

Commutative 2-cocycles on Lie algebras

On Lie algebras, we study commutative 2-cocycles, i.e., symmetric bilinear forms satisfying the usual cocycle equation. We note their relationship with antiderivations and compute them for some classes of Lie algebras, including finite-dimensional semisimple, current and Kac-Moody algebras.

math.RA↗
arXiv2015

Path decompositions of digraphs and their applications to Weyl algebra

We consider decompositions of digraphs into edge-disjoint paths and describe their connection with the $n$-th Weyl algebra of differential operators. This approach gives a graph-theoretic combinatorial view of the normal ordering problem and helps to study skew-symmetric polynomials on certain subspaces of Weyl algebra. For instance, path decompositions can be used to study minimal polynomial identities on Weyl algebra, similar as Eulerian tours applicable for Amitsur--Levitzki theorem. We introduce the $G$-Stirling functions which enumerate decompositions by sources (and sinks) of paths.

math.CO↗
arXiv2014

Asssociative algebras under multi-commutator

For an associative algebra $A$ a skew-symmetric sum of $n!$ products of $n$ elements of $A$ in all possible order is called $n$-commutator. We consider $A$ as $n$-ary algebra under $n$-commutator. We prove that it has an identity of $ω$-degree $2$ (namely, homotopical $n$-Lie identity) if $n$ is even and an identity of $ω$-degree $3$ if $n$ is odd.

math.RA↗
arXiv2013

Stirling permutations on multisets

A permutation $σ$ of a multiset is called Stirling permutation if $σ(s)\ge σ(i)$ as soon as $σ(i)=σ(j)$ and $i<s<j.$ In our paper we study Stirling polynomials that arise in the generating function for descent statistics on Stirling permutations of any multiset. We develop generalizations of the classical Stirling numbers and present their combinatorial interpretations. Particularly, we apply the theory of $P$-partitions. Using certain specifications we also introduce the Stirling numbers of odd type and generalizations of the central factorial numbers.

math.CO↗
arXiv1998

Minimal polynomial identities for right-symmetric algebras

An algebra $A$ with multiplication $A\times A \to A, (a,b)\mapsto a\circ b$, is called right-symmetric, if $a\circ(b\circ c)-(a\circ b)\circ a\circ (c\circ b)-(a\circ c)\circ b,$ for any $a,b,c\in A$. The multiplication of right-symmetric Witt algebras $W_n=\{u\der_i: u\in U, U={\cal K}[x_1^{\pm 1},...,x_n^{\pm}$ or $={\cal K}[x_1,...,x_n], i=1,...,n\}, p=0,$ or $W_n({\bf m)}=\{u\der_i: u\in U, U=O_n({\bf m})\}$, are given by $u\der_i\circ v\der_j=v\der_j(u)\der_i.$ An analogue of the Amitsur-Levitzki theorem for right-symmetric Witt algebras is established. Right-symmetric Witt algebras of $ satisfy the standard right-symmetric identity of degree $2n+1:$ $\sum_{σ\in Sym_{2n}}sign(σ)a_{σ(1)}\circ(a_{σ(2)}\circ >...(a_{σ(2n)}\circ a_{2n+1})...)=0.$ The minimal deg$ left polynomial identities of $W_n^{rsym}, W_n^{+rsym}, p=0,$ i$ The minimal degree of multilinear left polynomial identity of $$ is also $2n+1.$ All left polynomial (also multilinear, if $p>0$) identities of right-symmetric Witt algebras of minimal $ combinations of left polynomials obtained from standard ones by permutations of arguments.

math.RT↗
arXiv1998

Cohomologies and deformations of right-symmetric algebras

An algebra $A$ with identity $(a\circ b)\circ c-a\circ(b\circ c)=(a\circ c)\circ b-a\circ(c\circ b),$ is called right-symmetric. Cohomology and deformation theory for right-symmetric algebras are developed. Cohomologies of $gl_n$ and half-Witt algebras $W_n^{rsym}, p=0,$ $W_n^{rsym}({\bf m}), p>0,$ are calculated. In particular, one right-symmetric central extension of $W_1^{rsym}$ is constructed.

math.DG↗