arXiv subjects

F. A. Mashurov

Publications and source records attributed to F. A. Mashurov.

4 recordsLinked to original sources

arXiv2026

Identities in differential perm algebras

Let $(P,\cdot,d)$ be a differential perm algebra over a field of characteristic zero, i.e. an associative algebra satisfying $(ab)c=(ba)c$ and equipped with a derivation $d$. We study polynomial identities in the algebras obtained by the derived operations \[ a\prec b=ab',\quad a\succ b=a'b,\quad a\blacklozenge b=ab'+ba',\quad a\bullet b=a'b+ab',\quad a\Diamond b=ab'-ba',\quad a\circ b=a'b-ab', \] where $a'=d(a)$. We first prove that any nontrivial differential polynomial identity that does not belong to the right annihilator of the free differential perm algebra implies a differential identity of the form $a_1'a_2'\cdots a_m'=0$ for some positive integer $m$. We then obtain explicit generating sets and determine the dimensions of the multilinear homogeneous components of the subalgebras of the free differential perm algebra generated by $X$ with respect to the products $\blacklozenge$ and $\bullet$. Finally, we construct perm-Witt type Lie and Leibniz algebras arising naturally from differential perm algebras.

math.RA↗

arXiv2026

A vanishing criterion for Lie elements in a free Novikov algebra

Every Novikov algebra is Lie-admissible: the commutator turns it into a Lie algebra. We give a finite criterion for a multilinear element of the free Novikov algebra to belong to the Lie subalgebra generated by the free generators. By the differential realization of free Novikov algebras, the multilinear component of degree $n$ is identified with the space of homogeneous polynomials of degree $n-1$ in $n$ variables. We prove that a multilinear element is a Lie element if and only if its symbol vanishes at every integer point $(a_1,\ldots,a_n)$ with $a_i\le 1$ and $a_1+\cdots+a_n\ge 2$. Equivalently, the symbol is annihilated by two explicit linear differential operators of orders two and three. The proof combines the Witt algebra, a specialization argument for a symmetric block of variables, homogeneous interpolation and Molev's description of the multilinear component of the Lie algebra generated by the free generators as a module over the symmetric group. As applications we show that a nonzero multilinear Lie element is never a total derivative, recover the dimension of the multilinear Lie component, and illustrate the criterion by examples in degrees $4$, $5$ and $6$.

math.RA↗

arXiv2026

Nonsymmetric versions of binary quadratic operads

In this paper, we study the white Manin product of the associative operad $\As$ with a binary quadratic operad $\Var$. We introduce the notion of a nonsymmetric version of $\Var$ and provide a criterion for determining when the operad $\As\circ\Var$ has this property. We illustrate the construction with several examples and counterexamples. Finally, for some operads admitting nonsymmetric versions, we describe their combinatorial properties.

math.RA↗

arXiv2022

Metabelian Lie and perm algebras

It is well known that any Lie algebra can be embedded into an associative algebra. We prove that any metabelian Lie algebra can be embedded into an algebra in the subvariety of perm algebras, i.e., associative algebras with the identity $abc-acb =0$. In addition, a technical method to construct the universal enveloping perm algebra for a metabelian Lie algebra is given.

math.RA↗