arXiv subjects

A. S. Panasenko

Publications and source records attributed to A. S. Panasenko.

10 recordsLinked to original sources

arXiv2026

Right representations of Novikov algebras

We consider the concept of a right representation for Novikov algebras. We introduce the concept of a right modular ideal of a Novikov algebra and obtain a description of irreducible right representations of a Novikov algebra as quotients by maximal modular right ideals. We prove that the class of Novikov algebras without irreducible representations is a hereditary radical. We introduce the concept of a primitive Novikov algebra and prove that a Novikov algebra is primitive if and only if it has an almost faithful irreducible representation. We introduce the concept of the quasi-kernel of a representation as the largest ideal contained in the kernel of the representation. We prove that the Jacobson radical of a Novikov algebra is the intersection of the quasi-kernels of all irreducible representations of this algebra.

math.RT↗

arXiv2026

Radicals of Lie-solvable Novikov algebras

We prove that in a Lie-solvable Novikov algebra, the Baer radical coincides with the set of all right-nilpotent elements, and the Andrunakievich radical coincides with the largest left-quasiregular ideal. We investigate the stability of some properties of commutative algebras with derivation after applying the Gelfand-Dorfman construction.

math.RA↗

arXiv2025

Central orders in simple right-alternative superalgebras and right-symmetric algebras

We consider some recently constructed examples of simple finite-dimensional right-alternative superalgebras and right-symmetric algebras. We prove that the central order in any of these algebras and superalgebras is embedded in a finite module over its center (or over the even part of its center in the case of superalgebras)

math.RA↗

arXiv2025

Rota-Baxter operators of nonzero weight on the split Cayley-Dickson algebra

We describe Rota-Baxter operators on split octonions. It turns out that up to some transformations there exists exactly one such non-splitting operator over any field. We also obtain a description of all decompositions of split octonions over a quadratically closed field of characteristic different from 2 into a sum of two subalgebras, which describes the splitting Rota-Baxter operators. It completes the classification of Rota-Baxter operators on composition algebras of any weight.

math.RA↗

arXiv2024

Rota-Baxter operators of weight zero on Cayley-Dickson algebra

All Rota-Baxter operators of weight zero on split octonion algebra over a~field of characteristic not 2 are classified up to conjugation by automorphisms and antiautomorphisms. Thus, the classification of Rota-Baxter operators on composition algebras is finished. There are two descriptions: a~common description over arbitratry field of characteristic not 2 and more accurate description over a~quadratically closed field of characteristic not 2.

math.RA↗

arXiv2023

On radicals of Novikov algebras

We show that in a prime nonassociative Novikov algebra every nonzero ideal is non-associative. We prove that Baer (and Andrunakievich) radical and left quasiregular radical coincide in finite dimensional Novikov algebras over a field of characteristic 0 or algebraically closed field of odd characteristic. We show non-existence of right quasiregular radical in finite dimensional Novikov algebras.

math.RA↗

arXiv2022

Semiprime Novikov Algebras

We study prime and semiprime Novikov algebras. We prove that prime nonassociative Novikov algebra has zero nucleus and center. It is well known that an ideal of an alternative (semi)prime algebra is (semi)prime algebra. We proved this statement for Novikov algebras. It implies that a Baer radical exists in a class of Novikov algebras. Also, we proved that a minimal ideal of Novikov algebra is either trivial, or a simple algebra.

math.RA↗

arXiv2021

Quadratic Lie conformal superalgebras related to Novikov superalgebras

We study quadratic Lie conformal superalgebras associated with No\-vikov superalgebras. For every Novikov superalgebra $(V,\circ)$, we construct an enveloping differential Poisson superalgebra $U(V)$ with a derivation $d$ such that $u\circ v = ud(v)$ and $\{u,v\} = u\circ v - (-1)^{|u||v|} v\circ u$ for $u,v\in V$. The latter means that the commutator Gelfand--Dorfman superalgebra of $V$ is special. Next, we prove that every quadratic Lie conformal superalgebra constructed on a finite-dimensional special Gel'fand--Dorfman superalgebra has a finite faithful conformal representation. This statement is a step toward a solution of the following open problem: whether a finite Lie conformal (super)algebra has a finite faithful conformal representation.

math.QA↗

arXiv2020

Defining relations and Gröbner--Shirshov bases of Poisson algebras as of conformal modules

We study the relation between Poisson algebras and representations of Lie conformal algebras. We establish a setting for the calculation of a Gröbner--Shirshov basis in a module over an associative conformal algebra and apply this technique to Poisson algebras considered as conformal modules over appropriate associative envelopes of current Lie conformal algebras. As a result, we obtain a setting for the calculation of a Gröbner--Shirshov basis in a Poisson algebra.

math.RA↗

arXiv2019

Central orders in simple finite dimensional superalgebras

The well-known Formanek's module finiteness theorem states that every unital prime PI-algebra (i.e. a central order in a matrix algebra by Posner's theorem) embeds into a finitely generated module over its center. An analogue of this theorem for alternative and Jordan algebras was earlier proved by V.N.Zhelyabin and the author. In this paper we discuss this problem for associative, classical Jordan and some alternative superalgebras.

math.RA↗