arXiv subjects

O. O. Pypka

Publications and source records attributed to O. O. Pypka.

7 recordsLinked to original sources

arXiv2026

Dedekind Poisson Algebras over Arbitrary Fields

We study Poisson algebras in which every Poisson subalgebra is a Poisson ideal, called Dedekind Poisson algebras, over arbitrary fields and without any finite-dimensionality assumption. In characteristic different from 2, combining the two operations by $x*y=xy+[x,y]$ connects the problem with Outcalt's classical classification of power-associative H-algebras and yields the same structural type. We obtain a complete classification by a direct two-operation argument valid in every characteristic. This approach also shows that characteristic 2 retains additional Poisson information which cannot be recovered from the single product $*$.

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arXiv2026

On the Structure of Low-Dimensional Poisson Algebras over Arbitrary Fields

We investigate the structure of Poisson algebras of dimensions at most three over an arbitrary field. Our approach is based on the internal structure of the associated commutative associative and Lie algebras, with particular emphasis on the associative square $P^2$, the derived Lie algebra $[P,P]$, the Lie center, the associative annihilator, idempotents, ideals and decomposability. We obtain a complete classification in dimensions one and two and give a structural classification in dimension three. In dimension two, we prove that the associative and Lie multiplications cannot be simultaneously non-zero. In dimension three, the classification is organized according to the dimension and position of the derived Lie algebra and, in the case of trivial Lie multiplication, according to the dimension of $P^2$. The arbitrary-field setting leads to phenomena which do not occur over the complex field. In particular, quadratic and cubic field extensions appear naturally in the classification, some families depend on equivalence classes of symmetric bilinear forms and on the structure of three-dimensional Lie algebras over the ground field, and characteristic $2$ gives an additional family of Poisson algebras with both multiplications non-zero. Over the complex field, the resulting classification specializes, up to changes of basis and notation, to the known classifications in dimensions at most three.

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arXiv2025

Leibniz rings: some basic and structural results

In this paper, we study the fundamental properties of Leibniz rings. Special attention is given to the structure of Leibniz rings whose additive group is "small". The results obtained illustrate a significant difference between the classes of Leibniz rings and Lie rings.

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arXiv2024

On some relationships between the centers and the derived ideal in Leibniz 3-algebras

One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group $G/ζ(G)$ of a group $G$ is finite, then its derived subgroup $[G,G]$ is also finite. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures. In 2016, L.A. Kurdachenko, J. Otal and O.O. Pypka proved an analogue of Schur theorem for Leibniz algebras: if central factor-algebra $L/ζ(L)$ of Leibniz algebra $L$ has finite dimension, then its derived ideal $[L,L]$ is also finite-dimensional. Moreover, they also proved a slightly modified analogue of Schur theorem: if the codimensions of the left $ζ^{l}(L)$ and right $ζ^{r}(L)$ centers of Leibniz algebra $L$ are finite, then its derived ideal $[L,L]$ is also finite-dimensional. One of the generalizations of Leibniz algebras is the so-called Leibniz $n$-algebras. Therefore, the question of proving analogs of the above results for this type of algebras naturally arises. In this article, we prove the analogues of the two mentioned theorems for Leibniz 3-algebras.

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arXiv2024

On some relationships between the center and the derived subalgebra in Poisson (2-3)-algebras

One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group $G/ζ(G)$ of a group $G$ is finite, then its derived subgroup $[G,G]$ is also finite. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, $n$-groups, associative algebras, Lie algebras, Lie $n$-algebras, Lie rings, Leibniz algebras. In 2021, L.A. Kurdachenko, O.O. Pypka and I.Ya. Subbotin proved an analogue of Schur theorem for Poisson algebras: if the center of the Poisson algebra $P$ has finite codimension, then $P$ includes an ideal $K$ of finite dimension such that $P/K$ is abelian. In this paper, we continue similar studies for another algebraic structure. An analogue of Schur theorem for Poisson (2-3)-algebras is proved.

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arXiv2023

On the automorphism groups of some nilpotent 3-dimensional Leibniz algebras

Let $L$ be an algebra over a field $F$ with the binary operations $+$ and $[,]$. Then $L$ is called a left Leibniz algebra if it satisfies the left Leibniz identity: $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$ for all elements $a,b,c\in L$. The structure of the automorphism group of $3$-dimensional Leibniz algebras, which have nilpotency class $2$ and a one-dimensional center, is studied.

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arXiv2023

Automorphism groups of some 3-dimensional Leibniz algebras

Let $L$ be an algebra over a field $F$ with the binary operations $+$ and $[,]$. Then $L$ is called a left Leibniz algebra if it satisfies the left Leibniz identity: $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$ for all elements $a,b,c\in L$. A linear transformation $f$ of $L$ is called an endomorphism of $L$, if $f([a,b])=[f(a),f(b)]$ for all elements $a,b\in L$. A bijective endomorphism of $L$ is called an automorphism of $L$. It is easy to show that the set of all automorphisms of the Leibniz algebra is a group with respect to the operation of multiplication of automorphisms. The description of the structure of the automorphism groups of Leibniz algebras is one of the natural and important problems of the general Leibniz algebra theory. The main goal of this article is to describe the structure of the automorphism group of a certain type of nilpotent three-dimensional Leibniz algebras.

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