2019
Том 71
№ 8

All Issues

Petravchuk A. P.

Articles: 9
Article (Ukrainian)

Lie algebras associated with modules over polynomial rings

Petravchuk A. P., Sysak K. Ya.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1232-1241

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero. Let $V$ be a module over the polynomial ring $K[x, y]$. The actions of $x$ and $ y$ determine linear operators P and Q on V as a vector space over $\mathbb{K}$. Define the Lie algebra $L_V = K\langle P,Q\rangle \rightthreetimes V$ as the semidirect product of two abelian Lie algebras with the natural action of $\mathbb{K}\langle P,Q\rangle$ on $V$. We show that if $\mathbb{K}[x, y]$-modules $V$ and $W$ are isomorphic or weakly isomorphic, then the corresponding associated Lie algebras $L_V$ and $L_W$ are isomorphic. The converse is not true: we construct two $\mathbb{K}[x, y]$-modules $V$ and $W$ of dimension 4 that are not weakly isomorphic but their associated Lie algebras are isomorphic. We characterize such pairs of $\mathbb{K}[x, y]$-modules of arbitrary dimension over K. We prove that indecomposable modules $V$ and $W$ with $\mathrm{d}\mathrm{i}\mathrm{m}\mathbb{K} V = \mathrm{d}\mathrm{i}\mathrm{m}KW \geq 7$ are weakly isomorphic if and only if their associated Lie algebras $L_V$ and $L_W$ are isomorphic.

Brief Communications (English)

Finite-dimensional subalgebras in polynomial Lie algebras of rank one

Arzhantsev I. V., Makedonskii E. A., Petravchuk A. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 708-712

Let $W_n(\mathbb{K})$ be the Lie algebra of derivations of the polynomial algebra $\mathbb{K}[X] := \mathbb{K}[x_1,... ,x_n]$ over an algebraically closed field $K$ of characteristic zero. A subalgebra $L \subseteq W_n(\mathbb{K})$ is called polynomial if it is a submodule of the $\mathbb{K}[X]$-module $W_n(\mathbb{K})$. We prove that the centralizer of every nonzero element in $L$ is abelian provided that $L$ is of rank one. This fact allows to classify finite-dimensional subalgebras in polynomial Lie algebras of rank one.

Article (English)

Closed polynomials and saturated subalgebras of polynomial algebras

Arzhantsev I. V., Petravchuk A. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1587–1593

The behavior of closed polynomials, i.e., polynomials $f ∈ k[x_1,…,x_n]∖k$ such that the subalgebra $k[f]$ is integrally closed in $k[x_1,…,x_n]$, is studied under extensions of the ground field. Using some properties of closed polynomials, we prove that, after shifting by constants, every polynomial $f ∈ k[x_1,…,x_n]∖k$ can be factorized into a product of irreducible polynomials of the same degree. We consider some types of saturated subalgebras $A ⊂ k[x_1,…,x_n]$, i.e., subalgebras such that, for any $f ∈ A∖k$, a generative polynomial of $f$ is contained in $A$.

Brief Communications (Russian)

On Strongly Inert Subalgebras of an Infinite-Dimensional Lie Algebra

Petravchuk A. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2002. - 54, № 7. - pp. 1025-1028

We study infinite-dimensional Lie algebras L over an arbitrary field that contain a subalgebra A such that dim(A + [A, L])/A < ∞. We prove that if an algebra L is locally finite, then the subalgebra A is contained in a certain ideal I of the Lie algebra L such that dimI/A <. We show that the condition of local finiteness of L is essential in this statement.

Article (Russian)

On the sum of an almost abelian Lie algebra and a Lie algebra finite-dimensional over its center

Petravchuk A. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 636–644

We consider a Lie algebraL over an arbitrary field that is decomposable into the sumL=A+B of an almost Abelian subalgebraA and a subalgebraB finite-dimensional over its center. We prove that this algebra is almost solvable, i.e., it contains a solvable ideal of finite codimension. In particular, the sum of the Abelian and almost Abelian Lie algebras is an almost solvable Lie algebra.

Article (Russian)

On the sum of two Lie algebras with finite-dimensional commutants

Petravchuk A. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1089–1096

We prove that an infinite-dimensional Lie algebra over an arbitrary field which is decomposable into the sum of two of its subalgebras with finite-dimensional commutants is almost solvable.

Article (Ukrainian)

The solubility of a Lie algebra which decomposes into a direct sum of an abelian and a nilpotent subalgebra

Petravchuk A. P.

Full text (.pdf)

Ukr. Mat. Zh. - 1991. - 43, № 7-8. - pp. 986–991

Article (Ukrainian)

Lie algebras, decomposable into a sum of an Abelian and a nilpotent subalgebra

Petravchuk A. P.

Full text (.pdf)

Ukr. Mat. Zh. - 1988. - 40, № 3. - pp. 385–388

Article (Ukrainian)

A characterization of periodic locally solvable groups whose Sylow subgroups are solvable or have a finite exponent

Chernikov N. S., Petravchuk A. P.

Full text (.pdf)

Ukr. Mat. Zh. - 1987. - 39, № 6. - pp. 761–767