# Petravchuk A. P.

### Lie algebras associated with modules over polynomial rings

Petravchuk A. P., Sysak K. Ya.

↓ Abstract

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.

### Finite-dimensional subalgebras in polynomial Lie algebras of rank one

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

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.

### Closed polynomials and saturated subalgebras of polynomial algebras

Arzhantsev I. V., Petravchuk A. P.

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 $A$.

### On Strongly Inert Subalgebras of an Infinite-Dimensional Lie Algebra

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 dim*I*/*A* <. We show that the condition of local finiteness of *L* is essential in this statement.

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

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

We consider a Lie algebra*L* over an arbitrary field that is decomposable into the sum*L=A+B* of an almost Abelian subalgebra*A* and a subalgebra*B* 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.

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

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

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

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

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

Chernikov N. S., Petravchuk A. P.

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