arXiv subjects

Y. Chapovskyi

Publications and source records attributed to Y. Chapovskyi.

5 recordsLinked to original sources

arXiv2026

Maximal subalgebras of the Lie algebra $W_n(\mathbb{K})$

Let $K$ be an algebraically closed field of characteristic zero, $A= K[x_1, \dots, x_n]$ the polynomial ring in $n$ variables, and let $W_n(K)$ be the Lie algebra of all $K$-derivations of $A.$ This Lie algebra also is the free $A$-module of rank $n$ over the ring $A,$ so every subalgebra of $W_n(K)$ has a rank $\leq n$ over $A.$ We prove that every maximal subalgebra of rank $\leq n$ of $W_n(K)$ is a simple Lie algebra. If a maximal subalgebra $L\subset W_n(K)$ has rank $n$ and is a submodule of $W_n(K)$ then $L$ is not simple. Moreover, $L$ is of the form $L=\{ D\in W_n(K) \ | \ D(I)\subseteq I\} $ for some ideal $I$ of the ring $A.$ It is also proved that, for a simple derivation $D$ on the ring $K[x, y]$, the subalgebra $K[x, y]D$ is a maximal subalgebra of $W_2(K).$

math.RA↗

arXiv2026

A family of maximal subalgebras of the Lie algebra~$W_n(K)$

Let $K$ be an algebraically closed field of characteristic zero and ${P_n=K[x_1,\ldots,x_n]}$ the polynomial ring. Any $K$-derivation $D$ on $P_n$ is of the form ${ D=\sum_{i=1}^n f_i(x_1,\ldots,x_n)\frac{\partial}{\partial x_i} },$ where $f_i\in P_n.$ All such derivations form the Lie algebra $W_n(K)$ over the field $K$. We prove that for $s=1,\ldots,n-1$ the subalgebra $ m_s(K)=\left\{ \sum_{i=1}^s f_i\frac{\partial}{\partial x_i} +\sum_{j=s+1}^n g_j\frac{\partial}{\partial x_j} \mid f_i\in P_s,\ g_j\in P_n \right\} $ is a maximal subalgebra of~$W_n(K)$. The ideal $ I_s=\left\{\sum_{j=s+1}^n g_j\frac{\partial}{\partial x_j}\right\} $ of $m_s(K)$ is isomorphic to the Lie algebra $P_s\otimes \mathrm{Der}(K[x_{s+1},\ldots,x_n])$ and $m_s(K)/I_s\simeq W_s(K)$. The Lie algebra $W_n(K)$ is also the free module over the ring $P_n.$ Therefore, for any set $S\subseteq W_n(K)$ the rank $rk(S)$ (over $P_n$) is defined. Some properties of maximal subalgebras of rank $n$ in $W_n(K)$ are pointed out.

math.RA↗

arXiv2025

Module structure of the Lie algebra $W_n(K)$ over $sl_n(K)$

Let $\mathbb K$ be an algebraically closed field of characteristic zero, $A = \mathbb K[x_1,\dots,x_n]$ the polynomial ring, and let $W_n(\mathbb K)$ denote the Lie algebra of all $\mathbb K$-derivations on $A$. The Lie algebra $W_n := W_n(\mathbb K)$ admits a natural grading $W_n = \bigoplus_{i \ge -1} W^{[i]}_n$, where $W^{[i]}_n$ consists of all homogeneous derivations whose coefficients are homogeneous polynomials of degree $i+1$ or zero. The component $W^{[0]}_n$ is a subalgebra of $W_n$ and is isomorphic to $\mathfrak{gl}_n(\mathbb K).$ Moreover, each $W_n^{[i]}$ for $i \ge -1$ is a finite-dimensional module over $W_n^{[0]}$. We prove that $W^{[i]}_n,\; i \ge 0$ is a sum of two irreducible submodules $W^{[i]}_n = M_i \oplus N_i$, where $M_i$ consists of all divergence-free derivations, and $N_i$ consists of derivations that are polynomial multiples of the Euler derivation $E_n = \sum_{i=1}^n x_i \frac{\partial}{\partial x_i}$. As a consequence, we show that the standard grading is exact in certain sense, namely: $[W^{[i]}_n, W^{[j]}_n] = W^{[i+j]}_n$ for all $i,j,$ except when $i = j = 0$. We also address the question of when the subalgebra of $W_n$ generated by $W_n^{[-1]} \oplus W_n^{[0]},$ together with an additional element from $W_n,$ equals the entire Lie algebra $W_n$.

math.RA↗

arXiv2024

Decomposition of matrices from $SL_ 2(K[x, y])$

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $\mathbb{K}[x,y]$ the polynomial ring. The group $\text{SL}_{2}\left(\mathbb{K}[x,y]\right)$ of all matrices with determinant equal to $1$ over $\mathbb{K}[x,y]$ can not be generated by elementary matrices. The known counterexample was pointed out by P.M. Cohn. Conversely, A.A.Suslin proved that the group $\text{SL}_{r}\left(\mathbb{K}[x_{1},\dots,x_{n}]\right)$ is generated by elementary matrices for $r\ge 3$ and arbitrary $n\geq 2$, the same is true for $n=1$ and arbitrary $r.$ It is proven that any matrix from $\text{SL}_{2}\left(\mathbb{K}[x,y]\right)$ with at least one entry of degree $\le 2$ is either a product of elementary matrices or a product of elementary matrices and of a matrix similar to the one pointed out by P. Cohn. For any matrix $\begin{pmatrix}\begin{array}{cc} f & g\\ -Q & P \end{array}\end{pmatrix}\in\text{SL}_{2}\left(\mathbb{K}[x,y]\right)$, we obtain formulas for the homogeneous components $P_i , Q_i$ for the unimodular row $(-Q, P) $ as combinations of homogeneous components of the polynomials $f, g, $ respectively, with the same coefficients.

math.GR↗

arXiv2023

Centralizers of linear and locally nilpotent derivations

Let $K$ be an algebraically closed field of characteristic zero, $A = K[x_1,\dots,x_n]$ the polynomial ring, $R = K(x_1,\dots,x_n)$ the field of rational functions, and let $W_n(K) = \Der_{K}A$ be the Lie algebra of all $K$-derivations on $A$. If $D \in W_n(K),$ $D\not =0$ is linear (i.e. of the form $D = \sum_{i,j=1}^n a_{ij}x_j \frac{\partial}{\partial x_i}$) we give a description of the centralizer of $D$ in $W_n(K)$ and point out an algorithm for finding generators of $C_{W_n(K)}(D)$ as a module over the ring of constants in case when $D$ is the basic Weitzenboeck derivation. In more general case when the ring $A$ is a finitely generated domain over $K$ and $D$ is a locally nilpotent derivation on $A,$ we prove that the centralizer $C_{{\rm Der}A}(D)$ is a "large" \ subalgebra in ${\rm Der}_{K} A$, namely $\rk_A C_{\Der A}(D) := \dim_R RC_{\Der A}(D)$ equals ${\rm tr}.°_{K}R,$ where $R$ is the field of fraction of the ring $A.

math.AC↗