RESEARCH ARTICLE

Lie algebras of derivations with large abelian ideals

I. S. Klymenko, S. V. Lysenko, A. P. Petravchuk

Taras Shevchenko National University of Kyiv, 64, Volodymyrska street, 01033 Kyiv, Ukraine

Abstract: Let $\mathbb K$ be a field of characteristic zero, $A=\mathbb{K}[x_{1},\dots,x_{n}]$ the polynomial ring and $R=\mathbb{K}(x_{1},\dots,x_{n})$ the field of rational functions. The Lie algebra ${\widetilde W}_{n}(\mathbb{K}):=\operatorname{Der}_{\mathbb{K}}R$ of all $\mathbb{K}$-derivation on $R$ is a vector space (of dimension n) over $R$ and every its subalgebra $L$ has rank $\operatorname{rk}_{R}L=\dim_{R}RL$. We study subalgebras $L$ of rank $m$ over $R$ of the Lie algebra $\widetilde{W}_{n}(\mathbb{K})$ with an abelian ideal $I\subset L$ of the same rank $m$ over $R$. Let $F$ be the field of constants of $L$ in $R$. It is proved that there exist a basis $D_1,\dots,D_m$ of $FI$ over $F$, elements $a_1,\dots,a_k\in R$ such that $D_i(a_j)=\delta_{ij}$, $i=1,\dots,m$, $j=1,\dots,k$, and every element $D\in FL$ is of the form $D=\sum_{i=1}^{m}f_i(a_1,\dots,a_k)D_i$ for some $f_i\in F[t_1,\dots,t_k]$, $\deg f_i\leq 1$. As a consequence it is proved that $L$ is isomorphic to a subalgebra (of a very special type) of the general affine Lie algebra $\mathrm{aff}_{m}(F)$.

Keywords: Lie algebra, vector field, polynomial ring, abelian ideal, derivation.

MSC: Primary 17B66; Secondary 17B05, 13N15

Language: English