Lie algebras of derivations with large abelian ideals

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

Algebra and discrete mathematics, Tome 28 (2019) no. 1, pp. 123-129.

Résumé

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 ${\tilde{W}}_n(\mathbb{K}):={\mathrm{Der}}_{\mathbb{K}}R$ of all $\mathbb{K}$-derivation on $R$ is a vector space (of dimension n) over $R$ and every its subalgebra $L$ has rank ${\mathrm{rk}}_{R}L={\dim }_{R}RL$. We study subalgebras $L$ of rank $m$ over $R$ of the Lie algebra ${\tilde{W}}_n(\mathbb{K})$ 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 $f_i\in F[t_1,\dots ,t_k]$, $\mathrm{deg}{f}_i\le 1$. As a consequence it is proved that $L$ is isomorphic to a subalgebra (of a very special type) of the general affine Lie algebra ${\mathrm{aff}}_m(F)$.

Détail

Mots-clés : Lie algebra, vector field, polynomial ring, abelian ideal, derivation.

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