On the annihilators of rational functions in the Lie algebra of derivations of k[x, y]

Details On the annihilators of rational functions in the Lie algebra of derivations of k[x, y] On the annihilators of rational functions in the Lie algebra of derivations of k[x, y]

2009-10-23

arxiv.org/abs/0910.4465v1

Mathematics

Rings and Algebras

Scientific paper

Let k be an algebraically closed field of zero characteristic. The Lie algebra W_2 of all k-derivations of the polynomial ring k[x, y] naturally acts on the polynomial ring k[x, y] and also on the field of rational functions k(x, y). For a fixed non-constant rational function u from k(x,y) we consider the set A_{W_2}(u) of all derivations D from W_2 such that D(u)=0. We prove that A_{W_2}(u) is a free submodule of rank 1 of the k[x,y]-module W_2. A description of the maximal abelian subalgebras as well of the centralizers of elements in the Lie algebra A_{W_2}(u) has been obtained.

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