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

2009/10/23 by Iena, O. G., Petravchuk, A. P., Regeta, A. O.
#13N15 #16W25 #17B65 #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.0910.4465

Abstract

Let k be an algebraically closed field of zero characteristic. The Lie algebra W₂ 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_W2(u) of all derivations D from W₂ such that D(u)=0. We prove that A_W2(u) is a free submodule of rank 1 of the k[x,y]-module W₂. A description of the maximal abelian subalgebras as well of the centralizers of elements in the Lie algebra A_W2(u) has been obtained.

Related

• The group of automorphisms of the Lie algebra of derivations of a field of rational functions
• Centralizers of linear and locally nilpotent derivations
• Centralizers of Jacobian derivations
• The group of automorphisms of the Lie algebra of derivations of a polynomial algebra
• On noncommutative bases of the free module W_n(\mathbb K)