Seminar Algebra and Geometry: Andriy Regeta (Technische Universität Kaiserslautern)

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

(joint work with A. Petravchuk and O. Iena)

Let us consider the Lie algebra W_n=W_n(K) of all K-derivations of the polynomial ring K[x₁,...,x_n]. We discuss here a class
of subalgebras in W₂(K) over an algebraically closed field of characteristic zero.

Let us consider natural action of the Lie algebra W₂(K) on the field of rational functions K(x,y). Recall that every derivation D
of W₂(K) of the ring K[x,y] can be uniquely extended to a derivation of the field K(x,y). It is natural to consider for a
fixed rational function u in K(x,y)\K the set A_{W₂}(u) of all derivations D of W₂ such that D(u)=0. This set is called the annihilator
of u in W₂(K). It is a Lie subalgebra of W₂(K) and at the same time a K[x,y]-submodule of the K[x,y]-module W₂(K).

We show that it is a free K[x,y]-module of rank 1 and describe centralizers of elements and the maximal abelian subalgebras of the
Lie algebra A_{W₂}(u).

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