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Seminar Algebra & Geometry: Andriy Regeta
Andriy Regeta , TU 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 Wn=Wn(K) of all K-derivations
of the polynomial ring K[x1,...,xn]. 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).