On weakly semisimple derivations of the polynomial ring in two variables

Abstract

Let K be an algebraically closed field of characteristic zero and K[x, y] the polynomial ring. Every element f ∈ K[x, y] determines the Jacobian derivation Df of K[x, y] by the rule Df(h) = detJ(f, h), where J(f, h) is the Jacobian matrix of the polynomials f and h. A polynomial f is called weakly semisimple if there exists a polynomial g such that Df(g) = λg for some nonzero λ ∈ K. Ten years ago, Y. Stein posed a problem of describing all weakly semisimple polynomials (such a description would characterize all two dimensional nonabelian subalgebras of the Lie algebra of all derivations of K[x, y] with zero divergence). We give such a description for polynomials f with the separated variables, i.e. which are of the form: f(x, y) = f₁(x)f₂(y) for some f₁(t), f₂(t) ∈ K[t].