On Lie Algebras Generated by Two Differential Operators

Abstract

We shall denote by K a field of characteristic 0, by K[x] the ring of polynomials in r variables x₁,...,x_r with coefficients in K, and by D. the K-derivation in K[x] defined by D_ix_j. = δ_ij.. for 1 ≦ i, j ≦ r; then the multiplications by x₁,...,x_r in K[x] and D₁.,...,D_r generate a subalgebra A of the associative K-algebra of all K-linear transformations in K[x]. An element X of A can be written uniquely in the form

$$x = \sum {{a_{{i_l} \cdot \cdot \cdot {i_r}{j_l} \cdot \cdot \cdot {j_r}}}x_l^{{i_l}} \cdot \cdot \cdot x_r^{{i_r}}} D_l^{{j_l}} \cdot \cdot \cdot D_r^{{j_r}}$$

with a ${a_{{i_l} \cdot \cdot \cdot {i_r}{j_l} \cdot \cdot \cdot {j_r}}}$ in K; it is a linear differential operator with polynomial coefficients.