Ideals in the Enveloping Algebra of the Positive Witt Algebra

Let W+ be thepositive Witt algebra, which has a C-basis {e(n): n is an element of Z >= 1}, with Lie bracket [e(i),e(j)] = (j-i)e(i+j). We study the two-sided ideal structure of the universal enveloping algebra U(W+) ofW(+). We show that ifIis a (two-sided) ideal of U(W+) generated by quadratic expressions in thee(i), then U(W+)/Ihas finite Gelfand-Kirillov dimension, and that such ideals satisfy the ascending chain condition. We conjecture that analogous facts hold for arbitrary ideals of U(W+), and verify a version of these conjectures for radical Poisson ideals of the symmetric algebra S(W+).