Abstract
Let k be an algebraically closed field of characteristic 0 and let A be a finitely generated k-algebra that is a domain whose Gelfand-Kirillov dimension is in [2, 3). We show that if A has a nonzero locally nilpotent derivation then A has quadratic growth. In addition to this, we show that A either satisfies a polynomial identity or A is isomorphic to a subalgebra of D(X), the ring of differential operators on an irreducible smooth affine curve X, and A is birationally isomorphic to D(X).
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