Proof of the Gelfand-Kirillov Conjecture for Solvable Lie Algebras

Abstract

Let $g$ be a solvable algebraic Lie algebra over the complex numbers ${\mathbf {C}}$. It is shown that the quotient field of the enveloping algebra of $g$ is isomorphic to one of the standard fields ${D_{n,k}}$, being defined as the quotient field of the Weyl algebra of degree $n$ over ${\mathbf {C}}$ extended by $k$ indeterminates. This proves the Gelfand-Kirillov conjecture for $g$ solvable.