Proof of the Gelfand-Kirillov conjecture for solvable Lie algebras

Abstract

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