Gel'fand-Kirillov Conjecture on the Lie Field of an Algebraic Lie Algebra

Abstract

This is an article written to review with sufficient detail the so‐called Gel'fand‐Kirillov conjecture concerning the isomorphisms between the quotient fields of the algebras generated by canonical variables [p_i, q_j] = δ_ij1 and the quotient fields of the universal enveloping algebras of algebraic Lie algebras. This conjecture sheds new light on the relation between the universal enveloping algebra of an algebraic Lie algebra and the Lie algebras of dynamic groups in quantum mechanics.