On Universal Fields of Fractions for Free Algebras

Let L be an arbitrary Lie algebra. Then by Cohnʹs Theorem its universal enveloping algebra can be embedded in a skew field D(L). We study this skew field in the case when L is a free Lie algebra and prove that in this case D(L) is isomorphic to the universal field of fractions for the free associative algebra U(L). We apply this theorem to obtain new results on free fields.