The Isomorphism Problem for Universal Enveloping Algebras of Lie Algebras

Let L be a Lie algebra with universal enveloping algebra U(L). We prove that if H is another Lie algebra with the property that U(L) ≅ U(H) then certain invariants of L are inherited by H. For example, we prove that if L is nilpotent then H is nilpotent with the same class as L. We also prove that if L is nilpotent of class at most two then L is isomorphic to H.