The Isomorphism Problem for Universal Enveloping Algebras of Lie Algebras

Abstract

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.

The isomorphism problem for universal enveloping algebras of nilpotent Lie algebras

“…Indeed, if L is a finite-dimensional Lie algebra of nilpotency class 2, then it is always isomorphic to Gr(L). The same assertion without a restriction on the dimension is proved in [RU,Section 5]. …”

Fox-type problems in enveloping algebras

“…LetX be a homogeneous basis of gr(L) and let X be a set of coset representatives ofX. The following lemma is stated as Corollary 3.2 in [10].…”

“…It is proved in [8] and [10] that L ∩ ω n (L) = γ n (L), the n-th term of the lower central series of L. The identification of the subalgebras L ∩ ω n (L)ω m (S) naturally arises in the context of enveloping algebras. The motivation for this sort of problem also comes from its group ring counterpart.…”

The restricted isomorphism problem for metacyclic restricted Lie algebras

“…This sort of isomorphism problem also makes sense for ordinary Lie algebras and has been considered by David Riley and the present author in [6].…”