On Lie Algebras Having a Primitive Universal Enveloping Algebra

Abstract

In his book “Structure of Rings” [7, p. 231 Professor Jacobson raised the following open question: “What are the conditions on a finite dimensional Lie algebra L over a field K that insure that its universal enveloping algebra U(L) is primitive ?” [Since U(L) h as an anti-automorphism the notions left and right primitive are the same for U(L).] If R is of characteristic p f 0, then U(L) cannot be primitive unless L = 0 [7, p. 2551. Th ere f ore we may assume from now on that L is a nonzero finite dimensional Lie algebra over a field Fz of characteristic zero. For each linear functional f EL* we denote by L[f] the set of all x EL such that f (Ex) = 0 for all E in the algebraic hull of ad L C End L. Clearly L[f] is a Lie subalgebra of L containing the center Z(L) of L. The aim of this paper is to prove the following. THEOREM. If U(L) is primitive then L[f] = 0 foT some f EL*. Moreover, the converse holds ifL is solvable and k is algebraically closed. If we denote by D(L) the division ring of quotients of U(L), Z(D(L)) its center, we shall prove that the condition that Llf] = 0 for some f EL* is equivalent with Z(D(L)) = k (which f orces the centers of both L and U(L) to be trivial). In particular, U(L) cannot be primitive if L is either nilpotent or semi-simple. Finally, we shall give some examples of Lie algebras (of which one is not solvable) that do have a primitive universal enveloping algebra.