X-inner automorphisms of enveloping rings

Abstract

In this paper, we determine the X-inner automorphisms of the smash product R # U( L) of a prime ring R by the universal enveloping algebra U( L) of a characteristic 0 Lie algebra L. Specifically, we show that any such automorphism σ stabilizing R can be written as a product σ = σ 1 σ 2, where σ 1 is induced by conjugation by a unit of Q 3 ( R), the symmetric Martindale ring of quotients of R, and σ 2 is induced by conjugation by a unit of Q 3 ( T). Here S = Q l ( R) is the left Martindale ring of quotients of R and T is the centralizer of S in S # U( L) ⊃- R # U( L). One of the subtleties of the proof is that we must work in several unrelated overrings of R # U( L).