The semicenter of an enveloping algebra is factorial

Abstract

Let L be a finite-dimensional Lie algebra over a field k of characteristic zero, and U(L) its universal enveloping algebra. We show that the scmicenter of U(L) is a UFD. More generally, the same result holds when k is replaced by any factorial ring R of characteristic zero. Introduction. Throughout this note, L will be a nonzero finite-dimensional Lie algebra over a field k of characteristic zero. Let U(L) be the universal enveloping algebra of L with center Z(U(L)) and D(L) will be the division ring of quotients of U(L) with center Z(D(L)). For each X g L*, we denote by D(L)X the set of those u g D(L) such that xu-ux = X(x)u for all x g L. Its elements are called the semi-invariants of D(L) relative to X. Clearly, D(L)