The Primitive Factor Rings of the Enveloping Algebra of sl (2, C )

Let R denote a non-artinian primitive factor ring of the enveloping algebra of s/(2, C). Arnal-Pinczon [1] and Roos [10] have shown that if R is simple then it has Krull dimension 1. Roos also shows that "most" of the simple R have global dimension 1. In this paper we prove that if R is not simple then it has Krull dimension 1 (thus all non-artinian primitive factor rings of s/(2, C) have Krull dimension 1) and does not have global dimension 1. Notation and the basic properties of these factor rings are described in §2. In particular, if R denotes such a non-simple primitive factor ring, then R has a unique proper two-sided ideal M of finite codimension, and R embeds in the Weyl algebra A{. In §3 we prove that R has Krull dimension 1. The proof illustrates and depends on the close relationship between R and Al. In §4 the relationship between certain /^-modules and certain A x -modules is examined more closely. The results in §4 are used in §5 to describe the generators of M as a left ideal. We also show in §5 that the grading on Ax (defined by the semi-simple element) induces a grading on R, and that both R and M are graded by the induced grading. Finally, in §6 it is proved that R is not hereditary. In particular, it is shown that R/M has projective dimension 2. (The primitive ideal of the enveloping algebra corresponding to R is the annihilator of a Verma module of length two and both composition factors of this Verma module have projective dimension at most two as i?-modules.) The precise global dimension of R remains an open question. I would like to thank J. C. McConnell and J. C. Robson for bringing these problems to my attention. I am indebted to them both for their constant interest, and for their generous encouragement and advice.