Krull Dimension of the Enveloping Algebra of s / ( 2 , C )
نویسندگان
چکیده
Let U denote the enveloping algebra of the simple Lie algebra ~42, C). In this paper it is shown that the Krull dimension of U (denoted ] U]) is two. If U(g) is the enveloping algebra of a finite-dimensional solvable Lie algebra g then it is straightforward to show that ) U(g)] = dim g [5, 3.8.111. The problem as to the Krull dimension of U was first mentioned by Gabriel and Nouazt [9] they show that CJ has a chain of prime ideals of length two, and none of length greater than two. From this they conclude that the Krull dimension of U is two, although the correct conclusion is only that ] Uj > 2. Subsequent to [9], both Arnal and Pinczon [l] and Roos [lo] established that if R were a non-artinian simple primitive factor ring of U then ] R I= 1. More recently the author [ 111 proved that if R were a nonartinian primitive factor ring of U which was not simple then again IR ) = 1. The result in the present paper implies those in [ 1, 10, 111. The fundamental tool in the proof that (Ul = 2 is Gelfand-Kirillov dimension (GK-dimension). The proof is in two parts. In Section 2 a number of preliminary results (already known) concerning GK-dimension are recalled. In particular, Lemma 2.3 provides the basic connection between GK-dimension and Krull dimension. The more detailed analysis of U is carried out in Section 3. The crucial result is that any finitely generated U module of Krull dimension 1 has GK-dimension 2 the result then quickly follows from Lemma 2.3. The author would like to thank J. C. McConnell both for bringing this problem to his attention, and for many helpful conversations.
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تاریخ انتشار 2012