Sheets and hearts of prime ideals in enveloping algebras of semisimple Lie algebras
The authors consider the enveloping algebra \(U(\mathfrak g)\) of a complex semisimple Lie algebra \(\mathfrak g\). The heart of a prime ideal \(I\) of \(U(\mathfrak g)\) is the center of the total ring of fractions of \(U(\mathfrak g)/I\). This is an extension field of the field of fractions of the center of \(U(\mathfrak g)/I\). Let \(d\) be the degree of this field extension. The an old problem of J. Dixmier asked whether \(d=1\). A recent paper of the second author [\textit{R. Rentschler}, Commun. Algebra 36, No. 3, 1153--1170 (2008; Zbl 1204.17009), see also the preprint, 2004] gave a negative answer by an example in \(\mathfrak{sl}_4\). Now in the present paper the authors give many more examples, involving the so-called sheets of primitive ideals introduced and studied by the first author and \textit{A. Joseph} [J. Algebra 244, No. 1, 76--167 (2001); corrigendum 259, No. 1, 310--311 (2003); Zbl 1078.17005)]. A sheet corresponds to a prime ideal \(I\) which has a heart of degree \(d\). The main result of this paper is that \(d\) equals the covering degree of the sheet as introduced in [W. Borho, A. Joseph, loc.cit., 8.7].
scientific article; zbMATH DE number 5078828