### abstract

- Let k be a field and A a noetherian (noncommutative) k-algebra. The rigid dualizing complex of A was introduced by Van den Bergh. When A = U(g), the enveloping algebra of a finite dimensional Lie algebra g, Van den Bergh conjectured that the rigid dualizing complex is (U(g)circle times /\(n) g)[n], where n=dim g. We prove this conjecture, and give a few applications in representation theory and Hochschild cohomology. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: Primary 16D90; secondary 16E40; 16E30; 17B55.