The rigid dualizing complex of a universal enveloping algebra

Let k be a field and Aanoetherian (noncommutative) k-algebra. The rigid dualizing complex of A was introduced by Van den Bergh. When , the enveloping algebra of a finite dimensional Lie algebra , Van den Bergh conjectured that the rigid dualizing complex is , where . We prove this conjecture, and give a few applications in representation theory and Hochschildcohomology.