The rigid dualizing complex of a universal enveloping algebra

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)⊗ ∧n g)[n] where n=dim g. We prove this conjecture, and give a few applications in representation theory and Hochschild cohomology.