Rigid dualizing complex for quantum enveloping algebras and algebras of generalized differential operators

In the first part of this article, we compute the rigid dualizing complex of a quantum enveloping algebra. We consider the generic case and the case of a specialization at a non-root of unity. This answers a question of Yekutieli [J. Pure Appl. Algebra 150 (2000) 85]. In [Bull. Soc. Math. France 122 (1994) 371] and [Math. Z. 232 (1999) 367], we generalized -module theory to Lie algebroids. Using these results, we compute explicitly the rigid dualizing complex of the algebra of differential operators defined by an affine Lie algebroid. This generalizes results of Yekutieli [J. Pure Appl. Algebra 150 (2000) 85].