Interpreting Lie algebroid theory in terms of D-modules, we define a duality functor for a Lie algebroid as well as a direct image functor for a morphism of Lie algebroids. Generalizing the work of Schneiders (see also the work of Schapira-Schneiders) and making assumptions analog to his, we show that the duality functor and the direct image functor commute. As an application, we extend to Lie algebroids some duality properties already known for Lie algebras.

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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 D-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].

We explore special features of the pair (U (au),U (au)) formed by the right and left dual over a (left) bialgebroid U in case the bialgebroid is, in particular, a left Hopf algebroid. It turns out that there exists a bialgebroid morphism S (au) from one dual to another that extends the construction of the antipode on the dual of a Hopf algebra, and which is an isomorphism if U is both a left and right Hopf algebroid. This structure is derived from PhA(1)ng's categorical equivalence between left and right comodules over U without the need of a (Hopf algebroid) antipode, a result which we review and extend. In the applications, we illustrate the difference between this construction and those involving antipodes and also deal with dualising modules and their quantisations

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