An inverse image functor for Lie algebroids

✍ Scribed by Sophie Chemla

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 311 KB
• Volume: 269
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00284-9

✦ Synopsis

Extending material from the theory of D-modules to Lie algebroids (see also [S. Chemla, A duality property for complex Lie algebroids, Math. Z. (1999) 367-388]), we introduce an inverse image functor and show that it preserves coherence under appropriate circumstances. As in the case of D-modules, a suitable non characteristicity notion enables us to give a sufficient condition in order for the duality functor and the inverse image functor to commute. This generalizes a result for D-modules due to Kashiwara, Kawai, and Sato but, even in the case of D-modules, our proof is different from theirs. In particular, we obtain a new duality formula for complexes of modules over an ordinary Lie algebra and, as a special case, we get a new adjunction formula for modules over Lie algebras. Moreover, our result will shed some light on the behaviour of Poisson cohomology (a notion introduced by Lichnerowitz) under a Poisson map (in the analytic case).