Classical Lifting Processes and Multiplicative Vector Fields
Abstract
Lie algebroid A, a linear vector field ~ is a Lie algebroid morphism A --* T A if and only if the corresponding vector field on A* is Poisson (with respect to the dual Poisson structure on A *), and this is so if and only if D~ is a derivation of the Lie algebroid bracket. In §5 and §6 we give a comparable analysis for differential I-forms on Lie algebroids and their duals. For a Poisson groupoid G ~ P, the I-forms have a bracket structure reflecting the fact that T* G --* G is a Lie algebroid. In the final §7 we show that I-forms on a Poisson groupoid admit a calculus similar to that of §3, with the differential operators on AG now replaced by operators QI(p) --* QI(p); for example, if E QI (G) is multiplicative, then D4.> is a derivation of the Poisson bracket on QI (P). In these terms we obtain in Theorem 7.6 a complete description of the bracket structure on Ql(G). It is worth noting that the treatment we give here is entirely coordinate free, and so may offer something new even in the classical case. In the early stages