2 D ec 1 99 7 DOUBLE LIE ALGEBROIDS AND SECOND - ORDER GEOMETRY , II ∗ K . C . H . Mackenzie School of Mathematics and Statistics University of Sheffield Sheffield , S 3 7 RH England

We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We extend the construction of the tangent pro-longation of an abstract Lie algebroid to show that the Lie algebroid structure of any LA-groupoid may be prolonged to the Lie algebroid of its groupoid structure. In the case of a double groupoid, this prolonged structure for either LA-groupoid is canonically isomorphic to the Lie algebroid structure associated with the other; this extends many canonical isomorphisms associated with iterated tangent and cotangent structures. We calculate several examples from Poisson geometry. We show that the cotangent of any double Lie groupoid is a symplectic double groupoid and that the side groupoids of a symplectic double groupoid are Poisson groupoids in duality; thus the duals of the LA-groupoids of any double groupoid are a pair of Poisson groupoids in duality.