Prolongations of convenient Lie algebroids

Abstract

We first define the concept of Lie algebroid in the convenient setting. In reference to the finite dimensional context, we adapt the notion of prolongation of a Lie algebroid over a fibred manifold to a convenient Lie algebroid over a fibred manifold. Then we show that this construction is stable under projective and direct limits under adequate assumptions.

An integrability criterion for a projective limit of Banach distributions

“…is the fiber over x ∈ M , according to [3], the prolongation TA of the anchored Banach bundle (A, π, M, ρ…”

“…By induction, for i ≥ 1, again according to notations of Theorem 43 and Corollary 44 in [3] , we we set…”

“…Theorem 3.1 in § 3.1). In § 3.2, we give an application of Theorem 2.2 to a sequence of prolongations of a Banach-Lie algebroid (see [3]) and we end this paragraph by the announced contre-exemple of Theorem 2.2 and Theorem 2.3. The proof of the basic Theorem 2.2 is located in §4.…”