A category of locally convex Lie algebroids

Abstract

We study first-order locally convex Lie algebroids in the setting of Bastiani calculus. The first-order condition is automatic in finite dimensions, but is an additional regularity hypothesis for general locally convex vector bundles. Under this condition, we define sheaves of scalar-valued and vector-valued Lie algebroid forms as fiberwise continuous alternating maps with smooth local representatives. We define morphisms by requiring the induced pullback on inverse-image sheaves of scalar-valued forms to commute with the Lie algebroid differentials, and prove that first-order locally convex Lie algebroids form a category. We also study representations and the induced cohomology sheaves. We show that locally convex Lie groupoids have first-order Lie algebroids and that Lie groupoid morphisms induce morphisms in this category. As applications, we prove that the current algebroid associated with a first-order Banach Lie algebroid is a first-order Fréchet Lie algebroid, and we prove a Lie II theorem in the Banach setting: first-order morphisms between the Lie algebroids of Banach-Lie groupoids under source-connected and source-simply connected hypotheses integrate to unique Lie groupoid morphisms.