Integration of Singular Subalgebroids

Abstract

We establish a Lie theory for singular subalgebroids, objects which generalize singular foliations to the setting of Lie algebroids. First we carry out the longitudinal version of the theory. For the global one, a guiding example is provided by the holonomy groupoid, which carries a natural diffeological structure in the sense of Souriau. We single out a class of diffeological groupoids satisfying specific properties and introduce a differentiation-integration process under which they correspond to singular subalgebroids. In the regular case, we compare our procedure to the usual integration by Lie groupoids. We also specify the diffeological properties which distinguish the holonomy groupoid from the graph.

Coordinates for non-integrable Lie algebroids

“…• The foliation pM, F q integrates (in the sense of [AZ20]) to the holonomy groupoid HpF q constructed in [AS09]. This becomes a diffeological groupoid using the path-holonomy bi-submersions constructed in [AS09].…”

“…The map ψ : Z Ñ W pAq we constructed in §4.2 can probably be extended to a map GpAq Ñ W pAq, which "differentiates" to id : A Ñ A. This differentiation process is explained in [AZ20].…”

“…Although the topology of the groupoids constructed in [AS09] [Zam18] is extremely pathological, both functors can be applied on them (cf. [AZ20], [Zam18,Appendix]). The reason for this is that these are diffeological groupoids, namely they have well behaved (smooth) local coordinate systems, although their dimension varies.…”

Holonomy transformations for Lie subalgebroids

“…Eq. (3) shows that under the canonical identification Ψ t H (h) , composing L α with R Φ(h) −1 yields χ(h). On the other hand, the composition R Φ(h) −1 • L α is exactly χ cong (h), by definition.…”

“…Motivation. Our motivation to consider χ comes from the integration of singular subalgebroids, which we address in [3]. A singular subalgebroid is a submodule B of the compactly supported sections of A which is locally finitely generated and involutive.…”