Foliated groupoids and infinitesimal ideal systems

The main goal of this work is to introduce a natural notion of ideal in a Lie algebroid, the “infinitesimal ideal systems”. Ideals in Lie algebras and the Bott connection associated to involutive subbundles of tangent bundles are special cases. The definition of these objects is motivated by the infinitesimal description of involutive multiplicative distributions on Lie groupoids. In the Lie group case, such distributions correspond to ideals. Several examples of infinitesimal ideal systems are presented, and (under suitable regularity conditions) the quotient of a Lie algebroid by an infinitesimal ideal system is shown to be a Lie algebroid.