LieGruppeoid

LieGruppeoid is a mathematical structure in differential geometry that generalizes the notion of a Lie groupoid. It consists of two smooth manifolds: G, the arrows, and M, the objects, together with smooth structure maps s,t: G → M (source and target), a smooth partially defined multiplication m, an identity e, and an inversion i, satisfying the usual groupoid axioms. When a Lie group acts smoothly on G and M in a way that preserves s,t, m, e, and i, the resulting triple is called a LieGruppeoid with symmetry.

Typical examples include the pair groupoid M × M over a manifold M, where arrows are all pairs (x,y) and composition is given by (x,y)∘(y,z)=(x,z); and action groupoids H × M for a Lie group H acting on M. More generally, holonomy groupoids of foliations yield LieGruppeoids. These examples illustrate how LieGruppeoids encode both global symmetries and local geometric data.

Associated Lie algebroid: Every LieGruppeoid has an associated Lie algebroid A → M, defined as the restriction of the tangent bundle along the unit section, with anchor ρ: A → TM given by dt on A and Lie bracket induced by the groupoid multiplication. Integrability questions and connections to differentiable stacks arise naturally, connecting LieGruppeoids with stacky geometry and foliation theory.

Relation to other concepts: LieGruppeoids generalize Lie groupoids; they are central to the study of differentiable stacks, gerbes, and foliations. They provide a framework for describing symmetry in geometry, differential equations, and mathematical physics.