LieAlgebroide

LieAlgebroide, more commonly called a Lie algebroid, is a geometric object that generalizes both Lie algebras and tangent bundles. Formally, it consists of a vector bundle A → M over a smooth manifold M, a Lie bracket [ , ] on the space Γ(A) of smooth sections, and a bundle map ρ: A → TM known as the anchor. These data satisfy the Leibniz rule [α, fβ] = f[α,β] + (ρ(α)f)β for all α,β ∈ Γ(A) and f ∈ C∞(M). The anchor links the algebroid to the geometry of M.

Examples include: (i) the tangent bundle TM with the usual Lie bracket of vector fields and the identity anchor, recovering de Rham geometry as a special case; (ii) action algebroids for a Lie algebra g acting on M, given by A = M × g with anchor ρ(x,ξ) = ξM(x) (the vector field on M generated by ξ); (iii) the cotangent bundle T* M of a Poisson manifold (M, π), endowed with anchor π♯: T*M → TM and a Lie bracket on forms (the Koszul bracket), turning T*M into a Lie algebroid.

Cohomology and differential structures are part of the framework. There is a Lie algebroid exterior derivative d_A on Γ(∧•A*), yielding Lie algebroid cohomology H•(A), which generalizes both de Rham cohomology (A = TM) and Lie algebra cohomology (M a point).

Integration and applications are central themes. Lie algebroids can be integrated to Lie groupoids under suitable conditions; not every algebroid is integrable. The Crainic–Fernandes theory provides obstructions. Lie algebroids play a key role in Poisson geometry, foliation theory, and deformation problems, offering a unifying language for infinitesimal symmetries. In some texts and languages, the term LieAlgebroide is used to denote the same concept.