LieAlgebroid

A Lie algebroid is a geometric object that generalizes both Lie algebras and the tangent bundle of a manifold. It consists of a vector bundle A over a smooth manifold M together with a Lie bracket [ , ] on its space of sections Γ(A) and a vector bundle map called the anchor ρ: A → TM.

The bracket [ , ] on Γ(A) is bilinear, antisymmetric, and satisfies the Jacobi identity. The anchor connects A to the geometry of M by requiring the Leibniz rule: [s1, f s2] = f [s1, s2] + (ρ(s1)f) s2 for all sections s1, s2 and smooth functions f on M. The anchor is also required to be compatible with brackets in the sense that ρ([s1, s2]) = [ρ(s1), ρ(s2)], where the right-hand side is the usual Lie bracket of vector fields on M.

Examples illustrate the concept. The tangent bundle TM with the standard Lie bracket of vector fields and the identity anchor is a Lie algebroid. A Lie algebra g can be viewed as a Lie algebroid over a point with zero anchor. More generally, if a Lie algebra g acts on M, the action algebroid A = M × g has anchor ρ(x, ξ) = ξ_M(x), the generated vector field at x.

Lie algebroids naturally appear in questions of integrability and symmetry. They form the infinitesimal counterpart to Lie groupoids. Associated to a Lie algebroid is its cohomology, the Lie algebroid cohomology, defined from the exterior algebra of A*. Morphisms of Lie algebroids preserve brackets and anchors, providing a category-theoretic view of the structure.