algebroid

An algebroid is an anchored vector bundle A → M equipped with a bilinear bracket [ , ] on its space of sections Γ(A) and an anchor map a: A → TM to the tangent bundle of M. The defining compatibility is a Leibniz-type rule: for all X, Y in Γ(A) and f in C∞(M), [X, fY] = f[X, Y] + (a(X)f) Y. In many treatments, an algebroid is allowed to be a Leibniz or Loday algebroid, in which the bracket need not be skew-symmetric or satisfy Jacobi; the same Leibniz rule and anchor compatibility are required. When the bracket is skew-symmetric, satisfies Jacobi, and the anchor preserves brackets a([X, Y]) = [a(X), a(Y)], the structure is called a Lie algebroid.

Lie algebroids generalize Lie algebras and the tangent bundle. The tangent bundle TM with a = identity and the usual Lie bracket of vector fields is a canonical example. Other examples include the action algebroid associated to a Lie algebra acting on M, and the Atiyah algebroid of a principal G-bundle, which encodes infinitesimal gauge symmetries.

Poisson geometry provides a notable construction: on a Poisson manifold (M, π), the cotangent bundle T*M carries a Lie algebroid structure with anchor π♯: T*M → TM defined by α ↦ π♯(α) and a bracket on 1-forms given by a specific formula involving Lie derivatives and the differential of the Poisson tensor.

Generalizations such as Courant algebroids arise when the bracket is not skew-symmetric and when additional structures (like a bilinear form) are included, playing a central role in generalized geometry. Lie algebroids serve as infinitesimal models for Lie groupoids and appear in foliation theory, Poisson geometry, and deformation theory.