AktionsLieAlgebroid

AktionsLieAlgebroid is the Lie algebroid associated to a Lie algebra action on a smooth manifold. Given a smooth manifold M and a Lie algebra g acting on M through a Lie algebra homomorphism ρ: g → Γ(TM) (the infinitesimal action), the AktionsLieAlgebroid E is the trivial vector bundle E = M × g over M equipped with a Lie algebroid structure.

The anchor map a: E → TM is defined by a(x, ξ) = ρ(ξ)_x, i.e., at each point x ∈ M the element ξ ∈ g is sent to the tangent vector on M given by the action of ξ. If Γ(E) is identified with C∞(M, g), the Lie bracket on sections is given by

[s1, s2](x) = [s1(x), s2(x)]_g + ρ(s1(x)) s2 − ρ(s2(x)) s1,

where [ , ]_g is the Lie bracket in g and ρ(si) denotes the vector field on M obtained by applying the action to the value si(x) and then taking the directional derivative of the g-valued function s2 along that vector field. This bracket makes E a Lie algebroid with anchor a.

Special cases and relationships: If the action is trivial (ρ = 0), the anchor is zero and E ≅ M × g with the pointwise bracket. If the action integrates to a Lie group action of G on M, E is the Lie algebroid of the action groupoid G × M ⇒ M. The AktionsLieAlgebroid is transitive when the action is locally transitive, and it serves as an infinitesimal model for quotient constructions and symmetry in differential geometry.

Applications: appears in the study of symmetry reductions, gauge theories, connections on fiber bundles, and the formulation of infinitesimal actions within the framework of Lie algebroids.