LieAlgebroidMorphismus

LieAlgebroidMorphismus, in the context of differential geometry, denotes a morphism between Lie algebroids. A Lie algebroid is a vector bundle A → M endowed with a Lie bracket [ , ] on its space of sections Γ(A) and an anchor map ρ: A → TM satisfying the Leibniz rule [α, fβ] = f[α, β] + (ρ(α)f) β for all α, β ∈ Γ(A) and all smooth functions f on M.

A Lie algebroid morphism from a Lie algebroid (A, [ , ]_A, ρ_A) over M to another Lie algebroid (B, [ , ]_B, ρ_B) over N consists of a smooth map f: M → N and a vector bundle morphism Φ: A → B covering f, such that two compatibility conditions hold: Φ preserves the Lie bracket, Φ([α, β]_A) = [Φ(α), Φ(β)]_B for all α, β ∈ Γ(A); and Φ intertwines the anchors, ρ_B ∘ Φ = Tf ∘ ρ_A, where Tf is the differential of f. These requirements ensure the morphism respects both the bracket structure and the anchor.

Variants include morphisms over the same base (f = id_M) where Φ: A → B is a bundle map with preserved bracket and anchor, and morphisms over different bases, which are central in functorial treatments and in the study of representations and cohomology.

Examples often include the tangent bundle TM with its standard Lie algebroid structure: for a smooth map f: M → N, the differential df: TM → TN is a Lie algebroid morphism covering f. Correspondence with Lie algebra morphisms emerges when the bases are single points. LieAlgebroidMorphismus thus provides a framework for functorial maps between geometric structures carrying Lie algebroid data.