Formal Lie groupoids
Let by a differentiable manifold. The tangent Lie algebroid of is the Lie algebroid that corresponds – in the sense of Lie theory – to
When the tangent Lie algebroid is regarded as a Lie ∞-algebroid it corresponds to
The space of objects of is itself and its elements in degree 1 are the vectors on , i.e. the elements in the tangent bundle of . These are to be thouhght of as the infinitesimal paths in .
So to some extent the tangent Lie algebroid is the tangent bundle of . More precisely, when using the definition of a Lie algebroid over as a diagram
where is the map called the anchor map, the tangent Lie algebroid is that whose anchor map is the identity map
Therefore the tangent Lie algebroid of is usually denoted , just as the tangent bundle itself.
The Chevalley-Eilenberg algebra of is correspondingly fundamental: it is the deRham dg-algebra of differential forms on :
So regarded as an NQ-supermanifold the tangent Lie algebroid is the shifted tangent bundle equipped with its canonical odd vector field.
Relation to flat -Lie algebroid valued differential forms
For any other Lie algebroid or Lie infinity-algebroid, a morphism of Lie infinity-algebroids
is flat -valued differential form datum .
For instance if is an ordinary Lie algebra then a morphism is a flat Lie algebra valued differential form on , i.e. an element such that .
If is the abelian 1-dimensional Lie algebra, then a morphism is just a closed differential 1-form on .
More on -Lie algebroid-valued differential forms is (eventually) at ∞-Lie algebroid valued differential forms.
The higher-order version of tangent Lie algebroids are jet bundle D-schemes.
One of the earliest reference seems to be