Formal Lie groupoids
For a space, a group and a action of on , we have the corresponding action groupoid. If everything is sufficiently smooth, this is a Lie groupoid denoted .
The action Lie algebroid of is the Lie algebroid that corresponds to this Lie groupoid (under Lie integration).
The Chevalley-Eilenberg algebra of an action Lie algebroid is in physics known as a BRST complex.
Let be a Lie group, a smooth manifold and a smooth action. Write for the corresponding action groupoid, itself a Lie groupoid. The Lie algebroid corresponding to this is the action Lie algebroid.
The Chevalley-Eilenberg algebra of the action Lie algebroid is
where the differential acts on functions by
Explicitly, for this sends to the function which is the derivative along of the function .
Even more explicitly, if we choose local coordinates on a patch, and choose a basis of then we have that restricted to this patch the differential is on generators by
Specifically for a finite dimensional vector space, a linear action, a choice of basis of that vector space and a linear function , we have that are the components vector of the dual vector given by in this basis, and the above gives the matrix multiplication form of the action
Notice for completeness that the equation is equivalent to the Jacobi identity of the Lie bracket and the action property of :
These local formulas shall be useful below for recognizing from our general abstract definition of covariant derivative the formulas traditionally given in the literature. For that notice that in the above local coordinates further restricting attention to linear actions, the Weil algebra of the action Lie algebroid is given by
where the differential is given on generators by
and where the uniquely induced differential on the shifted generators – the one encoding Bianchi identities – is
Notice that we may identify the delooping Lie groupoid of with the action groupoid of the trivial action on the point, . On Lie algebroids this morphism is dually the inclusion
that is the identity on .