Accordingly, the notion of representation of a Lie--algebroid is a horizontal and vertical categorification of the ordinary notion of representation of a Lie algebra, which in turn is the linearization of the notion of representation of a Lie group.
While essentially equivalent, it is noteworthy that the first definition naturally takes place in the context of not-necessarily smooth (-)categories, while the second one usually remains within the context of smooth ()-groupoids:
namely for a Lie group, for definiteness and for simplicity, with corresponding one-object Lie groupoid – the delooping of the group –, a linear representation in terms of an action morphisms is a functor
from to the category of vector spaces. In fact, there is a canonical equivalence of the functor category with the category of linear representations of
Here is the action groupoid of the action of on the representation vector space , where is the single object of . This vector space, regarded as a discrete category on its underlying set, is the fiber of this fibration, so that the action gives rise to the fiber sequence
As described at generalized universal bundle, this may be thought of as (the groupoid incarnation of) the vector bundle which is associated via to the universal -bundle , which itself is the action groupoid of the fundamental representation? of on itself,
From this perspective a representation of a group is nothing but a -equivariant vector bundle over the point, or equivalently a vector bundle on the orbifold . So from this perspective the notion “representation” is not a primitive notion, but just a particular perspective on fibration sequences.
The definition of Lie- algebroid representation below is in this fibration sequence/fibration-theoretic/action groupoid spirit. The expected alternative definition in terms of action morphisms has been considered (and is well known) apparently only for special cases.
Recall that we take, by definition, Lie ∞-algebroids to be dual to non-negatively-graded, graded-commutative differential algebras, which are free as graded-commutative algebras (qDGCAs): we write for the qDGCA whose underlying graded-commutative algebra is the free (over the algebra ) graded commutative algebra for a non-postively graded cochain complex of -modules and its degree-wise dual over , to remind us that this is to be thought of as the Chevalley-Eilenberg algebra of the Lie ∞-algebroid whose space of objects is characterized dually by the algebra .
A representation of a Lie -algebroid on a co-chain complex of -modules is a cofibration sequence
in DGCAs, i.e. a homotopy pushout
What has been considered in the literature so far is the more restrictive version, where the pushout is taken to be strict (Urs: at least I think that this is the right way to say it):
A proper representation is a strict cofiber sequence of morphisms of DGCAs
i.e. such that
is the obvious surjection;
is the obvious injection;
the composite of both is the 0-map.
It follows that the differential on is given by a twisting map as
which may be thought of as the dual of the representation morphism (see the examples below).
Given two objects and in , the cochain complex
consist in degree of morphisms of degree
and the differential is the usual differential on hom-complexes .
For a fixed Lie -algebroid , the category
with Lie representations of as objects and chain comoplexes as above as hom-objects is a dg-category.
For a smooth complex manifold and the holomorphic tangent Lie algebroid of (so that the holomorphic deRham complex of ), and for taken to have as objects complexes of finitely generated and projective -modules (i.e. complexes of smooth vector bundles) the homotopy category of the dg-category is equivalent to the bounded derived category of complexes of sheaves with coherent cohomology on (see coherent sheaf).
This is Block, theorem 2.22.
The objects of are literally complexes of smooth vector bundles that are equipped with “half a flat connection”, namely with a flat covariant derivative only along holomorphic tangent vectors. It is an old result that holomorphic vector bundles are equivalent to such smooth vector bundles with “half a flat connection”. This is what the theorem is based on.
For the tangent Lie algebroid of a smooth manifold , it should be true, up to technicalities to be spelled out here eventually, that is equivelent to the derived category of D-modules on , or the like.
ordinary Lie algebra representation on a vector space: is essentially the Chevalley-Eilenberg complex that computes the cohomology of with coefficients in .
flat connections on bundles
The definition of representation of -algebras is discussed in section 5 of
The general definition of representation of -Lie algebroids as above appears in
The definition of the dg-category of representation of a tangent Lie algebroid and its equivalence in special cases to derived categories of complexes of coherent sheaves is in
For the case of Lie 1-algebroids essentially the same definition appears also in