This entry discusses ∞-actions/∞-representations of Lie-infinity algebroids, hence the infinitesimal version of ∞-actions of smooth ∞-groupoids.
Recall that an $L_\infty$-algebroid is both a
horizontal categorification as well as a vertical categorification of a Lie algebra: it is to Lie algebras as Lie ∞-groupoids are to Lie groups.
Accordingly, the notion of representation of a Lie-$\infty$-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.
In view of this notice that there are essentially two fundamental ways to express the notion of representation of a group or ∞-groupoid $Gr$:
as a morphism out of $Gr$: the action;
as a fibration sequence over $Gr$: the action groupoid.
While essentially equivalent, it is noteworthy that the first definition naturally takes place in the context of not-necessarily smooth ($\infty$-)categories, while the second one usually remains within the context of smooth ($\infty$)-groupoids:
namely for $G$ a Lie group, for definiteness and for simplicity, with corresponding one-object Lie groupoid $\mathbf{B} G$ – the delooping of the group $G$ –, a linear representation in terms of an action morphisms is a functor
from $\mathbf{B} G$ to the category of vector spaces. In fact, there is a canonical equivalence of the functor category $[\mathbf{B}G, Vect]$ with the category $Rep(G)$ of linear representations of $G$
Every such functor $\rho$ induces a fibration sequence $V//G \to \mathbf{B}G$ over $\mathbf{B}G$, obtained as the pullback of the generalized universal bundle $Vect_* \to Vect$ along $\rho$
Here $V//G$ is the action groupoid of the action of $\rho$ on the representation vector space $V := \rho(\bullet)$, where $\bullet$ is the single object of $\mathbf{B}G$. 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 $\rho$ to the universal $G$-bundle $\mathbf{E}G \to \mathbf{B}G$, which itself is the action groupoid of the fundamental representation of $G$ on itself,
From this perspective a representation of a group $G$ is nothing but a $G$-equivariant vector bundle over the point, or equivalently a vector bundle on the orbifold $\bullet//G$. So from this perspective the notion “representation” is not a primitive notion, but just a particular perspective on fibration sequences.
The definition of Lie-$\infty$ 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 $CE_A(g)$ for the qDGCA whose underlying graded-commutative algebra is the free (over the algebra $A$) graded commutative algebra $\wedge^\bullet g^*$ for $g$ a non-postively graded cochain complex of $A$-modules and $g^*$ its degree-wise dual over $A$, to remind us that this is to be thought of as the Chevalley-Eilenberg algebra of the Lie ∞-algebroid $g$ whose space of objects is characterized dually by the algebra $A$.
A representation $\rho$ of a Lie $\infty$-algebroid $(g, A)$ on a co-chain complex $V$ of $A$-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 $\rho$ is a strict cofiber sequence of morphisms of DGCAs
i.e. such that
$CE_\rho(g,V) = CE_A(g) \otimes \wedge^\bullet V$ as GCAs
$\wedge^\bullet V \leftarrow CE_\rho(g,V)$ is the obvious surjection;
$CE_\rho(g,V) \leftarrow CE_A(g)$ is the obvious injection;
the composite of both is the 0-map.
It follows that the differential $d_\rho$ on $CE_\rho(g,V)$ is given by a twisting map $\rho^* : V \to (\wedge^\bullet V) \wedge (g^*) \wedge (\wedge^\bullet g^*)$ as
$d_\rho|_{g^*} = d_g$
$d_\rho|_{V} = d_V + \rho^*$
which may be thought of as the dual of the representation morphism (see the examples below).
In (Block 05) the dg-category $Rep(g,A)$ of proper representations of a Lie-$\infty$-algebroid $(g,A)$ in the above sense – called dg-algebra modules there – is defined.
Given two objects $CE_\rho(g,V)$ and $CE_{\rho'}(g,V')$ in $Rep(g,A)$, the cochain complex
$Hom( CE_\rho(g,V), CE_{\rho'}(g,V') )$
consist in degree $k$ of morphisms of degree $k$
satisfying $\phi(v t) = (-1)^{k |a|} \phi(v) t$
and the differential $d_{Hom}$ is the usual differential on hom-complexes $d \phi = d_{\rho'} \circ \phi - (-1)^{|\phi|} \phi \circ d_\rho$.
For a fixed Lie $\infty$-algebroid $(g,A)$, the category
with Lie representations of $(g,A)$ as objects and chain comoplexes as above as hom-objects is a dg-category.
For an n ordinary Lie algebra representation $\rho$ on a vector space $V$ consider the Chevalley-Eilenberg algebra $CE_\rho(g,V)$ that computes the Lie algebra cohomology of $\mathfrak{g}$ with coefficients in $V$. This exhibits the action in the above sense.
A flat connections on a vector bundle exhibits a representation of the tangent Lie algebroid of the base manifold.
A holomorphic variant of this is below.
The following variant of example 2 is a homotopy-theoretic-refinement of the classical Koszul-Malgrange theorem.
For $X$ a smooth complex manifold and $(g,A) = T_{hol} X$ the holomorphic tangent Lie algebroid of $X$ (so that $CE_A(g) = \Omega^\bullet_{hol}(X) = \Omega^{\bullet,0}(X)$ the holomorphic part of the Dolbeault complex of $X$), and for $Rep(T_{hol} X)$ taken to have as objects complexes of finitely generated and projective $C^\infty(X)$-modules (i.e. complexes of smooth vector bundles) the homotopy category $Ho Rep(T_{hol} X)$ of the dg-category $Rep(T_{hol} X)$ is equivalent to the bounded derived category of chain complexes of abelian sheaves with coherent cohomology on $X$ (see at coherent sheaf).
This is (Block 05, theorem 2.22 (in the counting of version 1 on the arXiv!)).
The objects of $Rep(T_{hol} X)$ 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 (see there) are equivalent to such smooth vector bundles with “half a flat connection”. This is what the theorem is based on.
For $\mathfrak{g}$ any L-∞ algebra, and $\mathfrak{a}$ any other,then an L-∞ extension (see there) $\hat {\mathfrak{g}}$ of $\mathfrak{g}$ by $\mathfrak{a}$ is a homotopy fiber sequence
of L-∞ algebras (see at model structure for L-∞ algebras). Regarding this as sequence of L-∞ algebroids over the point
and then passing to Chevalley-Eilenberg algebras, this exhibits an action/representation of the $L_\infty$-algebra $\mathfrak{g}$ on the $L_\infty$-algebroid $\mathbf{B}\mathfrak{a}$.
For instance the string Lie 2-algebra is the $\mathbf{B} \mathbb{R}$-extension of a semisimple Lie algebra $\mathfrak{g}$ with bilinear invariant polynomial $\langle -,-\rangle$ corresponding to the 3-cocycle $\langle -,[-,-]\rangle \in CE(\mathfrak{g})$, hence exhibits an action/representation of $\mathfrak{g}$ on $\mathbf{B}\mathbb{R}$. This is the infinitesimal version of the ∞-action of a simply connected compact simple Lie group $G$ on the circle 2-group $\mathbf{B}U(1)$ which exhibits the String 2-group extension.
Analogous statements in various degrees hold for the $L_\infty$-algebra Fivebrane 6-group
exhibiting an $\infty$-action of the string Lie 2-algebra on $\mathbf{B}^7 \mathbb{R}$, and analogously for the supergravity Lie 3-algebra, the supergravity Lie 6-algebra and for all the other extensions in The brane bouquet.
The definition of representation of $L_\infty$-algebras is discussed in section 5 of
The general definition of representation of $\infty$-Lie algebroids (of finite type) as above appears as def. 4.9 in
(this discussion is not in the published version arXiv:0910.4001v2, for size reasons)
modeled after the geneal abstract definition of ∞-actions 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
Application of this to the description of B-branes is in
For the case of Lie 1-algebroids essentially the same definition appears also in
The Lie integration of representations of Lie 1-algebroids $\mathfrak{a} \to end(V)$ to morphisms of ∞-categories $A \to Ch_\bullet^\circ$ is discussed in