Contents

# Contents

## Idea

This entry discusses ∞-actions/∞-representations of Lie-infinity algebroids (generalizing Lie algebra actions), 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 $Gr$:

1. as a morphism out of $Gr$: the action;

2. 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

$\rho : \mathbf{B} G \to Vect$

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 $G$

$[\mathbf{B}G, Vect] \simeq Rep(G) \,.$

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 $\rho$

$\array{ V//G &\to& Vect_* \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& Vect } \,.$

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

$V \hookrightarrow V//G \to \mathbf{B}G \,.$

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,

$\array{ G &\hookrightarrow& \mathbf{E}G &\to& \mathbf{B}G \\ = && = && = \\ G &\hookrightarrow& G//G &\to& \mathbf{B}G } \,.$

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.

## Definition

### Representations

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 $A$.

###### Definition

A representation $\rho$ of a Lie $\infty$-algebroid $(g, A)$ on a co-chain complex $V$ of $A$-modules is a cofibration sequence

$\wedge^\bullet V \leftarrow CE_\rho(g,V) \leftarrow CE_A(g)$

in DGCAs, i.e. a homotopy pushout

$\array{ \wedge^\bullet V &\leftarrow& CE_\rho(g) \\ \uparrow && \uparrow \\ 0 &\leftarrow & CE_A(g) } \,.$

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

$\wedge^\bullet V \leftarrow CE_\rho(g,V) \leftarrow CE_A(g)$

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).

### dg-Category of representations

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.

###### Definition

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 $k$

$\phi : V \otimes \wedge^\bullet g \to V' \otimes \wedge^\bullet g^*$

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

$Rep(g,A)$

with Lie representations of $(g,A)$ as objects and chain comoplexes as above as hom-objects is a dg-category.

## Examples

### Basic

###### Example

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.

###### Example

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.

### Action of holomorphic tangent Lie algebroid on chain complexes of complex vector bundles

The following variant of example is a homotopy-theoretic-refinement of the classical Koszul-Malgrange theorem.

###### 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.

### Extensions of $L_\infty$-algebras

###### Example

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

$\mathfrak{a} \to \hat {\mathfrak{g}} \to \mathfrak{g}$

of L-∞ algebras (see at model structure for L-∞ algebras). Regarding this as sequence of L-∞ algebroids over the point

$\mathbf{B}\mathfrak{a} \to \mathbf{B}\hat {\mathfrak{g}} \to \mathbf{B}\mathfrak{g}$

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

$\mathbf{B}^6 \mathbb{R} \to \mathfrak{fivebrane}\to \mathfrak{string}$

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.

## Relations to (co-)modules over the CE (co-)algebra

Under identifying L-infinity algebras with graded (co-)commutative dg-coalgebras/dg-algebras (their Chevalley-Eilenberg algebras) then their representations correspond to dg-(co-)modules whose underlying graded (co-)algebras are (co-)free. See at

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