∞-Lie theory

# Contents

## Idea

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 $\mathrm{Gr}$:

1. as a morphism out of $\mathrm{Gr}$: the action;

2. as a fibration sequence over $\mathrm{Gr}$: 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 $BG$ – the delooping of the group $G$ –, a linear representation in terms of an action morphisms is a functor

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

from $BG$ to the category of vector spaces. In fact, there is a canonical equivalence of the functor category $\left[BG,\mathrm{Vect}\right]$ with the category $\mathrm{Rep}\left(G\right)$ of linear representations of $G$

$\left[BG,\mathrm{Vect}\right]\simeq \mathrm{Rep}\left(G\right)\phantom{\rule{thinmathspace}{0ex}}.$[\mathbf{B}G, Vect] \simeq Rep(G) \,.

Every such functor $\rho$ induces a fibration sequence $V//G\to BG$ over $BG$, obtained as the pullback of the generalized universal bundle ${\mathrm{Vect}}_{*}\to \mathrm{Vect}$ along $\rho$

$\begin{array}{ccc}V//G& \to & {\mathrm{Vect}}_{*}\\ ↓& & ↓\\ BG& \stackrel{\rho }{\to }& \mathrm{Vect}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $V:=\rho \left(•\right)$, where $•$ is the single object of $BG$. 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

$V↪V//G\to BG\phantom{\rule{thinmathspace}{0ex}}.$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 $EG\to BG$, which itself is the action groupoid of the fundamental representation? of $G$ on itself,

$\begin{array}{ccccc}G& ↪& EG& \to & BG\\ =& & =& & =\\ G& ↪& G//G& \to & BG\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $•//G$. 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 ${\mathrm{CE}}_{A}\left(g\right)$ for the qDGCA whose underlying graded-commutative algebra is the free (over the algebra $A$) graded commutative algebra ${\wedge }^{•}{g}^{*}$ 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 $\left(g,A\right)$ on a co-chain complex $V$ of $A$-modules is a cofibration sequence

${\wedge }^{•}V←{\mathrm{CE}}_{\rho }\left(g,V\right)←{\mathrm{CE}}_{A}\left(g\right)$\wedge^\bullet V \leftarrow CE_\rho(g,V) \leftarrow CE_A(g)

in DGCAs, i.e. a homotopy pushout

$\begin{array}{ccc}{\wedge }^{•}V& ←& {\mathrm{CE}}_{\rho }\left(g\right)\\ ↑& & ↑\\ 0& ←& {\mathrm{CE}}_{A}\left(g\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

${\wedge }^{•}V←{\mathrm{CE}}_{\rho }\left(g,V\right)←{\mathrm{CE}}_{A}\left(g\right)$\wedge^\bullet V \leftarrow CE_\rho(g,V) \leftarrow CE_A(g)

i.e. such that

• ${\mathrm{CE}}_{\rho }\left(g,V\right)={\mathrm{CE}}_{A}\left(g\right)\otimes {\wedge }^{•}V$ as GCAs

• ${\wedge }^{•}V←{\mathrm{CE}}_{\rho }\left(g,V\right)$ is the obvious surjection;

• ${\mathrm{CE}}_{\rho }\left(g,V\right)←{\mathrm{CE}}_{A}\left(g\right)$ is the obvious injection;

• the composite of both is the 0-map.

It follows that the differential ${d}_{\rho }$ on ${\mathrm{CE}}_{\rho }\left(g,V\right)$ is given by a twisting map ${\rho }^{*}:V\to \left({\wedge }^{•}V\right)\wedge \left({g}^{*}\right)\wedge \left({\wedge }^{•}{g}^{*}\right)$ as

• ${d}_{\rho }{\mid }_{{g}^{*}}={d}_{g}$

• ${d}_{\rho }{\mid }_{V}={d}_{V}+{\rho }^{*}$

which may be thought of as the dual of the representation morphism (see the examples below).

### dg-Category of representations

In Block the dg-category $\mathrm{Rep}\left(g,A\right)$ of proper representations of a Lie-$\infty$-algebroid $\left(g,A\right)$ in the above sense – called dg-algebra modules there – is defined.

Definition

Given two objects ${\mathrm{CE}}_{\rho }\left(g,V\right)$ and ${\mathrm{CE}}_{\rho \prime }\left(g,V\prime \right)$ in $\mathrm{Rep}\left(g,A\right)$, the cochain complex

$\mathrm{Hom}\left({\mathrm{CE}}_{\rho }\left(g,V\right),{\mathrm{CE}}_{\rho \prime }\left(g,V\prime \right)\right)$

consist in degree $k$ of morphisms of degree $k$

$\varphi :V\otimes {\wedge }^{•}g\to V\prime \otimes {\wedge }^{•}{g}^{*}$\phi : V \otimes \wedge^\bullet g \to V' \otimes \wedge^\bullet g^*

satisfying $\varphi \left(vt\right)=\left(-1{\right)}^{k\mid a\mid }\varphi \left(v\right)t$

and the differential ${d}_{\mathrm{Hom}}$ is the usual differential on hom-complexes $d\varphi ={d}_{\rho \prime }\circ \varphi -\left(-1{\right)}^{\mid \varphi \mid }\varphi \circ {d}_{\rho }$.

For a fixed Lie $\infty$-algebroid $\left(g,A\right)$, the category

$\mathrm{Rep}\left(g,A\right)$Rep(g,A)

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

## Properties

### Relation to coherent complexes of sheaves

Theorem

For $X$ a smooth complex manifold and $\left(g,A\right)={T}_{\mathrm{hol}}X$ the holomorphic tangent Lie algebroid of $X$ (so that ${\mathrm{CE}}_{A}\left(g\right)={\Omega }_{\mathrm{hol}}^{•}\left(X\right)$ the holomorphic deRham complex of $X$), and for $\mathrm{Rep}\left({T}_{\mathrm{hol}}X\right)$ taken to have as objects complexes of finitely generated and projective ${C}^{\infty }\left(X\right)$-modules (i.e. complexes of smooth vector bundles) the homotopy category $\mathrm{Ho}\mathrm{Rep}\left({T}_{\mathrm{hol}}X\right)$ of the dg-category $\mathrm{Rep}\left({T}_{\mathrm{hol}}X\right)$ is equivalent to the bounded derived category of complexes of sheaves with coherent cohomology on $X$ (see coherent sheaf).

This is Block, theorem 2.22.

The objects of $\mathrm{Rep}\left({T}_{\mathrm{hol}}X\right)$ 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.

### Relation to D-modules

For $\left(g,A\right)=TX$ the tangent Lie algebroid of a smooth manifold $X$, it should be true, up to technicalities to be spelled out here eventually, that $\mathrm{Ho}\mathrm{Rep}\left(TX\right)$ is equivelent to the derived category of D-modules on $X$, or the like.

## Examples

• ordinary representation of a Lie algebra on a vector space: ${\mathrm{CE}}_{\rho }\left(g,V\right)$ is essentially the Chevalley-Eilenberg complex that computes the cohomology of $g$ with coefficients in $V$.

• flat connections on bundles

• adjoint representation of ${L}_{\infty }$-algebras

## References

The definition of representation of ${L}_{\infty }$-algebras is discussed in section 5 of

The general definition of representation of $\infty$-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

• Jonathan Block, Duality and equivalence of module categories in noncommutative geometry I (arXiv)

For the case of Lie 1-algebroids essentially the same definition appears also in

The Lie integration of representations of Lie 1-algebroids $𝔞\to \mathrm{end}\left(V\right)$ to morphisms of ∞-categories $A\to {\mathrm{Ch}}_{•}^{\circ }$ is discussed in

Revised on April 2, 2011 13:06:06 by Urs Schreiber (89.204.153.118)