∞-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)