nLab Lie infinity-algebroid representation

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.

• (…) adjoint representation of $L_\infty$-algebras

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

• model structure on dg-comodules

• model structure on dg-modules

References

The definition of representation of $L_\infty$-algebras is discussed in section 5 of

• Tom Lada, Martin Markl, Strongly homotopy Lie algebras (arXiv:9406095)

The general definition of representation of $\infty$-Lie algebroids (of finite type) as above appears as def. 4.9 in

• Hisham Sati, Urs Schreiber, Jim Stasheff, Differential twisted string- and fivebrane structures (arXiv:0910.4001 version 1)

(this discussion is not in the published version arXiv:0910.4001v2, for size reasons)

following

• Urs Schreiber: On $\infty$-Lie (2008) [pdf, Schreiber-InfinityLie.pdf]

and modeled after the general abstract definition of ∞-actions in

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal ∞-bundles – theory, presentations and applications, Journal of Homotopy and Related Structures, 2014 (arXiv:1207.0248)

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:0509284)

Application of this to the description of B-branes is in

• Aaron Bergman, Topological D-branes from Descent (arXiv:0808.0168)

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

• Camilo Arias Abad, Marius Crainic, Representations up to homotopy of Lie algebroids (arXiv)

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

• Camilo Arias Abad, Florian Schätz, The $A_\infty$ de Rham theorem and integration of representations up to homotopy (arXiv:1011.4693)