Lie infinity-algebroid representation


\infty-Lie theory

∞-Lie theory


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids




\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras



Recall that an L 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 GrGr:

  1. as a morphism out of GrGr: the action;

  2. as a fibration sequence over GrGr: 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 GG a Lie group, for definiteness and for simplicity, with corresponding one-object Lie groupoid BG\mathbf{B} G – the delooping of the group GG –, a linear representation in terms of an action morphisms is a functor

ρ:BGVect \rho : \mathbf{B} G \to Vect

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

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

Every such functor ρ\rho induces a fibration sequence V//GBGV//G \to \mathbf{B}G over BG\mathbf{B}G, obtained as the pullback of the generalized universal bundle Vect *VectVect_* \to Vect along ρ\rho

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

Here V//GV//G is the action groupoid of the action of ρ\rho on the representation vector space V:=ρ()V := \rho(\bullet), where \bullet is the single object of BG\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

VV//GBG. 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 ρ\rho to the universal GG-bundle EGBG\mathbf{E}G \to \mathbf{B}G, which itself is the action groupoid of the fundamental representation? of GG on itself,

G EG BG = = = G G//G BG. \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 GG is nothing but a GG-equivariant vector bundle over the point, or equivalently a vector bundle on the orbifold //G\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)CE_A(g) for the qDGCA whose underlying graded-commutative algebra is the free (over the algebra AA) graded commutative algebra g *\wedge^\bullet g^* for gg a non-postively graded cochain complex of AA-modules and g *g^* its degree-wise dual over AA, to remind us that this is to be thought of as the Chevalley-Eilenberg algebra of the Lie ∞-algebroid gg whose space of objects is characterized dually by the algebra AA.


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

VCE ρ(g,V)CE A(g) \wedge^\bullet V \leftarrow CE_\rho(g,V) \leftarrow CE_A(g)

in DGCAs, i.e. a homotopy pushout

V CE ρ(g) 0 CE A(g). \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 ρ\rho is a strict cofiber sequence of morphisms of DGCAs

VCE ρ(g,V)CE A(g) \wedge^\bullet V \leftarrow CE_\rho(g,V) \leftarrow CE_A(g)

i.e. such that

  • CE ρ(g,V)=CE A(g) VCE_\rho(g,V) = CE_A(g) \otimes \wedge^\bullet V as GCAs

  • VCE ρ(g,V)\wedge^\bullet V \leftarrow CE_\rho(g,V) is the obvious surjection;

  • CE ρ(g,V)CE A(g)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 ρd_\rho on CE ρ(g,V)CE_\rho(g,V) is given by a twisting map ρ *:V( V)(g *)( g *)\rho^* : V \to (\wedge^\bullet V) \wedge (g^*) \wedge (\wedge^\bullet g^*) as

  • d ρ g *=d gd_\rho|_{g^*} = d_g

  • d ρ V=d V+ρ *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 the dg-category Rep(g,A)Rep(g,A) of proper representations of a Lie-\infty-algebroid (g,A)(g,A) in the above sense – called dg-algebra modules there – is defined.


Given two objects CE ρ(g,V)CE_\rho(g,V) and CE ρ(g,V)CE_{\rho'}(g,V') in Rep(g,A)Rep(g,A), the cochain complex

Hom(CE ρ(g,V),CE ρ(g,V))Hom( CE_\rho(g,V), CE_{\rho'}(g,V') )

consist in degree kk of morphisms of degree kk

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

satisfying ϕ(vt)=(1) kaϕ(v)t\phi(v t) = (-1)^{k |a|} \phi(v) t

and the differential d Homd_{Hom} is the usual differential on hom-complexes dϕ=d ρϕ(1) ϕϕd ρd \phi = d_{\rho'} \circ \phi - (-1)^{|\phi|} \phi \circ d_\rho.

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

Rep(g,A) Rep(g,A)

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


Relation to coherent complexes of sheaves


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

This is Block, theorem 2.22.

The objects of Rep(T holX)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 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 (g,A)=TX(g,A) = T X the tangent Lie algebroid of a smooth manifold XX, it should be true, up to technicalities to be spelled out here eventually, that HoRep(TX)Ho Rep(T X) is equivelent to the derived category of D-modules on XX, or the like.



The definition of representation of L 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 𝔞end(V)\mathfrak{a} \to end(V) to morphisms of ∞-categories ACh A \to Ch_\bullet^\circ is discussed in

Revised on July 19, 2013 10:30:09 by Urs Schreiber (