# Representation up to homotopy

A representation up to homotopy has several meanings. One of the earliest appeared in physics, in constrained Hamiltonian systems. The essential idea is lifting a non-representation on a quotient to a representation up to strong homotopy on a resolution of the quotient. As a concept in differential geometry, it generalizes the notion of representation of a Lie algebra to Lie algebroids and nontrivial vector bundles. As such, it was introduced by Abad and Crainic.
As a motivation consider a regular Lie algebroid (A,ρ,[.,.]) (regular meaning that the anchor ρ has constant rank) where we have two natural A-connections on g(A) = ker ρ and ν(A)= TM/im ρ respectively:
    $\nabla \colon \Gamma (A)\times \Gamma ({\mathfrak {g}}(A))\to \Gamma ({\mathfrak {g}}(A)):\nabla _{\!\phi \,}\psi :=[\phi ,\psi ],$
    $\nabla \colon \Gamma (A)\times \Gamma (\nu (A))\to \Gamma (\nu (A)):\nabla _{\!\phi \,}{\overline {X}}:={\overline {[\rho (\phi ),X]}}.$
In the deformation theory of the Lie algebroid A there is a long exact sequence
    $\dots \to H^{n}(A,{\mathfrak {g}}(A))\to H_{def}^{n}(A)\to H^{n-1}(A,\nu (A))\to H^{n-1}(A,{\mathfrak {g}}(A))\to \dots$
This suggests that the correct cohomology for the deformations (here denoted as H$_{def}$) comes from the direct sum of the two modules g(A) and ν(A) and should be called adjoint representation. Note however that in the more general case where ρ does not have constant rank we cannot easily define the representations g(A) and ν(A). Instead we should consider the 2-term complex A→TM and a representation on it. This leads to the notion explained here.

## Definition

Let (A,ρ,[.,.]) be a Lie algebroid over a smooth manifold M and let Ω(A) denote its Lie algebroid complex. Let further E be a ℤ-graded vector bundle over M and Ω(A,E) = Ω(A) ⊗ Γ(E) be its ℤ-graded A-cochains with values in E. A representation up to homotopy of A on E is a differential operator D that maps
    $D\colon \Omega ^{\bullet }(A,E)\to \Omega ^{\bullet +1}(A,E),$
fulfills the Leibniz rule
    $D(\alpha \wedge \beta )=(D\alpha )\wedge \beta +(-1)^{|\alpha |}\alpha \wedge (D\beta ),$
and squares to zero, i.e. D$^2$ = 0.

### Homotopy operators

A representation up to homotopy as introduced above is equivalent to the following data
* a degree 1 operator ∂: E → E that squares to 0,
* an A-connection ∇ on E compatible as $\nabla \circ \partial =\partial \circ \nabla$,
* an End(E)-valued A-2-form ω$_2$ of total degree 1, such that the curvature fulfills $\partial \omega _{2}+R_{\nabla }=0,$
* End(E)-valued A-p-forms ω$_p$ of total degree 1 that fulfill the homotopy relations....
The correspondence is characterized as
    $D=\partial +\nabla +\omega _{2}+\omega _{3}+\cdots .\,$

### Homomorphisms

A homomorphism between representations up to homotopy (E,D$_E$) and (F,D$_F$) of the same Lie algebroid A is a degree 0 map Φ:Ω(A,E) → Ω(A,F) that commutes with the differentials, i.e.
    $D_{F}\circ \Phi =\Phi \circ D_{E}.\,$
An isomorphism is now an invertible homomorphism. We denote Rep$^∞$ the category of equivalence classes of representations up to homotopy together with equivalence classes of homomorphisms.
In the sense of the above decomposition of D into a cochain map ∂, a connection ∇, and higher homotopies, we can also decompose the Φ as Φ$_0$ + Φ$_1$ + ... with
    $\Phi _{i}\in \Omega ^{i}(A,\mathrm {Hom} ^{-i}(E,F))$
and then the compatibility condition reads
    $\partial \Phi _{n}+d_{\nabla }(\Phi _{n-1})+[\omega _{2},\Phi _{n-2}]+\cdots +[\omega _{n},\Phi _{0}]=0.$

## Examples

Examples are usual representations of Lie algebroids or more specifically Lie algebras, i.e. modules.
Another example is given by a p-form ω$_p$ together with E = M × ℝ[0] ⊕ ℝ[p] and the operator D = ∇ + ω$_p$ where ∇ is the flat connection on the trivial bundle M × ℝ.
Given a representation up to homotopy as D = ∂ + ∇ + ω$_2$ + ... we can construct a new representation up to homotopy by conjugation, i.e.
    D' = ∂ − ∇ + ω$_2$ − ω$_3$ + −....

### Adjoint representation

Given a Lie algebroid (A,ρ,[.,.]) together with a connection ∇ on its vector bundle we can define two associated A-connections as follows
    $\nabla _{\!\phi \,}^{bas}\psi :=[\phi ,\psi ]+\nabla _{\!\rho (\psi )\,}\phi ,$
    $\nabla _{\!\phi \,}^{bas}X:=[\rho (\phi ),X]+\rho (\nabla _{\!X\,}\phi ).$
Moreover, we can introduce the mixed curvature as
    $R^{bas}(\phi ,\psi )(X):=\nabla _{\!X\,}[\phi ,\psi ]-[\nabla _{\!X\,}\phi ,\psi ]-[\phi ,\nabla _{\!X\,}\psi ]-\nabla _{\!\nabla _{\!\psi \,}^{bas}X\,}\phi +\nabla _{\!\nabla _{\!\psi \,}^{bas}X\,}\phi .$
This curvature measures the compatibility of the Lie bracket with the connection and is one of the two conditions of A together with TM forming a matched pair of Lie algebroids.
The first observation is that this term decorated with the anchor map ρ, accordingly, expresses the curvature of both connections ∇$^{bas}$. Secondly we can match up all three ingredients to a representation up to homotopy as:
    $D=\rho +\nabla ^{bas}+R^{bas}.$
Another observation is that the resulting representation up to homotopy is independent of the chosen connection ∇, basically because the difference between two A-connections is an (A − 1 -form with values in End(E).