Representation up to homotopy

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 ${\displaystyle \mathbb {Z} }$-graded vector bundle over M and Ω(A,E) = Ω(A) ⊗ Γ(E) be its ${\displaystyle \mathbb {Z} }$-graded A-cochains with values in E. A representation up to homotopy of A on E is a differential operator D that maps

${\displaystyle D\colon \Omega ^{\bullet }(A,E)\to \Omega ^{\bullet +1}(A,E),}$

fulfills the Leibniz rule

${\displaystyle D(\alpha \wedge \beta )=(D\alpha )\wedge \beta +(-1)^{|\alpha |}\alpha \wedge (D\beta ),}$

and squares to zero, i.e. D² = 0.

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 ${\displaystyle \nabla \circ \partial =\partial \circ \nabla }$,
• an End(E)-valued A-2-form ω₂ of total degree 1, such that the curvature fulfills ${\displaystyle \partial \omega _{2}+R_{\nabla }=0,}$
• End(E)-valued A-p-forms ω_p of total degree 1 that fulfill the homotopy relations....

The correspondence is characterized as

${\displaystyle D=\partial +\nabla +\omega _{2}+\omega _{3}+\cdots .\,}$

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.

${\displaystyle D_{F}\circ \Phi =\Phi \circ D_{E}.\,}$

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 Φ₀ + Φ₁ + ... with

${\displaystyle \Phi _{i}\in \Omega ^{i}(A,\mathrm {Hom} ^{-i}(E,F))}$

and then the compatibility condition reads

${\displaystyle \partial \Phi _{n}+d_{\nabla }(\Phi _{n-1})+[\omega _{2},\Phi _{n-2}]+\cdots +[\omega _{n},\Phi _{0}]=0.}$

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 ${\displaystyle E=M\times \mathbb {R} [0]\oplus \mathbb {R} [p]}$ and the operator D = ∇ + ω_p where ∇ is the flat connection on the trivial bundle ${\displaystyle M\times \mathbb {R} }$.

Given a representation up to homotopy as D = ∂ + ∇ + ω₂ + ... we can construct a new representation up to homotopy by conjugation, i.e.

D' = ∂ − ∇ + ω₂ − ω₃ + −....

Given a Lie algebroid (A,ρ,[.,.]) together with a connection ∇ on its vector bundle we can define two associated A-connections as follows^[3]

${\displaystyle \nabla _{\!\phi \,}^{bas}\psi$ :=[\phi ,\psi ]+\nabla _{\!\rho (\psi )\,}\phi ,}

${\displaystyle \nabla _{\!\phi \,}^{bas}X:=[\rho (\phi ),X]+\rho (\nabla _{\!X\,}\phi ).}$

Moreover, we can introduce the mixed curvature as

${\displaystyle R^{bas}(\phi ,\psi )(X):=\nabla _{\!X\,}[\phi ,\psi ]-[\nabla _{\!X\,}\phi ,\psi ]-[\phi ,\nabla _{\!X\,}\psi ]-\nabla _{\!\nabla _{\!\psi \,}^{bas}X\,}\phi +\nabla _{\!\nabla _{\!\psi \,}^{bas}X\,}\phi .}$

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:

${\displaystyle D=\rho +\nabla ^{bas}+R^{bas}.}$

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).