Double centralizer theorem

A Representation up to homotopy is a concept in differential geometry that generalizes the notion of representation of a Lie algebra to Lie algebroids and nontrivial vector bundles. 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 complexTemplate:Disambiguation needed 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. D² = 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 ω₂ 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.

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

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

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

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

D = ∂ − ∇ + ω₂ − ω₃ + −….

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).

References

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