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:

$\mathrm{\nabla }:\mathrm{\Gamma }(A)\times \mathrm{\Gamma }(\mathfrak{g}(A))\to \mathrm{\Gamma }(\mathfrak{g}(A)):{\mathrm{\nabla }}_{\varphi }\psi :=[\varphi ,\psi ],$

$\mathrm{\nabla }:\mathrm{\Gamma }(A)\times \mathrm{\Gamma }(\nu (A))\to \mathrm{\Gamma }(\nu (A)):{\mathrm{\nabla }}_{\varphi }\bar{X}:=\overline{[\rho (\varphi ),X]}.$

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

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

fulfills the Leibniz rule

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

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 $\mathrm{\nabla }\circ \partial =\partial \circ \mathrm{\nabla }$,
• an End(E)-valued A-2-form ω₂ of total degree 1, such that the curvature fulfills $\partial {\omega }_2+R_{\mathrm{\nabla }}=0,$
• End(E)-valued A-p-forms ω_p of total degree 1 that fulfill the homotopy relations….

The correspondence is characterized as

$D=\partial +\mathrm{\nabla }+{\omega }_2+{\omega }_3+\cdots .$

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.

$D_E\circ \mathrm{\Phi }=\mathrm{\Phi }\circ D_E.$

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

${\mathrm{\Phi }}_i\in {\mathrm{\Omega }}^i(A,{\mathrm{H}\mathrm{o}\mathrm{m}}^{-i}(E,F))$

and then the compatibility condition reads

$\partial {\mathrm{\Phi }}_n+d_{\mathrm{\nabla }}({\mathrm{\Phi }}_{n-1})+[{\omega }_2,{\mathrm{\Phi }}_{n-2}]+\cdots +[{\omega }_n,{\mathrm{\Phi }}_0]=0.$

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

$\underset{\varphi }{\overset{bas}{\mathrm{\nabla }}}\psi :=[\varphi ,\psi ]+{\mathrm{\nabla }}_{\rho (\psi )}\varphi ,$

$\underset{\varphi }{\overset{bas}{\mathrm{\nabla }}}X:=[\rho (\varphi ),X]+\rho ({\mathrm{\nabla }}_X\varphi ).$

Moreover we can introduce the mixed curvature as

$R^{bas}(\varphi ,\psi )(X):={\mathrm{\nabla }}_X[\varphi ,\psi ]-[{\mathrm{\nabla }}_X\varphi ,\psi ]-[\varphi ,{\mathrm{\nabla }}_X\psi ]-{\mathrm{\nabla }}_{\underset{\psi }{\overset{bas}{\mathrm{\nabla }}}X}\varphi +{\mathrm{\nabla }}_{\underset{\psi }{\overset{bas}{\mathrm{\nabla }}}X}\varphi .$

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:

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

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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