Representation up to homotopy

A Representation up to homotopy has several meanings. One of the earliest appeared in the `physical' context of 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. ^[1]

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:

${\displaystyle \nabla \colon \Gamma (A)\times \Gamma ({\mathfrak {g}}(A))\to \Gamma ({\mathfrak {g}}(A)):\nabla _{\!\phi \,}\psi$ :=[\phi ,\psi ],}

${\displaystyle \nabla \colon \Gamma (A)\times \Gamma (\nu (A))\to \Gamma (\nu (A)):\nabla _{\!\phi \,}{\overline {X}}:={\overline {[\rho (\phi ),X]}}.}$

In the deformation theory of the Lie algebroid A there is a long exact sequence ^[2]

${\displaystyle \dots \to H^{n}(A,{\mathfrak {g}}(A))\to H_{def}^{n}(A)\to H^{n-1}(A,\nu (A))\to H^{n-1}(A,{\mathfrak {g}}(A))\to \dots }$

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

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

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.

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 ${\displaystyle \nabla \circ \partial =\partial \circ \nabla }$,
• 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

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

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.

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

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

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 ^[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).

References

1. Abad, Camilo Arias; Crainic, Marius. "Representations up to homotopy of Lie algebroids". Journal für die reine und angewandte Mathematik. 2012 (663): 91–126. arXiv: 0901.0319 . doi:10.1515/CRELLE.2011.095.
2. Crainic, Marius; Moerdijk, Ieke (2008). "Deformations of Lie brackets: cohomological aspects". Journal of the European Mathematical Society . 10 (4): 1037–1059. arXiv: math/0403434 . doi: 10.4171/JEMS/139 .
3. Crainic, M.; Fernandes, R. L. (2005). "Secondary characteristic classes of Lie algebroids". Quantum field theory and noncommutative geometry. Lecture Notes in Physics. Vol. 662. Springer, Berlin. pp. 157–176. doi: 10.1007/11342786_9 .