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.[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:
\nabla\colon\Gamma(A) x \Gamma(ak{g}(A))\to\Gamma(ak{g}(A)):\nabla\phi\psi:=[\phi,\psi],
\nabla\colon\Gamma(A) x \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]
...\toHn(A,ak{g}(A))\toH
| n | |
| def |
(A)\toHn-1(A,\nu(A))\toHn-1(A,ak{g}(A))\to...
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.
A representation up to homotopy as introduced above is equivalent to the following data
\nabla\circ\partial=\partial\circ\nabla
\partial\omega2+R\nabla=0,
The correspondence is characterized as
D=\partial+\nabla+\omega2+\omega3+ … .
A homomorphism between representations up to homotopy (E,DE) and (F,DF) 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.
DF\circ\Phi=\Phi\circDE.
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
| i(A,Hom | |
| \Phi | |
| i\in\Omega |
-i(E,F))
and then the compatibility condition reads
\partial\Phin+d\nabla(\Phin-1)+[\omega2,\Phin-2]+ … +[\omegan,\Phi0]=0.
Examples are usual representations of Lie algebroids or more specifically Lie algebras, i.e. modules.
Another example is given by a p-form ωp together with E = M × ℝ[0] ⊕ ℝ[''p''] and the operator D = ∇ + ωp 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 + -....
Given a Lie algebroid (A,ρ,[.,.]) together with a connection ∇ on its vector bundle we can define two associated A-connections as follows[3]
| bas | |
| \nabla | |
| \phi |
\psi:=[\phi,\psi]+\nabla\rho(\psi)\phi,
| bas | |
| \nabla | |
| \phi |
X:=[\rho(\phi),X]+\rho(\nablaX\phi).
Rbas(\phi,\psi)(X):=\nablaX[\phi,\psi]-[\nablaX\phi,\psi]-[\phi,\nablaX\psi]
| -\nabla | ||||||||||
|
\phi
| +\nabla | ||||||||||
|
\phi.
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+\nablabas+Rbas.
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).