Lie algebroid

A Lie algebroid - introduced by Pradines - plays the same role for Lie groupoids as a Lie algebra plays for Lie groups. In the sense of abstract nonsense it is a Lie algebra with many objects. More concretely it is a vector bundle A→M whose module of sections is endowed with a Lie bracket and a linear map ρ: A→TM called the anchor where TM is the tangent bundle, subject to the Leibniz rule

$[\phi ,f\cdot \psi ]=\rho (\phi )[f]\cdot \psi +f\cdot [\phi ,\psi ]$

for all φ, ψ sections of A and f a smooth function.

To every Lie groupoid there is associated a Lie algebroid that encodes the infinitesimal structure of the Lie groupoid. But not every Lie algebroid can be integrated to a Lie groupoid.

Examples

1. Lie algebras are examples of Lie algebroids over a point.
2. The simplest example of a non-trivial Lie algebroid is the tangent bundle TM of a smooth manifold M. The anchor map is here the identity and the Lie bracket the commutator bracket of vector fields.
3. Given a principal bundle P→M with structure group G, then the G-action prolongs to TP. The quotient TP/G is again a vector bundle over P/G=M. The G-invariant vector fields on P correspond 1:1 with the sections of this vector bundle. Moreover the Lie bracket of two G-invariant vector fields is again G-invariant. The pull-back of a smooth function on M gives a G-invariant function on P and the application of a G-invariant vector field on a G-invariant function gives again a G-invariant function. Therefore we have an action of the sections of our vector bundle on the smooth functions on the base M. This construction is called Atiyah algebroid of a principal bundle.
4. Given a vector bundle V→M we can consider its covariant differential operators D(V) defined as follows. An R-linear map ψ: Γ(V)→Γ(V) for which exists a vector field X such that

   $\psi [f\cdot v]=X[f]\cdot v+f\cdot \psi [v]$

where v ∈ Γ(V) and f ∈ C^∞(M). Clearly D(V) form a projective module over M. Moreover the assignment ρ: ψ → X is C^∞-linear and thus a vector bundle morphism (once we find a vector bundle whose sections are D(V)). The kernel of ρ are the vertical endomorphisms of V, i.e. Γ(End(V)). Given a TM-connection ∇ on V we see that the map ρ is also surjective onto TM and thus fits into the short exact sequence

1. $0\to \mathrm {End} (V)\to {\mathfrak {D}}(V)\to TM\to 0$.

The realization of D(V) as a vector bundle is now given as a special case of the previous example. Let thus F(V) be the frame bundle of the vector bundle V. It is a principal bundle with structure group Gl_k(R) where k is the rank of V. Its associated Atiyah algebroid is the constructed algebroid D(V).

Lie algebroid of a Lie groupoid

Given a Lie groupoid G⇒M, then we can associate to it a Lie algebroid in the following way. The construction generalizes the association of a Lie algebra to a Lie group. Let 1: M→G be the unit elements and s,t: G⇒M be the source and target maps, then A:= 1^*T^sG with T^sG the tangent spaces to the source fibers s^-1(m) for every m∈M. The sections of A can uniquely be extended to G-left invariant tangent vector fields on G. The commutator bracket of two left-invariant vector fields is again left-invariant. Also smooth functions on the base can be pulled back to G-left invariant functions on G via the target map. Again the action of a left-invariant vector field on a left-invariant function gives a left-invariant function and therefore we have an action of the sections of A on the smooth functions on M.

Tangent bundle and pair groupoid

As a more particular example of the above construction, consider the pair groupoid MxM⇒M of a smooth manifold M. The source and target map are the projections to the first and second component respectively. Compatible pairs are those where the neighboring points coincide, i.e. ((x,y),(y,z)) for arbitrary x, y, and z in M. Multiplication is therefore omitting the common point, i.e. (x,y)o(y,z) = (x,z). The source fibers are (x,M) and thus the source tangent space is T^sG = ∪_x∈M (x,TM). The pull-back bundle under the unit map 1: x→(x,x) is just TM the tangent bundle and the commutator of left-invariant vector fields on the pair groupoid is just the ordinary commutator on the tangent bundle. Also the action on the smooth functions is the ordinary action of vector fields on functions.

Matched pair of Lie algebroids

This was introduced by Lu and studied by Mokri. Given two Lie algebroids A and B over the same base M, we can ask under which additional structures they can be added up to a Lie algebroid again. By addition we mean the Whitney sum A⊕B of its vector bundles. Obviously we need a representation ∇ of A on B as follows.

A representation of a Lie algebroid A on a vector bundle B is a linear map ∇: Γ(A)⊗Γ(B) → Γ(B) that is C^∞-linear in the section of A and fulfills the Leibniz rule

$\nabla _{\!\psi \,}[fv]=\rho (\psi )[f]v+f\nabla _{\!\psi \,}v$

for all ψ∈Γ(A), v∈Γ(B), and f∈C^∞(M). Moreover the connection ∇ needs to be flat, i.e.

$\nabla _{\!\phi \,}\nabla _{\!\psi \,}v-\nabla _{\!\psi \,}\nabla _{\!\phi \,}v-\nabla _{\![\phi ,\psi ]\,}v=0$

for all φ, ψ ∈ Γ(A) and v∈Γ(B).

For symmetry reasons we also need a representation of B on A. The sum bracket then reads

$[\phi \oplus \alpha ,\psi \oplus \beta ]=[\phi ,\psi ]+\nabla _{\!\alpha \,}\psi -\nabla _{\!\beta \,}\phi \oplus [\alpha ,\beta ]+\nabla _{\!\phi \,}\beta -\nabla _{\!\psi \,}\alpha$

where φ, ψ ∈ Γ(A) and α, β ∈ Γ(B). This bracket is skew-symmetric and certainly fulfills the Leibniz rule. However, for the bracket to fulfill the Jacobi identity, the two representations need to be compatible as

$\nabla _{\!\phi \,}[\alpha ,\beta ]=[\nabla _{\!\phi \,}\alpha ,\beta ]+[\alpha ,\nabla _{\!\phi \,}\beta ]-\nabla _{\!\nabla _{\!\alpha \,}\phi \,}\beta +\nabla _{\!\nabla _{\!\beta \,}\phi \,}\alpha$

and $\nabla _{\!\alpha \,}[\phi ,\psi ]=[\nabla _{\!\alpha \,}\phi ,\psi ]+[\phi ,\nabla _{\!\alpha \,}\psi ]-\nabla _{\!\nabla _{\!\phi \,}\alpha \,}\psi +\nabla _{\!\nabla _{\!\psi \,}\alpha \,}\phi$

where φ, ψ ∈ Γ(A) and α, β ∈ Γ(B) as before.

Examples of matched pairs arise e.g. on complex manifolds where the complexified tangent bundle is a matched pair of the holomorphic and anti-holomorphic tangent bundle. Another example comes from a holomorphic Lie algebroid where we require that the bracket as well as the anchor map are morphisms of the sheaf of holomorphic sections together with the anti-holomorphic tangent bundle.

A third example comes from Poisson structures where we can add up the tangent with the cotangent bundle endowed with the Koszul bracket from the Poisson structure.

Lie algebroid cohomology

Given a Lie algebroid A→M we can endow the cochains Ω_M(A) = Γ(∧A^*) with a differential d: Ω_M^p(A)→Ω_M^p+1(A) via the formula:

${\begin{aligned}\langle \mathrm {d} \alpha ,\psi _{0}\wedge \dots \psi _{p}\rangle =&\sum _{i}(-1)^{i}\rho (\psi _{i})\langle \alpha ,\psi _{0}\wedge \dots {\hat {\psi }}_{i}\dots \psi _{p}\rangle \\&+\sum _{i<j}(-1)^{i+j}\langle \alpha ,[\psi _{i},\psi _{j}]\wedge \psi _{0}\dots {\hat {\psi }}_{i}\dots {\hat {\psi }}_{j}\dots \psi _{p}\rangle \end{aligned}}$

where α∈Ω_M^p(A) and ψ_i∈Γ(A). This is a generalization of Chevalley–Eilenberg's construction of a differential for a Lie algebra with a representation. It also generalizes E. Cartan's construction of a differential on the forms on a manifold.

Due to the Leibniz rule of vector fields ρ(ψ_i) as well as the Lie bracket, the expression for d is indeed C^∞-linear and skew-symmetric in each ψ_i. Therefore d maps as claimed. In addition d fulfills the Leibniz rule

$\mathrm {d} (\alpha \wedge \beta )=\mathrm {d} \alpha \wedge \beta +(-1)^{|\alpha |}\alpha \wedge \mathrm {d} \beta$

where α∈Ω_M^|α|(A) and β∈Ω_M(A). Finally straight-forward computations for smooth functions f ∈ C^∞(M)=Ω_M⁰(A) and 1-forms α∈Ω_M¹(A) show that d²=0 follows from the morphism property of the anchor map

$\rho [\phi ,\psi ]=[\rho (\phi ),\rho (\psi )]$

where φ, ψ ∈ Γ(A) as well as the Jacobi identity of the Lie bracket. The morphism property of the anchor map in itself can be proven using the Leibniz rule and the Jacobi identity.

Representation up to homotopy

The problem is that a flat connection (i.e. a representation) of a transitive Lie algebroid (i.e. one with surjective anchor map) requires a trivial vector bundle. A generalization to non-trivial vector bundles was discovered by Abad and Crainic.

As a motivation consider the following construction for a regular Lie algebroid A→M (i.e. where the anchor map ρ has constant rank). The vector bundle ker ρ is endowed with a flat connection ∇ via

$\nabla _{\!\phi \,}v=[\phi ,v]$

where φ∈Γ(A) and v ∈ Γ(ker ρ). Also the vector bundle TM/im ρ is endowed with a flat connection ∇ via

$\nabla _{\!\phi \,}{\bar {X}}={\overline {[\rho (\phi ),X]}}$

where φ∈Γ(A) and ${\bar {X}}\in \Gamma (TM/\mathrm {im} \rho )$. Both connections are flat due to the Jacobi identity of the Lie bracket.

Given a Z-graded vector bundle V→M together with a Lie algebroid A→M, then a representation up to homotopy on V is the following structure: A differential ∂: V^p → V^p+1, an A-connection ∇ on V that preserves the grading, an A-2-form ω₂ with values in End(V) of total degree 1, an A-3-form ω₃ with values in End(V) of degree 1, …. The structure has to be subject to

$\partial \omega _{2}+R_{\nabla }=0$,

$\partial \omega _{i}+\mathrm {d} _{\nabla }\omega _{i-1}+\omega _{2}\circ \omega _{i-2}+\dots +\omega _{i-2}\circ \omega _{2}=0$ for all i.

Adjoint representation

As an example of a representation up to homotopy consider the following construction. Let A→M be an arbitrary Lie algebroid (not necessarily regular) and ∇ be any TM-connection on A. We define the connections

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

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

where φ, ψ ∈ Γ(A) and X ∈ Γ(TM). We can introduce the base curvature

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

with the same convention for φ, ψ, and X. Note that the curvature of each connection ∇^bas is just R^bas decorated with the anchor map ρ. Set finally V₀=A, V₁=TM, and ∂ = ρ and note that ω₂ = R^bas has the same total degree 1 as ∂. Therefore (V,∂,∇^bas,R^bas,0,…) form a representation up to homotopy called the adjoint representation.

Lie superalgebroids

The concept of Lie algebroids can be generalized to vector superbundles analog to Lie superalgebras. Let thus A→M be a vector superbundle (i.e. a Z/(2)-graded vector bundle) endowed with a graded skew-symmetric bracket [.,.]: A^pxA^q → A^p+q+d where d is the degree of the bracket and addition is performed modulo 2. It is a graded Lie bracket if it is subject to the graded Jacobi identity

$[\phi ,[\psi ,\chi ]]=[[\phi ,\psi ],\chi ]+(-1)^{(|\phi |+d)(|\psi |+d)}[\psi ,[\phi ,\chi ]]$

for all φ ∈ Γ(A^|φ|), ψ ∈ Γ(A^|ψ|), and χ ∈ Γ(A). Moreover we need a linear map ρ: A^d → TM called again the anchor subject to the graded Leibniz rule

$[\phi ,f\cdot \psi ]=\rho (\phi )[f]\cdot \psi +f\cdot [\phi ,\psi ]$

where φ ∈ Γ(A^|φ|), ψ ∈ Γ(A), and f ∈ C^∞(M).

If we generalize from smooth manifolds to supermanifolds, we can also widen the anchor map ρ to all of A and the Leibniz rule gets an additional sign when interchanging the order of f and φ.

Examples

As an example consider the following construction. Let V be a vector superbundle and consider the graded covariant differential operators D(V). The 00- and 11-differential operators are said to be even and the 01- as well as 10-linear maps are said to be odd. With this convention introduce the graded commutator

$[\phi ,\psi ]:=\phi \circ \psi -(-1)^{|\phi |\,|\psi |}\psi \circ \phi$

where φ ∈ Γ(D^|φ|(V)), ψ ∈ Γ(D^|ψ|(V)). This endows D(V) with the structure of a Lie superalgebroid. Note however that the odd derivations are all C^∞-linear.

In order to obtain odd derivations we need to generalize to the category of supermanifolds. In brief this is a ringed (Hausdorff second countable) space where the structure sheaf consists of Z/(2)-graded algebras that are locally isomorphic to the smooth functions on an open set of R^p tensor the exterior algebra on R^q. The dimension is thus denoted p|q. The graded derivations of the structure sheaf form a projective module over the structure sheaf. The underlying vector superbundle is called the tangent bundle and denoted TM for the supermanifold M.

An action of a Lie superalgebra on a supermanifold gives rise to the action algebroid that is a Lie superalgebroid.

References

1. M. Crainic and R.L. Fernandes: Integrability of Lie brackets, Ann. of Math. (2), vol. 157/2 (2003), pp.575–620, arXiv:math/0105033.
2. K.C.H. Mackenzie: Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge U. Press, 1987.
3. K.C.H. Mackenzie: General Theory of Lie Groupoids and Lie Algebroids, Cambridge U. Press, 2005.