Distributions and quotients on degree 1 NQ-manifolds and Lie algebroids

It is well-known that a Lie algebroid A is equivalently described by a degree 1 Q-manifold M. We study distributions on M, giving a characterization in terms of A. We show that involutive Q-invariant distributions on M correspond bijectively to IM-foliations on A (the infinitesimal version of Mackenzie's ideal systems). We perform reduction by such distributions, and investigate how they arise from non-strict actions of strict Lie 2-algebras on M.


Introduction
Lie algebroids play an important role in differential geometry and mathematical physics. It is known that integrable Lie algebroids are in bijection with source simply connected Lie groupoids Γ. There is an aboundant literature studying geometric structures on Γ which are compatible with the groupoid multiplication, and how they correspond to structures on the Lie algebroid. Examples of the former are multiplicative foliations on Γ, and examples of the latter are morphic foliations and IM-foliations, where "IM" stands for "infinitesimally multiplicative 1 " In this note we take a graded-geometric point of view on Lie algebroids, using their characterization as NQ-1 manifolds. The latter are graded manifolds with coordinates of degrees 0 and 1, endowed with a self-commuting vector field Q of degree 1. Given a Lie algebroid A, we denote the corresponging NQ-1 manifold by M.
In §2 we consider distributions on M which are involutive and Q-invariant. When regularity conditions are satisfied we perform reduction by distributions in the graded geometry setting, obtaining quotient NQ-1 manifolds. We show that, at the level of Lie algebroids, such distributions correspond exactly to ideal systems on A, and that this reduction is the reduction of A by ideal systems [12]. Further we show that involutive, Q-invariant distributions correspond bijectively to the IM-foliations on A introduced by Jotz-Ortiz [8]. To prepare the ground for the above results, at the beginning of §2 we consider distributions on degree 1 N-manifolds and characterize them in terms of ordinary differential geometry.
In §3 we consider distributions that arise form certain actions. As an NQ-1 manifold M has coordinates in degrees 1 and 0, its module of vector fields is generated in degrees −1 and 0, and hence it is natural to act on M by strict Lie-2 algebras L (differential graded Lie algebras concentrated in degrees −1 and 0). We define such actions as L ∞ -morphisms from L into the DGLA of vector fields on M. In general the "image" of an action fails to be an involutive distribution (however it is automatically preserved by Q). We give a sufficient condition for the involutivity, and show that performing reduction by the action one obtains a new NQ-1 manifold.
A possible explanation for the fact that the involutivity fails in general is the following. In ordinary geometry, when a Lie algebra action on a manifold is (globally) free and proper, the Lie groupoid of the transformation Lie algebroid is Morita equivalent to the quotient of the manifold by the action. Thus the transformation Lie algebroid (an NQ-1 manifold) may be taken as a good replacement of the quotient when the action is no longer free and proper. In our setup it is shown in [14] that L[1] × M inherits a degree 1 homological vector field (generalizing the notion of transformation Lie algebroid). The fact that L[1] × M is an NQ-2 manifold suggests that a suitably defined quotient M/L should not be an NQ-1 manifolds in general (it should be so only when certain "regularity" conditions are satisfied).
Notation and conventions: M always denotes a smooth manifold. For any vector bundle E, we denote by E [1] the N-manifold obtained from E by declaring that the fiber-wise linear coordinates on E have degree one. If M is an N-manifold, we denote by C(M) the graded commutative algebras of "functions on M". By χ(M) we denote graded Lie algebra of vector fields on M (i.e., graded derivations of C(M)). The symbol A always denotes a Lie algebroid over M .

Background
To be self-contained, we first recall some background material from [20, §1.1]. More precisely: we recall how the notion of degree 1 N-manifold is equivalent to the notion of vector bundle ( §1.1), and how the notion of NQ-1 manifolds is equivalent to that of Lie algebroid ( §1.2).
We use S • (V * ) to denote the graded symmetric algebra over V * , so its homogeneous elements anti-commute if they both have odd degree. S • (V * ) is a graded commutative algebra concentrated in positive degrees.
Ordinary manifolds are modeled on open subsets of R n , and N-manifolds modeled on the following graded charts: The local model for an N-manifold consists of a pair as follows: Definition 1.2. An N-manifold M consists of a pair as follows: • a topological space M (the "body") • a sheaf O M over M of graded commutative algebras, locally isomorphic to the above local model (the sheaf of "functions").
We  Since C(M) is a graded commutative algebra (concentrated in non-negative degrees), the space of vector fields χ(M), equipped with the graded commutator [−, −], is a graded Lie algebra (see Def. 1.9).
We will focus mainly on degree 1 N-manifolds, which we now describe in more detail. To do so we recall first Definition 1.5. Given a vector bundle E over M , a covariant differential operator 3 (CDO) is a linear map X : Γ(E) → Γ(E) such that there exists a vector field X on M (called symbol) with We denote the set of CDOs on E by CDO(E). If X ∈ CDO(E), then the dual X * ∈ CDO(E * ) is defined by Recall that if E → M is a (ordinary) vector bundle, E[1] denotes the graded vector bundle whose fiber over x ∈ M is (E x )[1] (a graded vector space concentrated in degree −1). Proof. See [20].
and α, β ≤ rk(E), and we adopt the Einstein summation convention.

NQ-manifolds and Lie algebroids
We will be interested in N-manifolds equipped with extra structure: Definition 1.8. An NQ-manifold is an N-manifold equipped with a homological vector field, i.e. a degree 1 vector field Q such that [Q, Q] = 0.
To shorten notation, we call a degree n NQ-manifold a NQ-n manifold. Before considering χ(M), we recall the notion of differential graded Lie algebra (DGLA): Definition 1.9. A graded Lie algebra consists of a graded vector space L = ⊕ i∈Z L i together with a bilinear bracket [·, ·] : L × L → L such that, for all homogeneous a, b, c ∈ L: -the bracket is degree-preserving: -the adjoint action [a, ·] is a degree |a| derivation of the bracket (Jacobi identity): A differential graded Lie algebra (DGLA) (L, [·, ·], δ) is a graded Lie algebra together with a linear δ : L → L such that δ is a degree 1 derivation of the bracket: is a negatively bounded DGLA with lowest degree −n.
A well-known example of NQ-manifolds is given by Lie algebroids [12].
Definition 1.11. A Lie algebroid A over a manifold M is a vector bundle over M , such that the global sections of A form a Lie algebra with Lie bracket [·, ·] A and Leibniz rule holds where ρ A : A → T M is a vector bundle morphism called the anchor.
We describe the correspondence using the derived bracket construction. By Lemma 1.6 there is a bijection between vector bundles and degree 1 N-manifolds. If A is a Lie algebroid, then the homological vector field is just the Lie algebroid differential acting on Γ(∧ • A * ) = C(A [1]). Conversely, if (M := A[1], Q) is an NQ-manifold, then the Lie algebroid structure on A can be recovered by the derived bracket construction [9, §4.3]: using the identification where a, a ′ ∈ Γ(A) and f ∈ C ∞ (M ).
In coordinates the correspondence is as follows. Choose coordinates x α on M and a frame of sections e i of A, inducing (degree 1) coordinates ξ i on the fibers of A [1]. Then where [e i , e j ] A = c k ij (x)e k and the anchor of e i is ρ α i (x) ∂ ∂xα .

Distributions
In this section we study distributions on degree 1 N-manifolds and NQ-manifolds. In §2.1 we characterize in classical terms distributions on degree 1 N-manifolds, and carry out reduction by involutive distributions. In §2.2 we consider involutive, Q-invariant distributions on an NQ-1 manifold. We show that in classical terms they correspond to IM-foliations. When regularity conditions are satisfied -classically the correspond to the notion of ideal system -we perform reduction obtaining quotient NQ-1 manifolds.

Distributions on degree 1 N manifolds
Let E → M be a vector bundle. We define distributions on E[1], following [3].  Recall that for any vector bundle E, there is an associated Lie algebroid D(E) whose sections are exactly CDO(E) (see Def. 1.5) endowed with the commutator bracket, and whose anchor s : D(E) → T M is given by the symbol [10, §1]. D(E) fits in an exact sequence of Lie algebroids where End(E) denotes the vector bundle endomorphisms of E that cover Id M . The following lemma gives a characterization of distributions on M in classical terms.
There is a bijection between distributions D on M := E[1] and the following data: The correspondence is where we identify is a Lie subalgebroid and the action of sections of C preserves Γ(B).
Proof. Let D be a distribution. By definition, locally there exist integers l ≤ dim(M ), λ ≤ rk(E) as well as homogeneous generators {X α −1 } α≤λ and {X i 0 } i≤l of D whose evaluations at points of M , which are given by {X α −1 } α≤λ and {s(X i 0 )} i≤l , are linearly independent.
The linear independence condition on the generators implies that ker( Conversely, given a pair (B, C) as in the statement, defining D −1 := Γ(B) and D 0 := Γ(C) we obtain a distribution.
Since a distribution D is generated (as a C(M)-module) by D −1 and D 0 , the involutivity of D is equivalent to The first condition means that C is a subalgebroid of D(E), the second (by Lemma 1.6) that the sections of C preserve Γ(B).
We rephrase the classical data associated to an involutive distributions as in Lemma 2.3. Here and in the following, if X 0 ∈ CDO(E), we denote its symbol by X 0 := s(X 0 ) ∈ χ(M ).
such that action of sections of C preserves Γ(B).
• pairs (F, ∇) where F is an integrable distribution on M and ∇ is a flat F -connection on the vector bundle E/B.
Proof. Let C be as above. Then F := s(C) has constant rank, and is involutive since C is a Lie subalgebroid. Define The above properties of C make clear that ∇ is well-defined, and further it is an F -connection. The flatness of ∇ follows from where in the first equality we used that C is a Lie subalgebroid of D(E).
Conversely, given a pair (F, B) as above, is the space of sections of a subbundle C of D(E), which fits in the short exact sequence of vector bundles To see this, notice that for any Y ∈ Γ(F ) there exists X 0 lying in (7) with X 0 = Y : choose locally a frame {e i } of E consisting of a local frame for B and lifts of ∇-flat local sections of E/B, then just define X 0 (e i ) = 0 for all i, and obtain a covariant differential operator X 0 by imposing eq. (1). Hence C is a subbundle of D(E), and one checks easily that it satisfies the required properties.
Let us introduce a piece of terminology: we say that an involutive distribution F on a manifold M is simple if there exists a smooth structure on M/F for which the projection π : M → M/F is a submersion.
Consider Assume that F is simple.
is a presheaf of graded commutative algebras over M/F . When it defines an N-manifold (with body M/F ), we say that the quotient of M by D is smooth, and denote the quotient N-manifold by M/D.
for sections ξ of B • and V of E/B. It is given by ∇ * X 0 ξ = X 0 (ξ) where X 0 ∈ D 0 and where we identify Γ(E * ) = C 1 (M) as in Lemma 1.6. Hence we conclude that By assumption, F is simple and ∇ has no holonomy. On one hand, this assures that the quotient of E/B by the action of ∇ is a smooth vector bundle. On the other hand this implies that the technical conditions i) and ii) of [7, Lemma 5.12] are satisfied and hence eq. (8) gives a sheaf of graded commutative algebras generated by its elements in degrees 0 and 1. Since by the above C 0 (M) ∼ = C ∞ (M/F ) and C 1 (M) D ∼ = Γ(Ẽ * ), this implies that the sheaf given by eq. (8) corresponds to the N-manifoldẼ [1]. Hence M/D ∼ =Ẽ [1] as N-manifolds.

Distributions on NQ-1 manifolds
In this subsection A is a Lie algebroid over M and M := A[1] the corresponding NQ-1 manifold (see Lemma 1.12), whose homological vector field we denote by Q.
This can be seen directly from the definition of distribution or from Lemma 2.3. The distribution D is automatically involutive. Further, if [Q, ·] preserves D, then B is an ideal in g. Indeed, for any index γ ≤ dim(B) we have [Q, ∂ ∂ξγ ] ∈ D 0 . Hence for any β ≤ dim(g), using the identification g ∼ = χ −1 (g [1]), we have for some constants c α ′ α .

Proof. B is a Lie subalgebroid of
If F is simple then R is a closed, embedded wide subgroupoid of M × M . If ∇ has no holonomy then the groupoid action of R on A/B is well-defined.
To check that we indeed have an ideal system we need to check (i)-(iv) in Def. 2.6. Recall that F and ∇ were defined in the proof of Lemma 2.4.
This is clear because Q preserves D.
(iv) If p, q are points of M lying in the same fiber of π and a p ∈ A p , then π * (ρ A (a p )) = π * (ρ A (a q )), where a q ∈ A q is a lift of the ∇-parallel translation of (a p mod B) from p to q.  The assumption Hence, by restricting the action of Q to C(M) D ∼ = C (Ã[1]), we obtain a homological vector fieldQ onÃ [1].
By construction the inclusion C(M) D → C(M) respects the action of the homological vector fields, so that the quotient Lie algebroid structure onÃ (obtained via the derived bracket construction usingQ) has the property that the projection A →Ã is a Lie algebroid morphism. Hence it agrees with the Lie algebroid structure obtained by the ideal system.
We present an example where D is singular, i.e. just a graded C(M)-submodule of χ(M) but not a distribution, and the quotient is nevertheless a smooth NQ-manifold (even though not concentrated in degrees 0 and 1).   The relevance of IM-foliations is that they are in bijective correspondence with morphic foliations on A and -when A is integrable -with multiplicative foliations on the source simply connected Lie groupoid integrating A [8]. We now show that they are also in correspondence with involutive distributions on A [1] which are preserved by Q. The space D 1 of degree 1 elements of D can be described as Using this we show that [Q, D 0 ] ⊂ D 1 : let X 0 ∈ D 0 and a ∈ Γ ∇ (A). We have The first term on the r.h.s. lies in D 0 since [X 0 , a] ∈ D −1 . The second term on the r.h.s. is [ad a , X 0 ], and is seen to lie in D 0 using i),ii) and iv) of Def. 2.11. Last, we prove eq. (9): the inclusion "⊂" follows immediately from D 1 = C 1 (M)D 0 . For the opposite inclusion, fix locally a frame a i of A consisting of elements of Γ ∇ (A), and denote by ξ i ∈ Γ(A * ) = C 1 (M) the dual frame. Any P ∈ χ 1 (M) can be written as This "Taylor expansion" identity is proven noticing that any P ∈ χ 1 (M) is determined by the values of [P, a i ] for all i (this is clear in coordinates), and checking by direct computation that these values coincide for both sides of the identity. Now, if P belongs to the r.h.s. of eq. (9), then X i ∈ D 0 and b ik ∈ Γ(B) = D −1 , showing that P ∈ C 1 (M)D 0 = D 1 .

Actions
In this section we consider distributions that arise from certain kinds of (infinitesimal) actions. In §3.1 we define (non-strict) actions of strict Lie-2 algebras on NQ-1 manifolds. In §3.2 we notice that such actions do not define involutive distributions in general. We give a sufficient condition for this to happen, and use Prop. 3.3 to perform reduction. Some examples are presented in §3.3.

Actions on NQ-1 manifolds
Recall that an L ∞ -algebra 5 is a graded vector space L = i∈Z L i endowed with a sequence of multi-brackets (n ≥ 1) [. . . ] n : ∧ n L → L of degree 2 − n, satisfying the quadratic relations specified in [11,Def. 2.1] 6 . Here ∧ n L denotes the n-th graded skew-symmetric product of L. When [. . . ] n = 0 for n ≥ 3 we recover the notion of DGLA (Def. 1.9), which is the one of interest in this note. An L ∞ -morphism φ : L L ′ between L ∞ -algebras is a sequence of maps (n ≥ 1) φ n : ∧ n L → L ′ of degree 1 − n, satisfying certain relations (see [11,Def. 5.2] in the case when L ′ is a DGLA).  From now on we will restrict ourselves to the case in which M is an NQ-1 manifold and (L = L −1 ⊕ L 0 , δ := [·] 1 , [·, ·] := [·, ·] 2 ) is a strict Lie 2-algebra. One motivation for having L concentrated in degrees −1 and 0 is that if a DGLA acts strictly and almost freely on M (see Def. 3.7) then it must be subject to this degree constraint.
An action φ : is an L ∞ -morphisms between DGLAs. We spell out its components. By degree reasons, the only non-zero components of φ are φ 1 =: µ and φ 2 =: η. More explicitly, subject to the constraints as well as an equation for x ∧ y ∧ z ∈ ∧ 3 L 0 : Notice that condition (11) says that µ is a map of complexes, (12) says that µ| L 0 is a morphism of Lie algebras up to homotopy, and (13) says that µ| L −1 is a morphism of Lie modules up to homotopy.

Quotients by actions
We define the distribution associated to an action of a strict Lie 2-algebra, and study the corresponding quotient.
All along this subsection we consider a strict Lie 2-algebra L (Def be an action as in (10). Notice that the C(M)-submodule of χ(M ) generated by µ(L) and η(∧ 2 L 0 ) is usually not a distribution on M (this problem arises already in the case of Lie algebra actions on ordinary manifolds). Moreover it has two defects: first it is not involutive (as suggested by eq. (12)), and second the operator d Q := [Q, ·] does not preserve its sections. A counterexample for both defects is given in Ex. 3.9 below. This hints to a possible up-tohomotopy version of the concept of involutivity, which we delay to later investigation. For the moment, we might consider a "completion" of the above C(M)-module: Unfortunately, the fact that φ is an L ∞ -morphism (the constraints (11)-(14)) does not imply that D is involutive. A counterexample is given by Ex. 3.10 below. Making an additional assumption, we can achieve that D is involutive and Q-invariant: Proof. i) follows from the facts that µ satisfies eq. (11) and that d 2 Q = 0. ii) Since [D 0 , D −1 ] ⊂ D −1 by assumption, and [D −1 , D −1 ] ⊂ D −2 = {0} by degree reasons, we just need to check that D 0 is closed under the bracket, i.e., we just need to consider µ(L 0 ) and d Q (η(∧ 2 L 0 )). Let x, y ∈ L 0 and m, n ∈ L 0 ∧ L 0 . We have [µ(x), µ(y)] ∈ D 0 by eq. (12). Further Proof. M/D is an NQ-1 manifold by Prop. 2.9, whose assumptions are satisfied because of Prop. 3.5. By the same Prop. 2.9 M/D corresponds to the quotient of A by the ideal system associated to D as in Prop. 2.8, which is the above ideal system.
We specialize further the action: When the action is strict, we showed in [20, §2.3.2] that there is an induced action Ψ : (L −1 ⋊ G) × A → A, which furthermore is an LA-group action 8 . The latter is called a morphic action [18, Def. 3.0.14]. From [18, Thm. 3.2.1] in Stefanini's thesis, it follows that, when the action Ψ is free and proper, A/(L −1 ⋊ G) → M/G is again a Lie algebroid with the property 9 that the projection of A onto the quotient is a Lie algebroid morphism.
Corollary 3.8. Let L be a strict Lie-2 algebra, M := A[1] a NQ-1 manifold, and let µ : L → χ(M) be a strict action. Assume that the action is locally free. Further assume that the induced Lie group action ψ : G × A → A obtained restricting Ψ is free and proper.
Then the Lie algebroid corresponding to the NQ-1 manifold M/D agrees with Stefanini's quotient of A by the LA-group action Ψ.
Proof. We have D = Span C(M) {µ(L)}. By the local freeness assumption, the image under µ of a basis of L provides a set of local homogeneous generators of D whose evaluations at points of M are linearly independent, hence D is a distribution. D is involutive since µ preserves brackets. The leaves of the distribution F on M are just the orbits of the free action ψ| M (the restriction of the G-action ψ to M ), so M/F is a smooth manifold. Let B = span{µ(L −1 )}, then the holonomy of the partial connection ∇ is given by the action of G on A/B induced by ψ, so by the freeness of ψ we conclude that the holonomy of ∇ is trivial.
Hence we can apply Cor. 3.6. The vector bundle obtained quotienting A by the ideal system of Cor. 3.6 agrees with the quotient of A by the action Ψ, and the induced Lie algebroid structures agree because in both cases the projection map from A is a Lie algebroid morphism.

Examples
We present two examples of actions (as in Def. 3.1) of a strict Lie 2-algebra L on an NQ-1 manifold M.
The first is an example where the image of the action is a distribution, which however is neither involutive nor preserved by [Q, ·].
Second, we display an example where the C(M)-module D is not involutive.