On the (3, N) Maurer-Cartan equation
Mauricio Angel, Jaime Camacaro and Rafael Dı́az
Abstract
Deformations of the 3-differential of 3-differential graded algebras are controlled by the
(3, N) Maurer-Cartan equation. We find explicit formulae for the coefficients appearing in
that equation, introduce new geometric examples of N -differential graded algebras, and use
these results to study N Lie algebroids.
AMS Subject Classification: 53B99, 18G99, 18G99.
Keywords: Lie algebroids, N -complexes, Higher differentials.
1 Introduction
In this work we study deformations of the N -differential of a N -differential graded algebra.
According to Kapranov [18] and Mayer [24, 25] a N -complex over a field k is a Z-graded k-
vector space V =
n∈Z Vn together with a degree one linear map d : V −→ V such that
dN = 0. Remarkably, there are at least two generalizations of the notion of differential graded
algebras to the context of N -complexes. A choice, introduced first by Kerner in [20, 21] and
further studied by Dubois-Violette [13, 14] and Kapranov [18], is to fix a primitive N -th root
of unity q and define a q-differential graded algebra A to be a Z-graded associative algebra
together with a linear operator d : A −→ A of degree one such that d(ab) = d(a)b + qāad(b)
and dN = 0. There are several interesting examples and constructions of q-differential graded
algebras [1, 2, 6, 8, 9, 15, 16, 19, 21].
We work within the framework of N -differential graded algebras (N -dga) introduced in [4].
This notion does not depend on the choice of a N -th primitive root of unity, and thus it is better
adapted for differential geometric applications. A N -differential graded algebra A consist of a
Z-graded associative algebra A =
n∈Z An together with a degree one linear map d : A −→ A
such that dN = 0 and d(ab) = d(a)b + (−1)āad(b) for a, b ∈ A. The main question regarding
this definition is whether there are interesting examples of N -differential graded algebras. Much
work still needs to be done, but already a variety of examples has been constructed in [4, 5].
These examples may be classified as follows:
• Deformations of 2-dga into N -dga. This is the simplest and most direct way to construct
N -differential graded algebras. Take a differential graded algebra A with differential d
and consider the deformed derivation d+e where e : A −→ A is a degree one derivation. It
http://arxiv.org/abs/0704.0824v3
is possible to write down explicitly the equations that determine under which conditions
d+ e is a N -differential, and thus turns A into a N -differential graded algebra. In other
words one can explicitly write down the condition (d+ e)N = 0.
• N flat connections. Let E be a vector bundle over a manifold M provided with a flat
connection ∂E . Differential forms on M with values in End(E) form a differential graded
algebra. An End(E)-valued one form T determines a deformation of this algebra into a
N -differential graded algebra with differential of the form ∂E + [T, ] if and only if T is a
N -flat connection, i.e., the curvature of T is N -nilpotent.
• Differential forms of depth N ≥ 3. Attached to each affine manifold M there is a
(dim(M)(N − 1) + 1)-differential graded algebra ΩN (M), called de algebra of differential
forms of depth N on M , constructed as the usual differential forms allowing higher order
differentials, i.e., for affine coordinates xi on M , there are higher order differentials d
for 1 ≤ j ≤ N − 1.
• Deformations of N -differential graded algebras into M -differential graded algebras. If we
are given a N -differential graded algebra A with differential d, one can study under which
condition a deformed derivation d + e, where e is a degree one derivation of A, turns
A into a M -differential graded algebra, i.e., one can determine conditions ensuring that
(d + e)M = 0. In [4] we showed that e must satisfy a system of non-linear equations,
which we called the (N,M) Maurer-Cartan equation.
• Algebras AN∞. This is not so much an example of N -differential graded algebras but rather
a homotopy generalization of such notion. AN∞ algebras are studied in [7].
This paper has three main goals. One is to introduce geometric examples of N - differential
graded algebras. We first review the constructions of N -differential graded algebras outlined
above and then proceed to consider the new examples:
• Differential forms on finitely generated simplicial sets. We construct a contravariant func-
tor ΩN : set
∆op −→ N ildga from the category of simplicial sets generated in finite dimen-
sions to N ildga, the category of nilpotent differential graded algebras, i.e., N -differential
graded algebras for some N ≥ 1. For a simplicial set s we let ΩN (s) be the algebra of
algebraic differential forms of depth N on the algebro-geometric realization of s. For each
integer K we define functor Sing≤K : Top −→ set
∆op , thus we obtain contravariant func-
tors ΩN ◦Sing≤K : Top −→ N
ildga assigning to each topological space X a nil-differential
graded algebra.
• Difference forms on finitely generated simplicial sets. We construct a contravariant functor
DN defined on set
∆op with values in a category whose objects are graded algebras which
are also N -complexes for some N , with the N -differential satisfying a twisted Leibnitz
rule. For a simplicial set s we let DN (s) be the algebra of difference forms of depth N
on the integral lattice in the algebro-geometric realization of s. Again, for each integer
K ≥ 0 we obtain a functor DN ◦ Sing≤K defined on Top assigning to each topological
space X a twisted nil-differential graded algebra.
Our second goal is to study the construction of N -differential graded algebras as defor-
mations of 3-differential graded algebras. Although in [4] a general theory solving this sort
of problem was proposed, our aim here is to provided a solution as explicit as possible. We
consider exact and infinitesimal deformations of 3-differentials in Section 3.
Our final goal in this work is to find applications of N -differential graded algebras to Lie
algebroids. In Section 4 we review the concept of Lie algebroids introduced by Pradines [27],
which generalizes both Lie algebras and tangent bundles of manifolds. A Lie algebroid E may
be defined as a vector bundle together with a degree one differential d on Γ(
E∗). We generalize
this notion to the world of N -complexes, that is we introduce the concept of N Lie algebroids
and construct several examples of such objects.
2 Examples of N-differential graded algebras
In this section we give a brief summary of the known examples of N -dgas and introduce new
examples of N -dgas of geometric nature.
Definition 1. Let N ≥ 1 be an integer. A N -complex is a pair (A, d), where A is a Z-graded
vector space and d : A −→ A is a degree one linear map such that dN = 0.
Clearly a N -complex is also a M -complex for M ≥ N . N -complexes are also referred to
as N -differential graded vector spaces. A N -complex (A, d) such that dN−1 6= 0 is said to be
a proper N -complex. Let (A, d) be a N -complex and (B, d) be a M -complex, a morphism
f : (A, d) −→ (B, d) is a linear map f : A −→ B such that df = fd. One of the most
interesting features of N -complexes is that they carry cohomological information. Let (A, d)
be a N -complex, a ∈ Ai is p-closed if dp(a) = 0, and is p-exact if there exists b ∈ Ai−N+p such
that dN−p(b) = a, for 1 ≤ p < N . The cohomology groups of (A, d) are the spaces
i(A) = Ker{dp : Ai −→ Ai+p}/Im{dN−p : Ai−N+p −→ Ai},
for i ∈ Z and p = 1, 2, ..., N−1.
Definition 2. A N -differential graded algebra (N -dga) over a field k, is a triple (A,m, d) where
m : A⊗A −→ A and d : A −→ A are linear maps such that:
1. dN = 0, i.e., (A, d) is a N -complex.
2. (A,m) is a graded associative algebra.
3. d satisfies the graded Leibnitz rule d(ab) = d(a)b+ (−1)āad(b).
The simplest way to obtain N -differential graded algebras is deforming differential graded
algebras. Let Der(A) be the Lie algebra of derivations on a graded algebra A. Recall that a
degree one derivation d on A, induces a degree one derivation, also denoted by d, on End(A).
Let A be a 2-dga and e ∈ Der(A). It is shown in [4] that e defines a deformation of A into
a N -differential graded algebra if and only if (d + e)N = 0, or equivalently, if and only if the
curvature Fe = d(e) + e
2 of e satisfies (Fe)
2 = 0 if N is even, or (Fe)
2 (d + e) = 0 if N is
odd. For example, consider the trivial bundle M × Rn over M . A connection on M × Rn is a
gl(n)-valued one form a on M , and its curvature is Fa = da +
[a, a]. Let Ω(M,gl(n)) be the
graded algebra of gl(n)-valued forms on M . Thus the pair (Ω(M,gl(n)), d + [a, ]) defines a
N -dga if and only if (Fa)
2 = 0 for N even, or (Fa)
2 (d+ a) = 0 for N odd.
Differential forms of depth N on simplicial sets
Fix an integer N ≥ 3. We are going to construct the (n(N − 1) + 1)-differential graded algebra
ΩN (R
n) of algebraic differential forms of depth N on Rn. Let x1, ..., xn be coordinates on R
and for 0 ≤ i ≤ n and 0 ≤ j < N, let djxi be a variable of degree j. We identify d
0xi with xi.
Definition 3. The (n(N − 1) + 1)-differential graded algebra ΩN (R
n) is given by
• ΩN (R
n) = R[djxi]/
djxid
kxi | j, k ≥ 1
as a graded algebras.
• The (n(N − 1) + 1)-differential d : ΩN (R
n) −→ ΩN (R
n) is given by d(djxi) = d
j+1xi, for
0 ≤ j ≤ N − 2, and d(dN−1xi) = 0.
One can show that d is (n(N − 1) + 1)-differential as follows:
1. It is easy to check that ΩN (R) is a N -dga.
2. If A is a N -dga and B is a P -dga, then A⊗B is a (N + P − 1)-dga.
3. ΩN (R
n) = ΩN (R)
We often write ΩN (x1, ..., xn) instead of ΩN (R
n) to indicate that a choice of affine coordi-
nates (x1, ..., xn) on R
n has been made.
Let ∆ be the category such that its objects are non-negative integers; morphisms in ∆(n,m)
are order preserving maps f : {0, ..., n} −→ {0, ...,m}. The category of simplicial sets Set∆
the category of contravariant functors ∆ −→ Set. Explicitly, a simplicial set s : ∆op −→ Set is
a functorial correspondence assigning:
• A set sn for each integer n ≥ 0. Elements of sn are called simplices of dimension n.
• A map s(f) : sm −→ sn for each f ∈ ∆(n,m).
Let Aff be the category of affine varieties, and let A : ∆ −→ Aff be the functor sending
n ≥ 0, into the affine variety A(x0, ..., xn) = ∆n = {(x0, ..., xn) ∈ R
n | x0 + ... + xn = 1}. A
sends f ∈ ∆(n,m) into A(f) : A(x0, ..., xn) → A(x0, ..., xm) given by A(f)
∗(xj) =
f(i)=j xi,
for 0 ≤ j ≤ m. Forms of depth N on the cosimplicial affine variety A are defined by the functor
ΩN : ∆
op −→ N ildga sending n ≥ 0 into
ΩN (n) = ΩN (x0, ..., xn)/ 〈x0 + ...+ xn − 1, dx0 + ...+ dxn〉 .
A map f ∈ ∆(n,m) induces a morphisms ΩN (f) : ΩN (m) −→ ΩN (n) given for 0 ≤ j ≤ m by
ΩN (f)(xj) =
f(i)=j
xi and ΩN (f)(dxj) =
f(i)=j
Let set∆
be the full subcategory of Set∆
whose objects are simplicial sets generated in
finite dimensions, i.e., simplicial sets s for which there is an integer K such that for each p ∈ si,
i ≥ K, there exists q ∈ sj, j ≤ K, with p = s(f)(q) for some f ∈ ∆(p, q). We are ready to
define the contravariant functor ΩN : set
∆op −→ N ildga announced in the introduction. The
nil-differential graded algebra ΩN (s) =
i=0 Ω
N (s) associated with s is given by
ΩiN (s) = {a ∈
ΩiN (n) | as(f)(p) = ΩN (f)(ap) for p ∈ sm and f ∈ ∆(n,m)}.
A natural transformation l : s −→ t induces a map ΩN(l) : ΩN (t) −→ ΩN (s) given by the rule
[ΩN (l)(a)]p = al(p) for a ∈ ΩN (t) and p ∈ sn.
For each integer K ≥ 0 there is functor ( )≤K : Set
∆op −→ set∆
sending a simplicial set
s, into the simplicial set s≤K generated by simplices in s of dimension lesser or equal to K.
The singular functor Sing : Top −→ Set∆
sends a topological space X into the simplicial set
Sing(X) such that
Singn(X) = {f : ∆n −→ X | f is continous }.
Thus, for each pair of integers N and K we have constructed a functor
ΩN ◦ ( )≤K ◦ Sing : Top −→ N
ildga
sending a topological space X into the nil-differential graded algebra ΩN (Sing≤K(X)).
Difference forms of depth N on simplicial sets
Next we construct difference forms of higher depth on finitely generated simplicial sets. Dif-
ference forms on discrete affine space were introduced by Zeilberger in [28]. We proceed to
construct a discrete analogue of the functors from topological spaces to nil-differential graded
algebras introduced above. First, we construct DN (Z
n) the algebra of difference forms of depth
N on Zn. Let F (Zn,R) be the algebra concentrated in degree zero of R-valued functions on the
lattice Zn. Introduce variables δjmi of degree j for 1 ≤ i ≤ n and 1 ≤ j < N . The graded
algebra of difference forms of depth N on Zn is given by
DN (Z
n) = F (Zn,R)⊗ R[δjmi]/
δjmiδ
kmi | j, k ≥ 1
A form ω ∈ DN (Z
n) can be written as ω =
I ωIdmI where I : {1, .., n} −→ N is any map
and dmI =
i=1 d
I(i)mi. The degree of dmI is |I| =
i=1 I(i). The finite difference ∆i(g) of
g ∈ F (Zn,R) along the i-direction is given by
∆i(g)(m) = g(m+ ei)− g(m),
where the vectors ei are the canonical generators of Z
n and m ∈ Zn. The difference operator δ
is defined for 1 ≤ j ≤ N − 2 by the rules
δ(g) =
∆i(g)δmi, δ(δ
jmi) = δ
j+1mi and δ(δ
N−1mi) = 0.
It is not difficult to check that if ω =
I ωIdmI , then δω =
J(δω)JdmJ where
(δω)J =
J(i)=1
(−1)|J<i|∆iωJ−ei +
J(i)≥2
(−1)|J<i|ωJ−ei .
From the later formula we see that (δω)J is a linear combination of (differences of) functions ωK
with |K| < |J |. This fact implies that δ is nilpotent, indeed, one can check that δn(N−1)+1 = 0.
All together we have proved the following result.
Theorem 4. DN (Z
n) is a graded algebra and the difference operator δ gives DN (Z
n) the
structure of a (n(N − 1) + 1)-complex.
One can check that δ satisfies a twisted Leibnitz rule, so DN (Z
n) is actually pretty close of
being a N -dga. Let Zn,1 ⊆ Zn+1 consists of tuples (m0, ...,mn) such that m0 + ... + mn = 1.
Consider the functor DN defined on ∆
op sending n ≥ 0 into
DN (n) = F (Z
n,1,R)⊗
δjmiδ
kmi, δm0 + ...+ δmn
A map f ∈ ∆(n, k) induces a morphisms DN (f) : DN (k) −→ DN (n) given for g ∈ F (Z
k,1,R)
and 0 ≤ j ≤ k by
DN (f)(g)(m0, ...,mn) = g
f(i)=0
mi, ...,
f(i)=k
 and DN (f)(δmj) =
f(i)=j
We extend DN to the functor defined on set
∆op sending a finitely generated simplicial set s into
DN (s) =
i=0 D
N (s) where
DiN (s) =
DiN (n) | as(f)(p) = DN (f)(ap) for p ∈ sk and f ∈ ∆(n, k)
A natural transformation l : s −→ t induces a map ΩN (l) : ΩN (t) −→ ΩN (s) by the rule
[DN (l)(a)]p = al(p) for a ∈ DN (t) and p ∈ sn. Thus for given integers N and K we have
constructed a functor DN ◦ ( )≤K ◦ Sing on Top sending a topological space X into a sort
of nil-differential graded algebra satisfying a twisted Leibnitz rule DN (Sing≤K(X)) . It would
be interesting to compute the cohomology groups of the algebra of difference forms of higher
depth on known simplicial sets. Even in the case of forms of depth 2 these groups have seldom
been studied.
3 On the (3,N) curvature
Recall that a discrete quantum mechanical system is given by the following data:
1. A directed graph with set of vertices V and set of directed edges E. The Hilbert space of
the system is H = CV .
2. A map ω : E −→ R assigning a weight to each edge.
3. Operators Un : H −→ H for n ∈ N given by (Unf)(y) =
ωn(y, x)f(x) where the
discretized kernel ωn(y, x) is given by
ωn(y, x) =
γ∈Pn(x,y)
ω(e).
Pn(x, y) denotes the set of paths in Γ from x to y of length n, i.e., sequences (e1, · · · , en)
of edges such that s(e1) = x, t(ei) = s(ei+1), for i = 1, ..., n − 1 and t(en) = y.
Let us introduce some notation. For s = (s1, ..., sn) ∈ N
n we set l(s) = n and |s| =
i si.
For 1 < i ≤ n we set s<i = (s1, ..., si−1); also we set s>n = s<1 = ∅. N
(∞) is equal to
where by convention N(0) = {∅}. Let A be a 3-dga and e be a degree one derivation on A. For
s ∈ Nn we let e(s) = e(s1) · · · e(sn), where e(l) = dl(e) if l ≥ 1, e(0) = e and e∅ = 1. For N ∈ N,
we set EN =
s ∈ N(∞) | s 6= ∅ and |s|+ l(s) ≤ N
and for s ∈ EN we let N(s) ∈ Z be given
by N(s) = N − |s| − l(s).
The following data defines a discrete quantum mechanical system:
1. The set of vertices is N(∞).
2. There is a unique directed edge from s to t if and only if t ∈ {(0, s), s, (s + ei)}, where
ei ∈ N
l(s) are the canonical vectors.
3. Edges are weighted according to the table:
source target weight
s (0, s) 1
s s (−1)|s|+l(s)
s (s+ ei) (−1)
|s<i|+i−1
PN (∅, s) consists of paths γ = (e1, ..., eN ), such that s(e1) = ∅, t(eN ) = s and s(el+1) = t(el).
The weight ω(γ) of a path γ ∈ PN (∅, s) is given by ω(γ) =
i=1 ω(ei). The following result,
proved in [4], tell us when d+ e defines a deformation of a 3-dga into a N -dga.
Theorem 5. d+ e defines a deformation of the 3-dga A into a N -dga if and only if the (3, N)
Maurer-Cartan equation holds co + c1d+ c2d
2 = 0, where for 0 ≤ k ≤ 2 we set
N(s)=k
c(s,N)e(s) and c(s,N) =
γ∈PN (∅,s)
ω(γ).
Exact deformations
Let us first consider the deformation of a 3-dga into a 3-dga. According to Theorem 5 the
derivation d+ e defines a 3-dga if and only if co + c1d+ c2d
2 = 0 where
N(s)=k
c(s, 3)e(s) and c(s, 3) =
γ∈P3(∅,s)
ω(γ).
Let us compute the coefficients ck. We have that
E3 = {(0), (1), (2), (0, 0), (1, 0), (0, 1), (0, 0, 0)} .
Let us first compute c0. There are four vectors s in E3 such that N(s) = 0, these are
(2), (1, 0), (0, 1) and (0, 0, 0). The only path from ∅ to (2) of length 3 is
∅ −→ (0) −→ (1) −→ (2)
of weight 1. Since e(2) = d2(e), then we have that c((2), 3) = d2(e). The unique path from ∅ to
(1, 0) of length 3 is
∅ −→ (0) −→ (0, 0) −→ (1, 0)
of weight 1. Since e(1,0) = d(e)e we have that c((1, 0), 3) = d(e)e. There are two paths from ∅
to (0, 1) of length 3, namely
∅ −→ (0) −→ (0, 0) −→ (0, 1)
∅ −→ (0) −→ (1) −→ (0, 1)
of weight −1 and 1, respectively. Thus c((1, 0), 3) = 0 since the sum of the weights vanishes.
The unique path from ∅ to (0, 0, 0) of length 3 is
∅ −→ (0) −→ (0, 0) −→ (0, 0, 0)
of weight 1. Since e(0,0,0) = e3, then c((0, 0, 0), 3) = e3. Thus we have shown that
c0 = d
2(e) + d(e)e + e3.
We proceed to compute c1. The vectors in E3 such that N(s) = 1 are (1) and (0, 0). Paths
from ∅ to (1) of length 3 are
∅ −→ ∅ −→ (0) −→ (1)
∅ −→ (0) −→ (0) −→ (1)
∅ −→ (0) −→ (1) −→ (1)
of weight 1, −1 and 1, respectively. Since e(1) = d(e), then c((1), 3) = d(e). Paths from ∅ to
(0, 0) of length 3 are
∅ −→ (0) −→ (0) −→ (0, 0)
∅ −→ ∅ −→ (0) −→ (0, 0)
∅ −→ (0) −→ (0, 0) −→ (0, 0).
The corresponding weights are, respectively, −1, 1 and 1. We have that e(0,0) = e2, thus
c((0, 0), 3) = e2 and c1 = d(e) + e
Finally we compute c2. (0) is the only vector in E3 such that N(s) = 2. The paths from ∅
to (0) of length 3 are
∅ −→ ∅ −→ ∅ −→ (0)
∅ −→ ∅ −→ (0) −→ (0)
∅ −→ (0) −→ (0) −→ (0).
The corresponding weights are, respectively, 1, −1 and 1. Since e(0) = e then c2 = c((0), 3) =
e. Altogether we have proven the following result.
Theorem 6. d+ e defines a deformation of the 3-dga A into a 3-dga if and only if
(d2(e) + d(e)e + e3) + (d(e) + e2)d+ ed2 = 0.
Consider now deformations of a 3-dga into a 4-dga. Again by Theorem 5 we must have
c0 + c1d+ c2d
2 = 0. We proceed to compute the coefficients ck. We have that
E4 = {(0), (1), (2), (0, 0), (1, 0), (0, 1), (2, 0), (0, 2), (1, 1),
(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 0, 0, 0)}.
(0) is the only vector in E4 such that N(s) = 3. Paths of length 4 from ∅ to (0) are of the
form ∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
with weight (−1)j , where i + j = 3, thus we have that
c3 = (1− 1 + 1− 1)e = 0.
We compute c2. Vectors in E4 with N(s) = 2 are (0, 0) and (1). Paths from ∅ to (0, 0)
of length 2 are of the form ∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
→ (0, 0) → · · · → (0, 0)︸ ︷︷ ︸
of weight
i+j+k=2(−1)
j = 2, thus c((0, 0), 4)e(0,0) = 2e2. Paths from ∅ to (1) of length 2 are of the
form ∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
→ (1) → · · · → (1)︸ ︷︷ ︸
with weight
i+j+k=2(−1)
j = 2, thus
c((1), 4)e(1) = 2d(e) and c2 = 2(e
2 + d(e)).
Let us now compute c1. Vectors in E4 with N(s) = 1 are (0, 0, 0), (1, 0), (0, 1) and (2).
Paths from ∅ to (0, 0, 0) are of 5 types. Paths of the form
∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
→ (0, 0) → · · · → (0, 0)︸ ︷︷ ︸
→ (0, 0, 0) → · · · → (0, 0, 0)︸ ︷︷ ︸
with weight
i+j+k+l=1(−1)
j(−1)l, so that c((0, 0, 0), 4)e(0,0,0) = (1− 1 + 1− 1)e3 = 0. Paths
of the form
∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
→ (1) → · · · → (1)︸ ︷︷ ︸
→ (0, 1) → · · · → (0, 1)︸ ︷︷ ︸
with weight ∑
i+j+k+l=1
(−1)j(−1)l = 1− 1 + 1− 1 = 0 .
Path of the form
∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
→ (0, 0) → · · · → (0, 0)︸ ︷︷ ︸
→ (0, 1) → · · · → (0, 1)︸ ︷︷ ︸
of weight
i+j+k+l=1(−1)
j(−1)(−1)l so that
c((0, 1), 4)e(0,1) = ((1 − 1 + 1− 1) + (1− 1 + 1− 1))ed(e) = 0.
Paths of the form
∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
→ (0, 0) → · · · → (0, 0)︸ ︷︷ ︸
(1, 0) → · · · → (1, 0)︸ ︷︷ ︸
of weight
i+j+k+l=1(−1)
j(−1)l, thus c((1, 0), 4)e(1,0) = (1 − 1 + 1 − 1)d(e)e = 0. There are
also paths of the form
∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
→ (1) → · · · → (1)︸ ︷︷ ︸
→ (2) → · · · → (2)︸ ︷︷ ︸
of weight
i+j+k+l=1(−1)
j(−1)l, so we have c((2), 4)e(2) = (1 − 1 + 1− 1)d2(e) = 0. We have
shown that
c1 = c((0, 0, 0), 4)e
(0,0,0) + c((0, 1), 4)e(0,1) + c((1, 0), 4)e(1,0) + c((2), 4)e(2) = 0.
Let us compute c0. There are several types of paths in this case. Path
∅ −→ (0) −→ (0, 0) −→ (0, 0, 0) −→ (0, 0, 0, 0)
of weight 1, thus cq((0, 0, 0, 0), 4)a
(0,0,0,0) = a4. Paths
∅ −→ (0) −→ (0, 0) −→ (0, 0, 0) −→ (0, 0, 1)
∅ −→ (0) −→ (1) −→ (0, 1) −→ (0, 0, 1)
∅ −→ (0) −→ (0, 0) −→ (0, 1) −→ (0, 0, 1)
of weight 1, thus we have that c((0, 0, 1), 4)e(0,0,1) = e2d(e). Paths
∅ −→ (0) −→ (0, 0) −→ (1, 0) −→ (0, 1, 0)
∅ −→ (0) −→ (0, 0) −→ (0, 0, 0) −→ (0, 1, 0)
of weight 0, thus c((0, 1, 0), 4)e(0,1,0) = 0ad(a)a = 0. Path
∅ −→ (0) −→ (0, 0) −→ (0, 0, 0) −→ (1, 0, 0)
of weight 1, thus c((1, 0, 0), 4)e(1,0,0) = d(e)e2. Path
∅ −→ (0) −→ (0, 0) −→ (1, 0) −→ (2, 0)
of weight 1, so c((2, 0), 4)e(2,0) = d2(e)e. Paths
∅ −→ (0) −→ (0, 0) −→ (0, 1) −→ (0, 2)
∅ −→ (0) −→ (1) −→ (0, 1) −→ (0, 2)
∅ −→ (0) −→ (1) −→ (2) −→ (0, 2)
of weight 1, so that c((0, 2), 4)e(0,2) = ed2(e). There are also paths
∅ −→ (0) −→ (0, 0) −→ (1, 0) −→ (1, 1)
∅ −→ (0) −→ (1) −→ (0, 1) −→ (1, 1)
of weight 2, so that c((1, 1), 4)e(1,1) = (d(e))2. We see that
c0 = e
4 + e2d(e) + d(e)e2 + d2(e)e + ed2(e) + (d(e))2.
All together we have shown the following result.
Theorem 7. d+ e defines a deformation of the 3-dga A into a 4-dga if and only if
(e4 + e2d(e) + d(e)e2 + d2(e)e + ed2(e) + (d(e))2) + 2(e2 + d(e))d2 = 0.
Infinitesimal deformations
Let t be a formal parameter such that t2 = 0.
Theorem 8. Let (A, d) be a N -dga and e a degree one derivation on A, then we have
(d+ te)N = t
p∈Par(k,N−k+1)
(−1)w(p)
 dN−k−1(e)dN−k−1,
where
Par(k,N − k + 1) = {p = (p1, · · · , pN−k+1) |
N−k+1∑
pi = k} and w(p) =
N−k+1∑
(i− 1)pi.
Proof. From Theorem 5 we know that DN =
k=0 ckd
k. Since t2 = 0, then
(te)(s) = (te)(s1) · · · (te)(sl(s)) = tl(s)e(s) = 0
unless l(s) ≤ 1. On the other hand we have that
EN = {(0), (1), · · · , (N − 1)}.
Suppose that l(s) = 1 and N(s) = N − |s| − l(s) = k, thus |s| = N − k − 1. The unique vector
s in EN of length 1 such that |s| = N − k − 1 is s = (N − k − 1). Therefore
N(s)=k
c(s,N)e(s) = c((N − k − 1), N)e(s) = c((N − k − 1), N)dN−k−1(e).
A path from ∅ to (N − k − 1) of length N must be of the form
∅→ · · · →︸ ︷︷ ︸
∅ → (0)→ · · · →︸ ︷︷ ︸
(0) → (1)→ · · · →︸ ︷︷ ︸
(1) → · · · → (N − k − 1)→ · · · →︸ ︷︷ ︸
pN−k+1
(N − k − 1)
with (p1 + 1) + (p1 + 1) + · · ·+ (pN−k + 1) + pN−k+1 = N , i.e.,
∑N−k+1
i=1 pi = k. The weight of
such path is
(−1)0p1(−1)(2−1)p2(−1)(3−1)p2 ...(−1)(N−k)pN−k+1 = (−1)w(p), thus we get that
c((N − k − 1), N) =
γ∈PN (∅,s)
ω(γ) =
p∈Par(k,N−k+1)
(−1)w(p).
Corollary 9. e defines an infinitesimal deformation of theN -dga (A, d) into theN -dga (A, d+e)
if and only if
p∈Par(k,N−k+1)
(−1)w(p)
 dN−k−1(e)dN−k−1 = 0. (1)
4 N Lie algebroids
In this section we introduce the notion of N Lie algebroids and construct examples of such
structures. We first review the notion of Lie algebroids, provide some examples, and write the
definition of Lie algebroids in a convenient way for our purposes.
Lie algebroids
We review basic ideas around the notion of Lie algebroids; the interested reader will find much
more information in [12, 23, 27]. The notion of Lie algebroids has gained much attention in
the last few years because of its interplay with various branches of mathematics and theoretical
physics, see [10, 11, 17]. We center our attention on the basic definitions and constructions of
Lie algebroids and its relation with graded manifolds and differential graded algebras.
Definition 10. A Lie algebroid is a vector bundle π : E −→ M together with:
• A Lie bracket [ , ] on the space Γ(E) of sections of E.
• A vector bundle map ρ : E −→ TM over the identity, called the anchor, such that the
induced map ρ : Γ(E) −→ Γ(TM) is a Lie algebra morphism.
• The identity [v, fw] = f [v,w]+(ρ(v)f)w must hold for sections v,w of E and f a smooth
function on M .
Let (x1, ..., xn) be coordinates on a local chart U ⊂ M , and let {eα | α = 1, . . . , r} be a basis
of local sections of π : E|U −→ U . Local coordinates on E|U are given by (x
i, yα). Locally the
Lie bracket and the anchor are given by [eα, eβ ]E = C
eγ and ρ(eα) = ρ
, respectively.
The smooth functions C
, ρiα are the structural functions of the Lie algebroid. The condition
for ρ to be a Lie algebra homomorphism is written in local coordinates as
= ρiγ C
The other compatibility condition between ρ and [ , ] is given by
cycl(α,β,γ)
+ Cµαν C
where the sum is over indices α, β, γ such that the map 1, 2, 3 −→ α, β, γ is a cyclic permuta-
tion. The simplest examples of Lie algebroids are described below; the reader will find further
examples in the references listed at the beginning of this section.
Example 11. A finite dimensional Lie algebra g may be regarded as a vector bundle over a
single point. Sections are elements of g, the Lie bracket is that of g, and the anchor map is
identically zero. The structural functions C
are the structural constants c
of g and ρiα = 0.
Example 12. The tangent bundle π : TM −→ M with anchor the identity map ITB on TB
and with the usual bracket on vector fields.
Exterior differential algebra of Lie algebroids
Sections Γ(
E) of a Lie algebroid E play the rôle of vector fields on a manifold and are called
E vector fields. Sections of the dual bundle π : E∗ −→ M are called E 1-forms. Similarly
sections Γ(
E∗) of
E∗ are called E forms. The degree of a E form in Γ(
E∗) is k. Let us
state and sketch the proof of a result of fundamental importance for the rest of this work.
Theorem 13. Let E be a vector bundle. E is a Lie algebroid if and only if Γ(
E∗) is a
differential graded algebra. A differential on
E∗ is the same as a degree one vector field v on
E[−1] such that v2 = 0.
Above E[−1] denotes the graded manifold whose underlying space is E with fibers placed
in degree one. If E is a Lie algebroid one defines a differential
d : Γ(∧kE∗) −→ Γ(∧k+1E∗)
as follows:
dθ(v1, . . . , vk+1) =
(−1)i+1ρ(vi)θ(v1, . . . , v̂i, . . . , vk+1)
(−1)i+jθ([vi, vj ], v1, . . . , v̂i, . . . , v̂j , . . . vk+1),
for v1, . . . , vk+1 ∈ Γ(E). The axioms for a Lie algebroid imply that:
1. d2 = 0;
2. If f ∈ C∞(M) and v ∈ Γ(E), then 〈df, v〉 = ρ(v)f ;
3. d is a derivation of degree 1, i.e., d(θ ∧ ζ) = dθ ∧ ζ + (−1)θθ ∧ ζ.
Conversely, assume that d is a degree one derivation on Γ(
E∗) satisfying d2 = 0. Then E
is a Lie algebroid with the structural maps ρ and [ , ] given by
ρ(v)f = df(v) ,
θ([v,w]) = ρ(v)θ(w) − ρ(w)θ(v)− dθ(v,w),
for v,w ∈ Γ(E), f ∈ C∞(M) and θ ∈ Γ(
E). In local coordinates d is determined by
dxi = ρiα e
α and deγ = C
eα ∧ eβ,
where {eα | α = 1, . . . , r} is the dual basis of {eα | α = 1, . . . , r}. It is not hard to see that
the conditions d2xi = 0 and d2eα = 0 are equivalent to the structural equations defining a Lie
algebroid. Let us compute the exterior algebra of a few Lie algebroids.
Example 14. To the trivial Lie algebroid structure on a vector bundle E corresponds to the
exterior algebra
E∗ with vanishing differential.
Example 15. Chevalley-Eilenberg differential on
∗ arises from the Lie algebroid g −→ {•}
of Example 11. The Chevalley-Eilenberg differential d takes the form
dθ(v1, . . . , vk+1) =
(−1)i+jθ([vi, vj ], v1, . . . , v̂i, . . . , v̂j , . . . vk+1),
for vi ∈ g and θ ∈
Example 16. The differential associated with the tangent bundle TM −→ M Lie algebroid is
de Rham differential.
N Lie algebroids
We are ready to introduce the main concept of this section. In the light of Theorem 13 it is
rather natural to define a N Lie algebroid as a vector bundle E together with a degree one
N -nilpotent vector field v on the graded manifold E[−1]. That definition, useful as it might
be, rules out some significant examples that we would not like to exclude, thus, we prefer the
more inclusive definition given below. Though not strictly necessary for our definition of N Lie
algebroids, the study of nilpotent vector fields on graded manifolds is of independent interest,
and we shall say a few words about them. Indeed our next result gives an explicit formula for
the N -th power of a graded vector field.
Let x1, ..., xm be local coordinates on a graded manifold and ∂1, ..., ∂m be the corresponding
vector fields. We recall that if xi is a variable of degree xi, then ∂i is of degree −xi, and dxi
is of degree xi + 1. Let a
1, ..., am be functions of homogeneous degree depending on x1, ..., xm.
For L a linearly ordered set and f : L −→ [m] a map we define
f(i) and ∂f =
∂f(i).
Also we define the sign s(f) by the rule
∂f = s(f)∂
|f−1(1)|
1 ... ∂
|f−1(m)|
Let p : N −→ Z2 be the map such that p(n) is 1 if n is even and −1 otherwise. Using induction
on N one can show that:
Theorem 17.
(ai∂i)
s(f, α)(
α−1(i)
af(i))∂f |
α−1(N+1)⊔N
where the sum runs over f : [N ] −→ [m] and α : [N − 1] −→ [2, N + 1] such that α(i) > i. The
sign s(f, α) is given by
s(f, α) = p(
s<j<α(s)
f(j) + xsf |α−1(j)∩[s+1,N−1] ).
Corollary 18.
(ai∂i)
cI∂I ,
where I : [m] −→ N is such that 1 ≤ |I| := I(1) + ...+ I(N) ≤ N , ∂I =
i=1 ∂
i , and
S(f, α)
(∂f(α−1(i))a
f(i))
where the sum runs over maps α : [N − 1] −→ [2, N + 1] with α(i) > i for i ∈ [N − 1], and
f : [N ] −→ [m] such that |{j ∈ α−1(N + 1) ⊔ {N} | f(j) = i}| = I(i), for i ∈ [m]. The sign
S(f, α) is given by
S(f, α) = s(f, α)s(f |α−1(N+1)⊔{N}) .
Corollary 19. (ai∂i)
N = 0 if and only if cI = 0 for I as above.
For example for N = 2 one gets
(ai∂i)
p(xiaj)aiaj∂i∂j + ai∂i(aj)∂j .
For N = 3 we get that
(ai∂i)
i,j,k
ai∂i(aj)∂j(ak)∂k + p(xiaj)aiaj∂i∂j(ak)∂k
+ p(xiak)aiaj∂j(ak)∂i∂k + p(xjak)ai∂i(aj)ak∂j∂k
+ xiaj)aiaj∂i(ak)∂j∂k + p(xjak + xiaj + xiak)aiajak∂i∂j∂k.
For N = 4 the corresponding expression have 24 terms and we won’t spell it out.
We return to the problem of defining N Lie algebroids. We need some general remarks on
differential operators on associative algebras. Given an associative algebra A we let DO(A)
be the algebra of differential operators on A, i.e., the subalgebra of End(A) generated by
A ⊂ End(A) and Der(A) ⊂ End(A), the space of derivations of A. Thus DO(A) is generated
as a vector space by operators of the form x1 ◦x2 ◦ · · · ◦xn ∈ End(A) where xi is in A⊔Der(A).
Notice that DO(A) admits a natural filtration
∅ = DO≤−1(A) ⊆ DO≤0(A) ⊆ DO≤1(A) ⊆ · · · ⊆ DO≤k(A) ⊆ · · · ⊆ DO(A),
whereDO≤k(A) ⊆ DO(A) is the subspace generated by operators x1◦x2◦· · ·◦xn, where at most
k operators among the xi belong to Der(A). Thus DO(A) admits the following decomposition
as graded vector space
DO(A) =
DOk(A) :=
DO≤k(A)/DO≤k−1(A).
Clearly DO0(A) = A and if A is either commutative or graded commutative, then
DO1(A) = Der(A).
The projection map π1 : DO(A) −→ DO1(A) induces a non-associative product
⋄ : DO1(A)⊗DO1(A) −→ DO1(A)
given by s ⋄ t = π1(s ◦ t) for s, t ∈ DO1(A). In particular if A is commutative or graded
commutative we obtain a non-associative product
⋄ : Der(A)⊗Der(A) −→ Der(A).
To avoid unnecessary use of parenthesis we assume that in the iterated applications of ⋄ we
associate in the minimal form from right to left.
Definition 20. A N Lie algebroid is a vector bundle E together with a degree one derivation
d : Γ(
E∗) −→ Γ(
E∗), such that the result of N ⋄-compositions of d with itself vanishes,
i.e., d ⋄ d ⋄ · · · ⋄ d = 0.
The notions of Lie algebroids and 2 Lie algebroids agree; indeed it is easy to check that
d ◦ d = d ⋄ d for any degree one derivation d : Γ(
E∗) −→ Γ(
E∗). Let us now illustrate with
an example the difference between the condition d◦d◦· · ·◦d = 0 and the much weaker condition
d ⋄ d ⋄ · · · ⋄ d = 0. Let C[x1, ..., xn] be the free graded algebra generated by graded variables xi
for 1 ≤ i ≤ n. A derivation on C[x1, ..., xn] is a vector field ∂ =
ai∂i where ai ∈ C[x1, ..., xn].
The condition ∂N = 0 is rather strong and restrictive, it might be tackled with the methods
provided above. In contrast, the condition ∂ ⋄ ∂ ⋄ · · · ⋄ ∂ = 0 is much simpler and indeed it is
equivalent to the condition ∂N (xi) = 0 for 1 ≤ i ≤ n.
Definition 21. A N Lie algebra is a vector space g together with a degree one derivation d on
∗ such that the N -th ⋄-composition of d with itself vanishes.
Our next result characterizes 3 Lie algebras in more familiar terms. For integers k1, k2, ..., kl
such that k1 + k2 + · · ·+ kl = n, we let Sh(k1, k2, · · · , kl) be the set of permutations
σ : {1, · · · , n} −→ {1, · · · , n}
such that σ is increasing on the intervals [ki + 1, ki+1] for 0 ≤ i ≤ l, k0 = 1 and kl+1 = n.
Assume we are given a map [ , ] :
g −→ g.
Theorem 22. The pair (g, [ , ]) is a 3 Lie algebra if and only if for v1, v2, v3, v4 ∈ g we have
σ∈Sh(2,1,1)
sgn(σ)[[[vσ(1), vσ(2)], vσ(3)]vσ(4)] =
σ∈Sh(2,2)
sgn(σ)[[vσ(1) , vσ(2)], [vσ(3), vσ(4)]],
Proof. One can show that a degree one differential on
∗ is necessarily the Chevalley-Eilenberg
operator
dθ(v1, . . . , vn+1) =
(−1)i+jθ([vi, vj ], v1, . . . , v̂i, . . . , v̂j , . . . vn+1) ,
where [ , ] :
g −→ g is an antisymmetric operator. We remark that we are not assuming,
at this point, that the bracket [ , ] satisfies any further identity. Jacobi identity arises when
the square of d is set to be equal to zero, but we do not do that since we want to investigate
the weaker condition that the third ⋄-power of d be equal to zero. For θ ∈
∗ = g∗ the
Chevalley-Eilenberg operator takes the simple form
dθ(v1, v2) = −θ([v1, v2]).
Moreover a further application of d to dθ yields
d2θ(v1, v2, v3) =
σ∈Sh(2,1)
sgn(σ)θ([[vσ(1), vσ(2)], vσ(3)]).
From the last equation it is evident that Jacobi identity is equivalent to the condition d2 = 0.
We do not assume assume that Jacobi identity holds and proceed to compute the third ⋄-power
of d. We obtain that
d ⋄ d ⋄ dθ(v1, v2, v3, v4) =
σ∈Sh(2,1,1)
sgn(σ)θ([[[vσ(1), vσ(2)], vσ(3)]vσ(4)])
σ∈Sh(2,2)
sgn(σ)θ([[vσ(1), vσ(2)], [vσ(3), vσ(4)]]).
Thus d ⋄ d ⋄ d = 0 if and only if the condition from the statement of the Theorem holds.
Using local coordinates θ1, ..., θm on the graded manifold g[−1], it is not hard to show that
a vector field of degree one on g[−1] can be written as
where the constants C
may be identified with the structural constants of [·, ·]. The square of
the vector field ∂ is given by
Cσδ εθ
Cσγ εθ
αθβθε
Cσδ γθ
αθβθδ
θαθβCσδ εθ
Using the antisymmetry properties of C
and the commutation rules for θα one can write
together the first to terms. We find that
∂ ⋄ ∂ =
Cσγ εθ
αθβθε
The condition ∂ ⋄ ∂ = 0 is equivalent to Jacobi identity. We assume that ∂ ⋄ ∂ 6= 0 and proceed
to compute consider the condition ∂ ⋄ ∂ ⋄ ∂ = 0. We have that
∂ ◦ (∂ ⋄ ∂) =
Cνλµθ
Cσγ εθ
αθβθε
Using carefully the properties of C
and θα we find that
∂ ◦ (∂ ⋄ ∂) =
CνλµC
Cσγ εθ
λθµθβθε
CνλµC
Cσγ νθ
λθµθαθβ
CνλµC
Cσγ εθ
λθµθαθβθε
Therefore we have shown that
∂ ⋄ (∂ ⋄ ∂) =
CνλµC
Cσγ ǫ −
CσγαC
θλθµθβθε
Thus the condition ∂ ⋄ (∂ ⋄ ∂) = 0 is equivalent to the following equations for fixed σ:
λ,µ,β,ε
CνλµC
Cσγ ǫ −
CσγαC
θλθµθβθε = 0 .
Let us now go back to the case of Lie algebroids as opposed to Lie algebras. There is a
natural degree one vector field on the graded manifold T[−1]R
n, namely, de Rham differential.
We now investigate whether it is possible to deform, infinitesimally, de Rham differential into
a 3-differential. In local coordinates (x1, . . . , xn, θ1, . . . , θn) on T[−1]R
n, with xi of degree zero
and θi of degree 1, de Rham operator takes the form
∂ = δiαθ
Let t be a formal infinitesimal parameter such that t2 = 0. We are going to show that any set
of functions aiα of degree zero on T[−1]R
n determine a deformation of de Rham operator into a
3-⋄ nilpotent operator given by
δiα + ta
Theorem 23. ∂a ⋄ ∂a = t
θα θβ
and ∂a ⋄ (∂a ⋄ ∂a) = 0.
Proof.
∂2a =
δiα + ta
θα θβ
θα θβ
θα θβ
Since t2 = 0 the third term on the right hand side of the expression above vanishes. The second
term also vanishes because it is a contraction of even and odd indices. So we get that
∂a ⋄ ∂a = t
θα θβ
The third power of ∂a is given by
∂a ⋄ (∂a ⋄ ∂a) = t
∂xγ∂xα
θγ θα θβ
It also vanishes because it includes a contraction of even and odd indices.
The nilpotency condition for the operator ∂a ⋄∂a is
θα θβ = 0 for j = 1, . . . , n. It is not
hard to find examples of matrices a
such that ∂a ⋄ ∂a = 0, for example

(x4)2
x1 x1
x2 x2 x3 x2
x3 x3 x2 x4
x4 x4 x1 x4 x3

More importantly there are also matrices a
such that ∂a ⋄ ∂a 6= 0, for example

x1 x4 x1 x1 x1
x2 x2 x4 x2 x2
x3 x3 x3 x4 x3
x4 x4 x4 x1 x4

We now consider full deformations as opposed to infinitesimal ones. Let
δiα + a
be a vector field. We think of ∂a as a deformation of de Rham differential with deformation
parameters aiα.
Theorem 24.
∂a ⋄ (∂a ⋄ ∂a) =
δlγ + a
){∂aiα
+ aiα
∂xl∂xi
θγθαθβ
Proof. Since
∂2a =
δiα + a
∂a ⋄ ∂a =
δiα + a
) ∂ajβ
we get
∂a ⋄ (∂a ⋄ ∂a) =
δlγ + a
δiα + a
) ∂ajβ
δlγ + a
){∂aiα
+ aiα
∂xl∂xi
θγθαθβ
Corollary 25. ∂a ⋄ (∂a ⋄ ∂a) = 0 if for fixed indices α, β, λ, j the following identity holds
δlγ + a
){∂aiα
+ aiα
∂xl∂xi
θγθαθβ = 0.
Corollary 26. Each matrix A = (A
) ∈ Mn(R) such that A
2 = 0 determines a 3 Lie algebroid
structure on TRn with differential given by (δiα +A
αxα)dx
Our final result describes explicitly the conditions defining a 3 Lie algebroid. Let E be a
vector bundle over M . A vector field on E[−1] of degree one is given in local coordinates by
∂ = ρiαθ
where ρiα and C
are functions of the bosonic variables only.
Theorem 27. ∂ ⋄ (∂ ⋄ ∂) = 0 if and only if for fixed γ and i the following identity holds:
Cασ µ)
Cλµσ −
CαλµC
CαβµC
θνθσθµθβ = 0 ,
Cǫσν−
Cǫσν + ρ
θσθνθγ = 0 .
Proof. We sketch the rather long proof. For ∂ = ρiαθ
θαθβ ∂
, we have
∂ ⋄ ∂ =
θλθµθβ
As in the previous theorem one finds that the condition ∂ ⋄ (∂ ⋄ ∂) = 0 is equivalent to the
following identities
Cασ µ)
Cλµσ −
Cλνσ −
θνθσθµθβ
= 0 ,
Cǫσν − ρ
θσθνθγ
Needless to say further research is necessary in order to have a better grasp of the meaning
and applications of the notion of N Lie algebroids. We expect that this approach will lead
towards new forms of infinitesimal symmetries, and for that reason alone it should find appli-
cations in various problems in mathematical physics. In our forthcoming work [3] we are going
to discuss some applications of N Lie algebroids in the context of Batalin-Vilkovisky algebras
and the master equation.
Acknowledgment
Thanks to Takashi Kimura, Juan Carlos Moreno and Jim Stasheff.
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mangel@euler.ciens.ucv.ve, jcama@usb.ve, ragadiaz@gmail.com
	Introduction
	Examples of N-differential graded algebras
	On the (3,N) curvature
	 N Lie algebroids

  Deformations of the 3-differential of 3-differential graded algebras are
controlled by the (3,N) Maurer-Cartan equation. We find explicit formulae for
the coefficients appearing in that equation, introduce new geometric examples
of N-differential graded algebras, and use these results to study N Lie
algebroids.

Introduction
In this work we study deformations of the N -differential of a N -differential graded algebra.
According to Kapranov [18] and Mayer [24, 25] a N -complex over a field k is a Z-graded k-
vector space V =
n∈Z Vn together with a degree one linear map d : V −→ V such that
dN = 0. Remarkably, there are at least two generalizations of the notion of differential graded
algebras to the context of N -complexes. A choice, introduced first by Kerner in [20, 21] and
further studied by Dubois-Violette [13, 14] and Kapranov [18], is to fix a primitive N -th root
of unity q and define a q-differential graded algebra A to be a Z-graded associative algebra
together with a linear operator d : A −→ A of degree one such that d(ab) = d(a)b + qāad(b)
and dN = 0. There are several interesting examples and constructions of q-differential graded
algebras [1, 2, 6, 8, 9, 15, 16, 19, 21].
We work within the framework of N -differential graded algebras (N -dga) introduced in [4].
This notion does not depend on the choice of a N -th primitive root of unity, and thus it is better
adapted for differential geometric applications. A N -differential graded algebra A consist of a
Z-graded associative algebra A =
n∈Z An together with a degree one linear map d : A −→ A
such that dN = 0 and d(ab) = d(a)b + (−1)āad(b) for a, b ∈ A. The main question regarding
this definition is whether there are interesting examples of N -differential graded algebras. Much
work still needs to be done, but already a variety of examples has been constructed in [4, 5].
These examples may be classified as follows:
• Deformations of 2-dga into N -dga. This is the simplest and most direct way to construct
N -differential graded algebras. Take a differential graded algebra A with differential d
and consider the deformed derivation d+e where e : A −→ A is a degree one derivation. It
http://arxiv.org/abs/0704.0824v3
is possible to write down explicitly the equations that determine under which conditions
d+ e is a N -differential, and thus turns A into a N -differential graded algebra. In other
words one can explicitly write down the condition (d+ e)N = 0.
• N flat connections. Let E be a vector bundle over a manifold M provided with a flat
connection ∂E . Differential forms on M with values in End(E) form a differential graded
algebra. An End(E)-valued one form T determines a deformation of this algebra into a
N -differential graded algebra with differential of the form ∂E + [T, ] if and only if T is a
N -flat connection, i.e., the curvature of T is N -nilpotent.
• Differential forms of depth N ≥ 3. Attached to each affine manifold M there is a
(dim(M)(N − 1) + 1)-differential graded algebra ΩN (M), called de algebra of differential
forms of depth N on M , constructed as the usual differential forms allowing higher order
differentials, i.e., for affine coordinates xi on M , there are higher order differentials d
for 1 ≤ j ≤ N − 1.
• Deformations of N -differential graded algebras into M -differential graded algebras. If we
are given a N -differential graded algebra A with differential d, one can study under which
condition a deformed derivation d + e, where e is a degree one derivation of A, turns
A into a M -differential graded algebra, i.e., one can determine conditions ensuring that
(d + e)M = 0. In [4] we showed that e must satisfy a system of non-linear equations,
which we called the (N,M) Maurer-Cartan equation.
• Algebras AN∞. This is not so much an example of N -differential graded algebras but rather
a homotopy generalization of such notion. AN∞ algebras are studied in [7].
This paper has three main goals. One is to introduce geometric examples of N - differential
graded algebras. We first review the constructions of N -differential graded algebras outlined
above and then proceed to consider the new examples:
• Differential forms on finitely generated simplicial sets. We construct a contravariant func-
tor ΩN : set
∆op −→ N ildga from the category of simplicial sets generated in finite dimen-
sions to N ildga, the category of nilpotent differential graded algebras, i.e., N -differential
graded algebras for some N ≥ 1. For a simplicial set s we let ΩN (s) be the algebra of
algebraic differential forms of depth N on the algebro-geometric realization of s. For each
integer K we define functor Sing≤K : Top −→ set
∆op , thus we obtain contravariant func-
tors ΩN ◦Sing≤K : Top −→ N
ildga assigning to each topological space X a nil-differential
graded algebra.
• Difference forms on finitely generated simplicial sets. We construct a contravariant functor
DN defined on set
∆op with values in a category whose objects are graded algebras which
are also N -complexes for some N , with the N -differential satisfying a twisted Leibnitz
rule. For a simplicial set s we let DN (s) be the algebra of difference forms of depth N
on the integral lattice in the algebro-geometric realization of s. Again, for each integer
K ≥ 0 we obtain a functor DN ◦ Sing≤K defined on Top assigning to each topological
space X a twisted nil-differential graded algebra.
Our second goal is to study the construction of N -differential graded algebras as defor-
mations of 3-differential graded algebras. Although in [4] a general theory solving this sort
of problem was proposed, our aim here is to provided a solution as explicit as possible. We
consider exact and infinitesimal deformations of 3-differentials in Section 3.
Our final goal in this work is to find applications of N -differential graded algebras to Lie
algebroids. In Section 4 we review the concept of Lie algebroids introduced by Pradines [27],
which generalizes both Lie algebras and tangent bundles of manifolds. A Lie algebroid E may
be defined as a vector bundle together with a degree one differential d on Γ(
E∗). We generalize
this notion to the world of N -complexes, that is we introduce the concept of N Lie algebroids
and construct several examples of such objects.
2 Examples of N-differential graded algebras
In this section we give a brief summary of the known examples of N -dgas and introduce new
examples of N -dgas of geometric nature.
Definition 1. Let N ≥ 1 be an integer. A N -complex is a pair (A, d), where A is a Z-graded
vector space and d : A −→ A is a degree one linear map such that dN = 0.
Clearly a N -complex is also a M -complex for M ≥ N . N -complexes are also referred to
as N -differential graded vector spaces. A N -complex (A, d) such that dN−1 6= 0 is said to be
a proper N -complex. Let (A, d) be a N -complex and (B, d) be a M -complex, a morphism
f : (A, d) −→ (B, d) is a linear map f : A −→ B such that df = fd. One of the most
interesting features of N -complexes is that they carry cohomological information. Let (A, d)
be a N -complex, a ∈ Ai is p-closed if dp(a) = 0, and is p-exact if there exists b ∈ Ai−N+p such
that dN−p(b) = a, for 1 ≤ p < N . The cohomology groups of (A, d) are the spaces
i(A) = Ker{dp : Ai −→ Ai+p}/Im{dN−p : Ai−N+p −→ Ai},
for i ∈ Z and p = 1, 2, ..., N−1.
Definition 2. A N -differential graded algebra (N -dga) over a field k, is a triple (A,m, d) where
m : A⊗A −→ A and d : A −→ A are linear maps such that:
1. dN = 0, i.e., (A, d) is a N -complex.
2. (A,m) is a graded associative algebra.
3. d satisfies the graded Leibnitz rule d(ab) = d(a)b+ (−1)āad(b).
The simplest way to obtain N -differential graded algebras is deforming differential graded
algebras. Let Der(A) be the Lie algebra of derivations on a graded algebra A. Recall that a
degree one derivation d on A, induces a degree one derivation, also denoted by d, on End(A).
Let A be a 2-dga and e ∈ Der(A). It is shown in [4] that e defines a deformation of A into
a N -differential graded algebra if and only if (d + e)N = 0, or equivalently, if and only if the
curvature Fe = d(e) + e
2 of e satisfies (Fe)
2 = 0 if N is even, or (Fe)
2 (d + e) = 0 if N is
odd. For example, consider the trivial bundle M × Rn over M . A connection on M × Rn is a
gl(n)-valued one form a on M , and its curvature is Fa = da +
[a, a]. Let Ω(M,gl(n)) be the
graded algebra of gl(n)-valued forms on M . Thus the pair (Ω(M,gl(n)), d + [a, ]) defines a
N -dga if and only if (Fa)
2 = 0 for N even, or (Fa)
2 (d+ a) = 0 for N odd.
Differential forms of depth N on simplicial sets
Fix an integer N ≥ 3. We are going to construct the (n(N − 1) + 1)-differential graded algebra
ΩN (R
n) of algebraic differential forms of depth N on Rn. Let x1, ..., xn be coordinates on R
and for 0 ≤ i ≤ n and 0 ≤ j < N, let djxi be a variable of degree j. We identify d
0xi with xi.
Definition 3. The (n(N − 1) + 1)-differential graded algebra ΩN (R
n) is given by
• ΩN (R
n) = R[djxi]/
djxid
kxi | j, k ≥ 1
as a graded algebras.
• The (n(N − 1) + 1)-differential d : ΩN (R
n) −→ ΩN (R
n) is given by d(djxi) = d
j+1xi, for
0 ≤ j ≤ N − 2, and d(dN−1xi) = 0.
One can show that d is (n(N − 1) + 1)-differential as follows:
1. It is easy to check that ΩN (R) is a N -dga.
2. If A is a N -dga and B is a P -dga, then A⊗B is a (N + P − 1)-dga.
3. ΩN (R
n) = ΩN (R)
We often write ΩN (x1, ..., xn) instead of ΩN (R
n) to indicate that a choice of affine coordi-
nates (x1, ..., xn) on R
n has been made.
Let ∆ be the category such that its objects are non-negative integers; morphisms in ∆(n,m)
are order preserving maps f : {0, ..., n} −→ {0, ...,m}. The category of simplicial sets Set∆
the category of contravariant functors ∆ −→ Set. Explicitly, a simplicial set s : ∆op −→ Set is
a functorial correspondence assigning:
• A set sn for each integer n ≥ 0. Elements of sn are called simplices of dimension n.
• A map s(f) : sm −→ sn for each f ∈ ∆(n,m).
Let Aff be the category of affine varieties, and let A : ∆ −→ Aff be the functor sending
n ≥ 0, into the affine variety A(x0, ..., xn) = ∆n = {(x0, ..., xn) ∈ R
n | x0 + ... + xn = 1}. A
sends f ∈ ∆(n,m) into A(f) : A(x0, ..., xn) → A(x0, ..., xm) given by A(f)
∗(xj) =
f(i)=j xi,
for 0 ≤ j ≤ m. Forms of depth N on the cosimplicial affine variety A are defined by the functor
ΩN : ∆
op −→ N ildga sending n ≥ 0 into
ΩN (n) = ΩN (x0, ..., xn)/ 〈x0 + ...+ xn − 1, dx0 + ...+ dxn〉 .
A map f ∈ ∆(n,m) induces a morphisms ΩN (f) : ΩN (m) −→ ΩN (n) given for 0 ≤ j ≤ m by
ΩN (f)(xj) =
f(i)=j
xi and ΩN (f)(dxj) =
f(i)=j
Let set∆
be the full subcategory of Set∆
whose objects are simplicial sets generated in
finite dimensions, i.e., simplicial sets s for which there is an integer K such that for each p ∈ si,
i ≥ K, there exists q ∈ sj, j ≤ K, with p = s(f)(q) for some f ∈ ∆(p, q). We are ready to
define the contravariant functor ΩN : set
∆op −→ N ildga announced in the introduction. The
nil-differential graded algebra ΩN (s) =
i=0 Ω
N (s) associated with s is given by
ΩiN (s) = {a ∈
ΩiN (n) | as(f)(p) = ΩN (f)(ap) for p ∈ sm and f ∈ ∆(n,m)}.
A natural transformation l : s −→ t induces a map ΩN(l) : ΩN (t) −→ ΩN (s) given by the rule
[ΩN (l)(a)]p = al(p) for a ∈ ΩN (t) and p ∈ sn.
For each integer K ≥ 0 there is functor ( )≤K : Set
∆op −→ set∆
sending a simplicial set
s, into the simplicial set s≤K generated by simplices in s of dimension lesser or equal to K.
The singular functor Sing : Top −→ Set∆
sends a topological space X into the simplicial set
Sing(X) such that
Singn(X) = {f : ∆n −→ X | f is continous }.
Thus, for each pair of integers N and K we have constructed a functor
ΩN ◦ ( )≤K ◦ Sing : Top −→ N
ildga
sending a topological space X into the nil-differential graded algebra ΩN (Sing≤K(X)).
Difference forms of depth N on simplicial sets
Next we construct difference forms of higher depth on finitely generated simplicial sets. Dif-
ference forms on discrete affine space were introduced by Zeilberger in [28]. We proceed to
construct a discrete analogue of the functors from topological spaces to nil-differential graded
algebras introduced above. First, we construct DN (Z
n) the algebra of difference forms of depth
N on Zn. Let F (Zn,R) be the algebra concentrated in degree zero of R-valued functions on the
lattice Zn. Introduce variables δjmi of degree j for 1 ≤ i ≤ n and 1 ≤ j < N . The graded
algebra of difference forms of depth N on Zn is given by
DN (Z
n) = F (Zn,R)⊗ R[δjmi]/
δjmiδ
kmi | j, k ≥ 1
A form ω ∈ DN (Z
n) can be written as ω =
I ωIdmI where I : {1, .., n} −→ N is any map
and dmI =
i=1 d
I(i)mi. The degree of dmI is |I| =
i=1 I(i). The finite difference ∆i(g) of
g ∈ F (Zn,R) along the i-direction is given by
∆i(g)(m) = g(m+ ei)− g(m),
where the vectors ei are the canonical generators of Z
n and m ∈ Zn. The difference operator δ
is defined for 1 ≤ j ≤ N − 2 by the rules
δ(g) =
∆i(g)δmi, δ(δ
jmi) = δ
j+1mi and δ(δ
N−1mi) = 0.
It is not difficult to check that if ω =
I ωIdmI , then δω =
J(δω)JdmJ where
(δω)J =
J(i)=1
(−1)|J<i|∆iωJ−ei +
J(i)≥2
(−1)|J<i|ωJ−ei .
From the later formula we see that (δω)J is a linear combination of (differences of) functions ωK
with |K| < |J |. This fact implies that δ is nilpotent, indeed, one can check that δn(N−1)+1 = 0.
All together we have proved the following result.
Theorem 4. DN (Z
n) is a graded algebra and the difference operator δ gives DN (Z
n) the
structure of a (n(N − 1) + 1)-complex.
One can check that δ satisfies a twisted Leibnitz rule, so DN (Z
n) is actually pretty close of
being a N -dga. Let Zn,1 ⊆ Zn+1 consists of tuples (m0, ...,mn) such that m0 + ... + mn = 1.
Consider the functor DN defined on ∆
op sending n ≥ 0 into
DN (n) = F (Z
n,1,R)⊗
δjmiδ
kmi, δm0 + ...+ δmn
A map f ∈ ∆(n, k) induces a morphisms DN (f) : DN (k) −→ DN (n) given for g ∈ F (Z
k,1,R)
and 0 ≤ j ≤ k by
DN (f)(g)(m0, ...,mn) = g
f(i)=0
mi, ...,
f(i)=k
 and DN (f)(δmj) =
f(i)=j
We extend DN to the functor defined on set
∆op sending a finitely generated simplicial set s into
DN (s) =
i=0 D
N (s) where
DiN (s) =
DiN (n) | as(f)(p) = DN (f)(ap) for p ∈ sk and f ∈ ∆(n, k)
A natural transformation l : s −→ t induces a map ΩN (l) : ΩN (t) −→ ΩN (s) by the rule
[DN (l)(a)]p = al(p) for a ∈ DN (t) and p ∈ sn. Thus for given integers N and K we have
constructed a functor DN ◦ ( )≤K ◦ Sing on Top sending a topological space X into a sort
of nil-differential graded algebra satisfying a twisted Leibnitz rule DN (Sing≤K(X)) . It would
be interesting to compute the cohomology groups of the algebra of difference forms of higher
depth on known simplicial sets. Even in the case of forms of depth 2 these groups have seldom
been studied.
3 On the (3,N) curvature
Recall that a discrete quantum mechanical system is given by the following data:
1. A directed graph with set of vertices V and set of directed edges E. The Hilbert space of
the system is H = CV .
2. A map ω : E −→ R assigning a weight to each edge.
3. Operators Un : H −→ H for n ∈ N given by (Unf)(y) =
ωn(y, x)f(x) where the
discretized kernel ωn(y, x) is given by
ωn(y, x) =
γ∈Pn(x,y)
ω(e).
Pn(x, y) denotes the set of paths in Γ from x to y of length n, i.e., sequences (e1, · · · , en)
of edges such that s(e1) = x, t(ei) = s(ei+1), for i = 1, ..., n − 1 and t(en) = y.
Let us introduce some notation. For s = (s1, ..., sn) ∈ N
n we set l(s) = n and |s| =
i si.
For 1 < i ≤ n we set s<i = (s1, ..., si−1); also we set s>n = s<1 = ∅. N
(∞) is equal to
where by convention N(0) = {∅}. Let A be a 3-dga and e be a degree one derivation on A. For
s ∈ Nn we let e(s) = e(s1) · · · e(sn), where e(l) = dl(e) if l ≥ 1, e(0) = e and e∅ = 1. For N ∈ N,
we set EN =
s ∈ N(∞) | s 6= ∅ and |s|+ l(s) ≤ N
and for s ∈ EN we let N(s) ∈ Z be given
by N(s) = N − |s| − l(s).
The following data defines a discrete quantum mechanical system:
1. The set of vertices is N(∞).
2. There is a unique directed edge from s to t if and only if t ∈ {(0, s), s, (s + ei)}, where
ei ∈ N
l(s) are the canonical vectors.
3. Edges are weighted according to the table:
source target weight
s (0, s) 1
s s (−1)|s|+l(s)
s (s+ ei) (−1)
|s<i|+i−1
PN (∅, s) consists of paths γ = (e1, ..., eN ), such that s(e1) = ∅, t(eN ) = s and s(el+1) = t(el).
The weight ω(γ) of a path γ ∈ PN (∅, s) is given by ω(γ) =
i=1 ω(ei). The following result,
proved in [4], tell us when d+ e defines a deformation of a 3-dga into a N -dga.
Theorem 5. d+ e defines a deformation of the 3-dga A into a N -dga if and only if the (3, N)
Maurer-Cartan equation holds co + c1d+ c2d
2 = 0, where for 0 ≤ k ≤ 2 we set
N(s)=k
c(s,N)e(s) and c(s,N) =
γ∈PN (∅,s)
ω(γ).
Exact deformations
Let us first consider the deformation of a 3-dga into a 3-dga. According to Theorem 5 the
derivation d+ e defines a 3-dga if and only if co + c1d+ c2d
2 = 0 where
N(s)=k
c(s, 3)e(s) and c(s, 3) =
γ∈P3(∅,s)
ω(γ).
Let us compute the coefficients ck. We have that
E3 = {(0), (1), (2), (0, 0), (1, 0), (0, 1), (0, 0, 0)} .
Let us first compute c0. There are four vectors s in E3 such that N(s) = 0, these are
(2), (1, 0), (0, 1) and (0, 0, 0). The only path from ∅ to (2) of length 3 is
∅ −→ (0) −→ (1) −→ (2)
of weight 1. Since e(2) = d2(e), then we have that c((2), 3) = d2(e). The unique path from ∅ to
(1, 0) of length 3 is
∅ −→ (0) −→ (0, 0) −→ (1, 0)
of weight 1. Since e(1,0) = d(e)e we have that c((1, 0), 3) = d(e)e. There are two paths from ∅
to (0, 1) of length 3, namely
∅ −→ (0) −→ (0, 0) −→ (0, 1)
∅ −→ (0) −→ (1) −→ (0, 1)
of weight −1 and 1, respectively. Thus c((1, 0), 3) = 0 since the sum of the weights vanishes.
The unique path from ∅ to (0, 0, 0) of length 3 is
∅ −→ (0) −→ (0, 0) −→ (0, 0, 0)
of weight 1. Since e(0,0,0) = e3, then c((0, 0, 0), 3) = e3. Thus we have shown that
c0 = d
2(e) + d(e)e + e3.
We proceed to compute c1. The vectors in E3 such that N(s) = 1 are (1) and (0, 0). Paths
from ∅ to (1) of length 3 are
∅ −→ ∅ −→ (0) −→ (1)
∅ −→ (0) −→ (0) −→ (1)
∅ −→ (0) −→ (1) −→ (1)
of weight 1, −1 and 1, respectively. Since e(1) = d(e), then c((1), 3) = d(e). Paths from ∅ to
(0, 0) of length 3 are
∅ −→ (0) −→ (0) −→ (0, 0)
∅ −→ ∅ −→ (0) −→ (0, 0)
∅ −→ (0) −→ (0, 0) −→ (0, 0).
The corresponding weights are, respectively, −1, 1 and 1. We have that e(0,0) = e2, thus
c((0, 0), 3) = e2 and c1 = d(e) + e
Finally we compute c2. (0) is the only vector in E3 such that N(s) = 2. The paths from ∅
to (0) of length 3 are
∅ −→ ∅ −→ ∅ −→ (0)
∅ −→ ∅ −→ (0) −→ (0)
∅ −→ (0) −→ (0) −→ (0).
The corresponding weights are, respectively, 1, −1 and 1. Since e(0) = e then c2 = c((0), 3) =
e. Altogether we have proven the following result.
Theorem 6. d+ e defines a deformation of the 3-dga A into a 3-dga if and only if
(d2(e) + d(e)e + e3) + (d(e) + e2)d+ ed2 = 0.
Consider now deformations of a 3-dga into a 4-dga. Again by Theorem 5 we must have
c0 + c1d+ c2d
2 = 0. We proceed to compute the coefficients ck. We have that
E4 = {(0), (1), (2), (0, 0), (1, 0), (0, 1), (2, 0), (0, 2), (1, 1),
(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 0, 0, 0)}.
(0) is the only vector in E4 such that N(s) = 3. Paths of length 4 from ∅ to (0) are of the
form ∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
with weight (−1)j , where i + j = 3, thus we have that
c3 = (1− 1 + 1− 1)e = 0.
We compute c2. Vectors in E4 with N(s) = 2 are (0, 0) and (1). Paths from ∅ to (0, 0)
of length 2 are of the form ∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
→ (0, 0) → · · · → (0, 0)︸ ︷︷ ︸
of weight
i+j+k=2(−1)
j = 2, thus c((0, 0), 4)e(0,0) = 2e2. Paths from ∅ to (1) of length 2 are of the
form ∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
→ (1) → · · · → (1)︸ ︷︷ ︸
with weight
i+j+k=2(−1)
j = 2, thus
c((1), 4)e(1) = 2d(e) and c2 = 2(e
2 + d(e)).
Let us now compute c1. Vectors in E4 with N(s) = 1 are (0, 0, 0), (1, 0), (0, 1) and (2).
Paths from ∅ to (0, 0, 0) are of 5 types. Paths of the form
∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
→ (0, 0) → · · · → (0, 0)︸ ︷︷ ︸
→ (0, 0, 0) → · · · → (0, 0, 0)︸ ︷︷ ︸
with weight
i+j+k+l=1(−1)
j(−1)l, so that c((0, 0, 0), 4)e(0,0,0) = (1− 1 + 1− 1)e3 = 0. Paths
of the form
∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
→ (1) → · · · → (1)︸ ︷︷ ︸
→ (0, 1) → · · · → (0, 1)︸ ︷︷ ︸
with weight ∑
i+j+k+l=1
(−1)j(−1)l = 1− 1 + 1− 1 = 0 .
Path of the form
∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
→ (0, 0) → · · · → (0, 0)︸ ︷︷ ︸
→ (0, 1) → · · · → (0, 1)︸ ︷︷ ︸
of weight
i+j+k+l=1(−1)
j(−1)(−1)l so that
c((0, 1), 4)e(0,1) = ((1 − 1 + 1− 1) + (1− 1 + 1− 1))ed(e) = 0.
Paths of the form
∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
→ (0, 0) → · · · → (0, 0)︸ ︷︷ ︸
(1, 0) → · · · → (1, 0)︸ ︷︷ ︸
of weight
i+j+k+l=1(−1)
j(−1)l, thus c((1, 0), 4)e(1,0) = (1 − 1 + 1 − 1)d(e)e = 0. There are
also paths of the form
∅ → · · · → ∅︸ ︷︷ ︸
→ (0) → · · · → (0)︸ ︷︷ ︸
→ (1) → · · · → (1)︸ ︷︷ ︸
→ (2) → · · · → (2)︸ ︷︷ ︸
of weight
i+j+k+l=1(−1)
j(−1)l, so we have c((2), 4)e(2) = (1 − 1 + 1− 1)d2(e) = 0. We have
shown that
c1 = c((0, 0, 0), 4)e
(0,0,0) + c((0, 1), 4)e(0,1) + c((1, 0), 4)e(1,0) + c((2), 4)e(2) = 0.
Let us compute c0. There are several types of paths in this case. Path
∅ −→ (0) −→ (0, 0) −→ (0, 0, 0) −→ (0, 0, 0, 0)
of weight 1, thus cq((0, 0, 0, 0), 4)a
(0,0,0,0) = a4. Paths
∅ −→ (0) −→ (0, 0) −→ (0, 0, 0) −→ (0, 0, 1)
∅ −→ (0) −→ (1) −→ (0, 1) −→ (0, 0, 1)
∅ −→ (0) −→ (0, 0) −→ (0, 1) −→ (0, 0, 1)
of weight 1, thus we have that c((0, 0, 1), 4)e(0,0,1) = e2d(e). Paths
∅ −→ (0) −→ (0, 0) −→ (1, 0) −→ (0, 1, 0)
∅ −→ (0) −→ (0, 0) −→ (0, 0, 0) −→ (0, 1, 0)
of weight 0, thus c((0, 1, 0), 4)e(0,1,0) = 0ad(a)a = 0. Path
∅ −→ (0) −→ (0, 0) −→ (0, 0, 0) −→ (1, 0, 0)
of weight 1, thus c((1, 0, 0), 4)e(1,0,0) = d(e)e2. Path
∅ −→ (0) −→ (0, 0) −→ (1, 0) −→ (2, 0)
of weight 1, so c((2, 0), 4)e(2,0) = d2(e)e. Paths
∅ −→ (0) −→ (0, 0) −→ (0, 1) −→ (0, 2)
∅ −→ (0) −→ (1) −→ (0, 1) −→ (0, 2)
∅ −→ (0) −→ (1) −→ (2) −→ (0, 2)
of weight 1, so that c((0, 2), 4)e(0,2) = ed2(e). There are also paths
∅ −→ (0) −→ (0, 0) −→ (1, 0) −→ (1, 1)
∅ −→ (0) −→ (1) −→ (0, 1) −→ (1, 1)
of weight 2, so that c((1, 1), 4)e(1,1) = (d(e))2. We see that
c0 = e
4 + e2d(e) + d(e)e2 + d2(e)e + ed2(e) + (d(e))2.
All together we have shown the following result.
Theorem 7. d+ e defines a deformation of the 3-dga A into a 4-dga if and only if
(e4 + e2d(e) + d(e)e2 + d2(e)e + ed2(e) + (d(e))2) + 2(e2 + d(e))d2 = 0.
Infinitesimal deformations
Let t be a formal parameter such that t2 = 0.
Theorem 8. Let (A, d) be a N -dga and e a degree one derivation on A, then we have
(d+ te)N = t
p∈Par(k,N−k+1)
(−1)w(p)
 dN−k−1(e)dN−k−1,
where
Par(k,N − k + 1) = {p = (p1, · · · , pN−k+1) |
N−k+1∑
pi = k} and w(p) =
N−k+1∑
(i− 1)pi.
Proof. From Theorem 5 we know that DN =
k=0 ckd
k. Since t2 = 0, then
(te)(s) = (te)(s1) · · · (te)(sl(s)) = tl(s)e(s) = 0
unless l(s) ≤ 1. On the other hand we have that
EN = {(0), (1), · · · , (N − 1)}.
Suppose that l(s) = 1 and N(s) = N − |s| − l(s) = k, thus |s| = N − k − 1. The unique vector
s in EN of length 1 such that |s| = N − k − 1 is s = (N − k − 1). Therefore
N(s)=k
c(s,N)e(s) = c((N − k − 1), N)e(s) = c((N − k − 1), N)dN−k−1(e).
A path from ∅ to (N − k − 1) of length N must be of the form
∅→ · · · →︸ ︷︷ ︸
∅ → (0)→ · · · →︸ ︷︷ ︸
(0) → (1)→ · · · →︸ ︷︷ ︸
(1) → · · · → (N − k − 1)→ · · · →︸ ︷︷ ︸
pN−k+1
(N − k − 1)
with (p1 + 1) + (p1 + 1) + · · ·+ (pN−k + 1) + pN−k+1 = N , i.e.,
∑N−k+1
i=1 pi = k. The weight of
such path is
(−1)0p1(−1)(2−1)p2(−1)(3−1)p2 ...(−1)(N−k)pN−k+1 = (−1)w(p), thus we get that
c((N − k − 1), N) =
γ∈PN (∅,s)
ω(γ) =
p∈Par(k,N−k+1)
(−1)w(p).
Corollary 9. e defines an infinitesimal deformation of theN -dga (A, d) into theN -dga (A, d+e)
if and only if
p∈Par(k,N−k+1)
(−1)w(p)
 dN−k−1(e)dN−k−1 = 0. (1)
4 N Lie algebroids
In this section we introduce the notion of N Lie algebroids and construct examples of such
structures. We first review the notion of Lie algebroids, provide some examples, and write the
definition of Lie algebroids in a convenient way for our purposes.
Lie algebroids
We review basic ideas around the notion of Lie algebroids; the interested reader will find much
more information in [12, 23, 27]. The notion of Lie algebroids has gained much attention in
the last few years because of its interplay with various branches of mathematics and theoretical
physics, see [10, 11, 17]. We center our attention on the basic definitions and constructions of
Lie algebroids and its relation with graded manifolds and differential graded algebras.
Definition 10. A Lie algebroid is a vector bundle π : E −→ M together with:
• A Lie bracket [ , ] on the space Γ(E) of sections of E.
• A vector bundle map ρ : E −→ TM over the identity, called the anchor, such that the
induced map ρ : Γ(E) −→ Γ(TM) is a Lie algebra morphism.
• The identity [v, fw] = f [v,w]+(ρ(v)f)w must hold for sections v,w of E and f a smooth
function on M .
Let (x1, ..., xn) be coordinates on a local chart U ⊂ M , and let {eα | α = 1, . . . , r} be a basis
of local sections of π : E|U −→ U . Local coordinates on E|U are given by (x
i, yα). Locally the
Lie bracket and the anchor are given by [eα, eβ ]E = C
eγ and ρ(eα) = ρ
, respectively.
The smooth functions C
, ρiα are the structural functions of the Lie algebroid. The condition
for ρ to be a Lie algebra homomorphism is written in local coordinates as
= ρiγ C
The other compatibility condition between ρ and [ , ] is given by
cycl(α,β,γ)
+ Cµαν C
where the sum is over indices α, β, γ such that the map 1, 2, 3 −→ α, β, γ is a cyclic permuta-
tion. The simplest examples of Lie algebroids are described below; the reader will find further
examples in the references listed at the beginning of this section.
Example 11. A finite dimensional Lie algebra g may be regarded as a vector bundle over a
single point. Sections are elements of g, the Lie bracket is that of g, and the anchor map is
identically zero. The structural functions C
are the structural constants c
of g and ρiα = 0.
Example 12. The tangent bundle π : TM −→ M with anchor the identity map ITB on TB
and with the usual bracket on vector fields.
Exterior differential algebra of Lie algebroids
Sections Γ(
E) of a Lie algebroid E play the rôle of vector fields on a manifold and are called
E vector fields. Sections of the dual bundle π : E∗ −→ M are called E 1-forms. Similarly
sections Γ(
E∗) of
E∗ are called E forms. The degree of a E form in Γ(
E∗) is k. Let us
state and sketch the proof of a result of fundamental importance for the rest of this work.
Theorem 13. Let E be a vector bundle. E is a Lie algebroid if and only if Γ(
E∗) is a
differential graded algebra. A differential on
E∗ is the same as a degree one vector field v on
E[−1] such that v2 = 0.
Above E[−1] denotes the graded manifold whose underlying space is E with fibers placed
in degree one. If E is a Lie algebroid one defines a differential
d : Γ(∧kE∗) −→ Γ(∧k+1E∗)
as follows:
dθ(v1, . . . , vk+1) =
(−1)i+1ρ(vi)θ(v1, . . . , v̂i, . . . , vk+1)
(−1)i+jθ([vi, vj ], v1, . . . , v̂i, . . . , v̂j , . . . vk+1),
for v1, . . . , vk+1 ∈ Γ(E). The axioms for a Lie algebroid imply that:
1. d2 = 0;
2. If f ∈ C∞(M) and v ∈ Γ(E), then 〈df, v〉 = ρ(v)f ;
3. d is a derivation of degree 1, i.e., d(θ ∧ ζ) = dθ ∧ ζ + (−1)θθ ∧ ζ.
Conversely, assume that d is a degree one derivation on Γ(
E∗) satisfying d2 = 0. Then E
is a Lie algebroid with the structural maps ρ and [ , ] given by
ρ(v)f = df(v) ,
θ([v,w]) = ρ(v)θ(w) − ρ(w)θ(v)− dθ(v,w),
for v,w ∈ Γ(E), f ∈ C∞(M) and θ ∈ Γ(
E). In local coordinates d is determined by
dxi = ρiα e
α and deγ = C
eα ∧ eβ,
where {eα | α = 1, . . . , r} is the dual basis of {eα | α = 1, . . . , r}. It is not hard to see that
the conditions d2xi = 0 and d2eα = 0 are equivalent to the structural equations defining a Lie
algebroid. Let us compute the exterior algebra of a few Lie algebroids.
Example 14. To the trivial Lie algebroid structure on a vector bundle E corresponds to the
exterior algebra
E∗ with vanishing differential.
Example 15. Chevalley-Eilenberg differential on
∗ arises from the Lie algebroid g −→ {•}
of Example 11. The Chevalley-Eilenberg differential d takes the form
dθ(v1, . . . , vk+1) =
(−1)i+jθ([vi, vj ], v1, . . . , v̂i, . . . , v̂j , . . . vk+1),
for vi ∈ g and θ ∈
Example 16. The differential associated with the tangent bundle TM −→ M Lie algebroid is
de Rham differential.
N Lie algebroids
We are ready to introduce the main concept of this section. In the light of Theorem 13 it is
rather natural to define a N Lie algebroid as a vector bundle E together with a degree one
N -nilpotent vector field v on the graded manifold E[−1]. That definition, useful as it might
be, rules out some significant examples that we would not like to exclude, thus, we prefer the
more inclusive definition given below. Though not strictly necessary for our definition of N Lie
algebroids, the study of nilpotent vector fields on graded manifolds is of independent interest,
and we shall say a few words about them. Indeed our next result gives an explicit formula for
the N -th power of a graded vector field.
Let x1, ..., xm be local coordinates on a graded manifold and ∂1, ..., ∂m be the corresponding
vector fields. We recall that if xi is a variable of degree xi, then ∂i is of degree −xi, and dxi
is of degree xi + 1. Let a
1, ..., am be functions of homogeneous degree depending on x1, ..., xm.
For L a linearly ordered set and f : L −→ [m] a map we define
f(i) and ∂f =
∂f(i).
Also we define the sign s(f) by the rule
∂f = s(f)∂
|f−1(1)|
1 ... ∂
|f−1(m)|
Let p : N −→ Z2 be the map such that p(n) is 1 if n is even and −1 otherwise. Using induction
on N one can show that:
Theorem 17.
(ai∂i)
s(f, α)(
α−1(i)
af(i))∂f |
α−1(N+1)⊔N
where the sum runs over f : [N ] −→ [m] and α : [N − 1] −→ [2, N + 1] such that α(i) > i. The
sign s(f, α) is given by
s(f, α) = p(
s<j<α(s)
f(j) + xsf |α−1(j)∩[s+1,N−1] ).
Corollary 18.
(ai∂i)
cI∂I ,
where I : [m] −→ N is such that 1 ≤ |I| := I(1) + ...+ I(N) ≤ N , ∂I =
i=1 ∂
i , and
S(f, α)
(∂f(α−1(i))a
f(i))
where the sum runs over maps α : [N − 1] −→ [2, N + 1] with α(i) > i for i ∈ [N − 1], and
f : [N ] −→ [m] such that |{j ∈ α−1(N + 1) ⊔ {N} | f(j) = i}| = I(i), for i ∈ [m]. The sign
S(f, α) is given by
S(f, α) = s(f, α)s(f |α−1(N+1)⊔{N}) .
Corollary 19. (ai∂i)
N = 0 if and only if cI = 0 for I as above.
For example for N = 2 one gets
(ai∂i)
p(xiaj)aiaj∂i∂j + ai∂i(aj)∂j .
For N = 3 we get that
(ai∂i)
i,j,k
ai∂i(aj)∂j(ak)∂k + p(xiaj)aiaj∂i∂j(ak)∂k
+ p(xiak)aiaj∂j(ak)∂i∂k + p(xjak)ai∂i(aj)ak∂j∂k
+ xiaj)aiaj∂i(ak)∂j∂k + p(xjak + xiaj + xiak)aiajak∂i∂j∂k.
For N = 4 the corresponding expression have 24 terms and we won’t spell it out.
We return to the problem of defining N Lie algebroids. We need some general remarks on
differential operators on associative algebras. Given an associative algebra A we let DO(A)
be the algebra of differential operators on A, i.e., the subalgebra of End(A) generated by
A ⊂ End(A) and Der(A) ⊂ End(A), the space of derivations of A. Thus DO(A) is generated
as a vector space by operators of the form x1 ◦x2 ◦ · · · ◦xn ∈ End(A) where xi is in A⊔Der(A).
Notice that DO(A) admits a natural filtration
∅ = DO≤−1(A) ⊆ DO≤0(A) ⊆ DO≤1(A) ⊆ · · · ⊆ DO≤k(A) ⊆ · · · ⊆ DO(A),
whereDO≤k(A) ⊆ DO(A) is the subspace generated by operators x1◦x2◦· · ·◦xn, where at most
k operators among the xi belong to Der(A). Thus DO(A) admits the following decomposition
as graded vector space
DO(A) =
DOk(A) :=
DO≤k(A)/DO≤k−1(A).
Clearly DO0(A) = A and if A is either commutative or graded commutative, then
DO1(A) = Der(A).
The projection map π1 : DO(A) −→ DO1(A) induces a non-associative product
⋄ : DO1(A)⊗DO1(A) −→ DO1(A)
given by s ⋄ t = π1(s ◦ t) for s, t ∈ DO1(A). In particular if A is commutative or graded
commutative we obtain a non-associative product
⋄ : Der(A)⊗Der(A) −→ Der(A).
To avoid unnecessary use of parenthesis we assume that in the iterated applications of ⋄ we
associate in the minimal form from right to left.
Definition 20. A N Lie algebroid is a vector bundle E together with a degree one derivation
d : Γ(
E∗) −→ Γ(
E∗), such that the result of N ⋄-compositions of d with itself vanishes,
i.e., d ⋄ d ⋄ · · · ⋄ d = 0.
The notions of Lie algebroids and 2 Lie algebroids agree; indeed it is easy to check that
d ◦ d = d ⋄ d for any degree one derivation d : Γ(
E∗) −→ Γ(
E∗). Let us now illustrate with
an example the difference between the condition d◦d◦· · ·◦d = 0 and the much weaker condition
d ⋄ d ⋄ · · · ⋄ d = 0. Let C[x1, ..., xn] be the free graded algebra generated by graded variables xi
for 1 ≤ i ≤ n. A derivation on C[x1, ..., xn] is a vector field ∂ =
ai∂i where ai ∈ C[x1, ..., xn].
The condition ∂N = 0 is rather strong and restrictive, it might be tackled with the methods
provided above. In contrast, the condition ∂ ⋄ ∂ ⋄ · · · ⋄ ∂ = 0 is much simpler and indeed it is
equivalent to the condition ∂N (xi) = 0 for 1 ≤ i ≤ n.
Definition 21. A N Lie algebra is a vector space g together with a degree one derivation d on
∗ such that the N -th ⋄-composition of d with itself vanishes.
Our next result characterizes 3 Lie algebras in more familiar terms. For integers k1, k2, ..., kl
such that k1 + k2 + · · ·+ kl = n, we let Sh(k1, k2, · · · , kl) be the set of permutations
σ : {1, · · · , n} −→ {1, · · · , n}
such that σ is increasing on the intervals [ki + 1, ki+1] for 0 ≤ i ≤ l, k0 = 1 and kl+1 = n.
Assume we are given a map [ , ] :
g −→ g.
Theorem 22. The pair (g, [ , ]) is a 3 Lie algebra if and only if for v1, v2, v3, v4 ∈ g we have
σ∈Sh(2,1,1)
sgn(σ)[[[vσ(1), vσ(2)], vσ(3)]vσ(4)] =
σ∈Sh(2,2)
sgn(σ)[[vσ(1) , vσ(2)], [vσ(3), vσ(4)]],
Proof. One can show that a degree one differential on
∗ is necessarily the Chevalley-Eilenberg
operator
dθ(v1, . . . , vn+1) =
(−1)i+jθ([vi, vj ], v1, . . . , v̂i, . . . , v̂j , . . . vn+1) ,
where [ , ] :
g −→ g is an antisymmetric operator. We remark that we are not assuming,
at this point, that the bracket [ , ] satisfies any further identity. Jacobi identity arises when
the square of d is set to be equal to zero, but we do not do that since we want to investigate
the weaker condition that the third ⋄-power of d be equal to zero. For θ ∈
∗ = g∗ the
Chevalley-Eilenberg operator takes the simple form
dθ(v1, v2) = −θ([v1, v2]).
Moreover a further application of d to dθ yields
d2θ(v1, v2, v3) =
σ∈Sh(2,1)
sgn(σ)θ([[vσ(1), vσ(2)], vσ(3)]).
From the last equation it is evident that Jacobi identity is equivalent to the condition d2 = 0.
We do not assume assume that Jacobi identity holds and proceed to compute the third ⋄-power
of d. We obtain that
d ⋄ d ⋄ dθ(v1, v2, v3, v4) =
σ∈Sh(2,1,1)
sgn(σ)θ([[[vσ(1), vσ(2)], vσ(3)]vσ(4)])
σ∈Sh(2,2)
sgn(σ)θ([[vσ(1), vσ(2)], [vσ(3), vσ(4)]]).
Thus d ⋄ d ⋄ d = 0 if and only if the condition from the statement of the Theorem holds.
Using local coordinates θ1, ..., θm on the graded manifold g[−1], it is not hard to show that
a vector field of degree one on g[−1] can be written as
where the constants C
may be identified with the structural constants of [·, ·]. The square of
the vector field ∂ is given by
Cσδ εθ
Cσγ εθ
αθβθε
Cσδ γθ
αθβθδ
θαθβCσδ εθ
Using the antisymmetry properties of C
and the commutation rules for θα one can write
together the first to terms. We find that
∂ ⋄ ∂ =
Cσγ εθ
αθβθε
The condition ∂ ⋄ ∂ = 0 is equivalent to Jacobi identity. We assume that ∂ ⋄ ∂ 6= 0 and proceed
to compute consider the condition ∂ ⋄ ∂ ⋄ ∂ = 0. We have that
∂ ◦ (∂ ⋄ ∂) =
Cνλµθ
Cσγ εθ
αθβθε
Using carefully the properties of C
and θα we find that
∂ ◦ (∂ ⋄ ∂) =
CνλµC
Cσγ εθ
λθµθβθε
CνλµC
Cσγ νθ
λθµθαθβ
CνλµC
Cσγ εθ
λθµθαθβθε
Therefore we have shown that
∂ ⋄ (∂ ⋄ ∂) =
CνλµC
Cσγ ǫ −
CσγαC
θλθµθβθε
Thus the condition ∂ ⋄ (∂ ⋄ ∂) = 0 is equivalent to the following equations for fixed σ:
λ,µ,β,ε
CνλµC
Cσγ ǫ −
CσγαC
θλθµθβθε = 0 .
Let us now go back to the case of Lie algebroids as opposed to Lie algebras. There is a
natural degree one vector field on the graded manifold T[−1]R
n, namely, de Rham differential.
We now investigate whether it is possible to deform, infinitesimally, de Rham differential into
a 3-differential. In local coordinates (x1, . . . , xn, θ1, . . . , θn) on T[−1]R
n, with xi of degree zero
and θi of degree 1, de Rham operator takes the form
∂ = δiαθ
Let t be a formal infinitesimal parameter such that t2 = 0. We are going to show that any set
of functions aiα of degree zero on T[−1]R
n determine a deformation of de Rham operator into a
3-⋄ nilpotent operator given by
δiα + ta
Theorem 23. ∂a ⋄ ∂a = t
θα θβ
and ∂a ⋄ (∂a ⋄ ∂a) = 0.
Proof.
∂2a =
δiα + ta
θα θβ
θα θβ
θα θβ
Since t2 = 0 the third term on the right hand side of the expression above vanishes. The second
term also vanishes because it is a contraction of even and odd indices. So we get that
∂a ⋄ ∂a = t
θα θβ
The third power of ∂a is given by
∂a ⋄ (∂a ⋄ ∂a) = t
∂xγ∂xα
θγ θα θβ
It also vanishes because it includes a contraction of even and odd indices.
The nilpotency condition for the operator ∂a ⋄∂a is
θα θβ = 0 for j = 1, . . . , n. It is not
hard to find examples of matrices a
such that ∂a ⋄ ∂a = 0, for example

(x4)2
x1 x1
x2 x2 x3 x2
x3 x3 x2 x4
x4 x4 x1 x4 x3

More importantly there are also matrices a
such that ∂a ⋄ ∂a 6= 0, for example

x1 x4 x1 x1 x1
x2 x2 x4 x2 x2
x3 x3 x3 x4 x3
x4 x4 x4 x1 x4

We now consider full deformations as opposed to infinitesimal ones. Let
δiα + a
be a vector field. We think of ∂a as a deformation of de Rham differential with deformation
parameters aiα.
Theorem 24.
∂a ⋄ (∂a ⋄ ∂a) =
δlγ + a
){∂aiα
+ aiα
∂xl∂xi
θγθαθβ
Proof. Since
∂2a =
δiα + a
∂a ⋄ ∂a =
δiα + a
) ∂ajβ
we get
∂a ⋄ (∂a ⋄ ∂a) =
δlγ + a
δiα + a
) ∂ajβ
δlγ + a
){∂aiα
+ aiα
∂xl∂xi
θγθαθβ
Corollary 25. ∂a ⋄ (∂a ⋄ ∂a) = 0 if for fixed indices α, β, λ, j the following identity holds
δlγ + a
){∂aiα
+ aiα
∂xl∂xi
θγθαθβ = 0.
Corollary 26. Each matrix A = (A
) ∈ Mn(R) such that A
2 = 0 determines a 3 Lie algebroid
structure on TRn with differential given by (δiα +A
αxα)dx
Our final result describes explicitly the conditions defining a 3 Lie algebroid. Let E be a
vector bundle over M . A vector field on E[−1] of degree one is given in local coordinates by
∂ = ρiαθ
where ρiα and C
are functions of the bosonic variables only.
Theorem 27. ∂ ⋄ (∂ ⋄ ∂) = 0 if and only if for fixed γ and i the following identity holds:
Cασ µ)
Cλµσ −
CαλµC
CαβµC
θνθσθµθβ = 0 ,
Cǫσν−
Cǫσν + ρ
θσθνθγ = 0 .
Proof. We sketch the rather long proof. For ∂ = ρiαθ
θαθβ ∂
, we have
∂ ⋄ ∂ =
θλθµθβ
As in the previous theorem one finds that the condition ∂ ⋄ (∂ ⋄ ∂) = 0 is equivalent to the
following identities
Cασ µ)
Cλµσ −
Cλνσ −
θνθσθµθβ
= 0 ,
Cǫσν − ρ
θσθνθγ
Needless to say further research is necessary in order to have a better grasp of the meaning
and applications of the notion of N Lie algebroids. We expect that this approach will lead
towards new forms of infinitesimal symmetries, and for that reason alone it should find appli-
cations in various problems in mathematical physics. In our forthcoming work [3] we are going
to discuss some applications of N Lie algebroids in the context of Batalin-Vilkovisky algebras
and the master equation.
Acknowledgment
Thanks to Takashi Kimura, Juan Carlos Moreno and Jim Stasheff.
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mangel@euler.ciens.ucv.ve, jcama@usb.ve, ragadiaz@gmail.com
	Introduction
	Examples of N-differential graded algebras
	On the (3,N) curvature
	 N Lie algebroids
