A Survey of Huebschmann and Stasheff’s Paper:
Formal solution of the master equation via HPT
and deformation theory
Füsun Akman and Lucian M. Ionescu
October 22, 2018
1 Introduction
These notes, based on the paper [8] by Huebschmann and Stasheff, were pre-
pared for a series of talks at Illinois State University with the intention of ap-
plying Homological Perturbation Theory (HPT) to the construction of derived
brackets [11, 16], and eventually writing Part II of the paper [1].
Derived brackets are obtained by deforming the initial bracket via a deriva-
tion of the bracket. In [3] it was demonstrated that such deformations corre-
spond to solutions of the Maurer-Cartan equation, and the role of an “almost
contraction” was noted. This technique (see also [9]) is very similar to the itera-
tive procedure of [8] for finding the most general solution of the Maurer-Cartan
equation, i.e. the deformation of a given structure in a prescribed direction.
The present article, besides providing additional details of the condensed
article [8], forms a theoretical background for understanding and generalizing
the current techniques that give rise to derived brackets. The generalization,
which will be the subject matter of [2], will be achieved by using Stasheff and
Huebschmann’s universal solution. A second application of the universal solu-
tion will be in deformation quantization and will help us find the coefficients of
star products in a combinatorial manner, rather than as a byproduct of string
theory which underlies the original solution given by Kontsevich [10].
HPT is often used to replace given chain complexes by homotopic, smaller,
and more readily computable chain complexes (to explore “small” or “minimal”
models). This method may prove to be more efficient than “spectral sequences”
in computing (co)homology. One useful tool in HPT is
Lemma 1 (Basic Perturbation Lemma (BPL)). Given a contraction of N onto
M and a perturbation ∂ of dN , under suitable conditions there exists a pertur-
bation d∂ of dM such that H(M,dM + d∂) = H(N, dN + ∂).
The main question is: under what conditions does the BPL allow the preser-
vation of the data structures (DGA’s, DG coalgebras, DGLA’s etc.)? (We will
use the self-explanatory abbreviations such as DG for “differential graded”,
http://arxiv.org/abs/0704.0432v1
DGA for “differential graded (not necessarily associative) algebra”, and DGLA
for “differential graded Lie algebra”.)
Another prominent idea is that of a “(universal) twisting cochain” as a so-
lution of the “master equation”:
Proposition 1. Given a contraction of N onto M and a twisting cochain N →
A (A some DGA), there exists a unique twisting cochain M → A that factors
through the given one and which can be constructed inductively.
The explicit formulas are reminiscent of the Kuranishi map [13] (p.17), and
the relationship will be investigated elsewhere.
Note: we will assume that the ground ring is a field F of characteristic zero.
We will denote the end of an example with the symbol ♦ and the end of a proof
by �.
2 Perturbations of (co)differentials
2.1 Derivations of the tensor algebra
For any vector space V over F we have the isomorphismDer(TV ) ∼= Hom(V, TV )
where TV denotes the (augmented) tensor algebra on V . Namely, every linear
map f from V into TV extends uniquely into a derivation of the algebra TV
via the formula
f̂(v1 ⊗ · · · vn) =
v1 ⊗ · · · ⊗ f(vi)⊗ · · · vn.
Equivalently, every derivation of TV is determined by its restriction to V .
2.2 Coderivations of the tensor coalgebra
Similarly, we have the isomorphism Coder(T cV ) ∼= Hom(T cV, V ) where T cV is
the (coaugmented) coassociative tensor coalgebra of V , with counit η : T cV → F
(projection onto F ), and comultiplication
∆(v1 ⊗ · · · ⊗ vn) =
(v1 ⊗ · · · ⊗ vi)⊗ (vi+1 ⊗ · · · ⊗ vn).
Every linear map f = f1 + f2 + · · ·+ fn + · · · : T
cV → V (where fi : V
⊗i → V )
factors uniquely through a coderivation f̂ of T cV defined via the formula
f̂(v1 ⊗ · · · ⊗ vn) =
v1 ⊗ · · · ⊗ f1(vi)⊗ · · · vn
v1 ⊗ · · · ⊗ f2(vi ⊗ vi+1)⊗ · · · vn
+fn(v1 ⊗ · · · ⊗ vn).
That is, each coderivation on T cV is determined by itself followed by the projec-
tion onto V . Recall that the condition for f̂ to be a coderivation can be written
as ∆f̂ = (1⊗ f̂ + f̂ ⊗ 1)∆.
2.3 Coderivations of the symmetric coalgebra
Let us consider the cofree cocommutative counital coassociative algebra ST cV
on the vector space V as a subspace of T cV . The symmetric group Σn acts on
the left on V ⊗n via σ(v1 ⊗ · · · ⊗ vn) = vσ−1(1) ⊗ · · · ⊗ vσ−1(n). Then
ST cV =
(V ⊗n)Σn
is the space of invariants of this action. The action is compatible with the
coproduct on ST cV , so ST cV is a subcoalgebra of T cV which is cocommutative.
Note that ST (V ) is not a subalgebra with respect to the tensor multiplication
in T (V ); the product has to be symmetrized so that it projects back onto this
subspace (reminiscent of what T. Voronov does with derived brackets). The
projection (symmetrization) map P : T cV → ST cV is given by
P (v1 ⊗ · · · ⊗ vn) =
σ(v1 ⊗ · · · ⊗ vn).
This is not a coalgebra map, but is a retraction of the canonical inclusion
ST cV →֒ T cV . Now a coderivation D : T cV → T cV induces a coderivation
DS : ST
cV → ST cV by the composition ST cV →֒ T cV
→ T cV
→ ST cV .
In particular, a coderivation of T cV induces one of ΛcV = ST c[sV ] where V is
thought of as living in degree zero: we introduce the graded symmetric coalgebra
below. Once again, coderivations of ST cV are determined by their projections
onto ST c1V ; a map f = f1 + f2 + · · · : ST
c(V ) → V determines a coderivation
f̂ as in the tensor coalgebra case.
In the remaining part of this survey, we choose to identify ST cV with the
abstract symmetric coalgebra ScV under the isomorphism
v1 · · · vn 7→ P (v1 ⊗ · · · ⊗ vn).
The coproduct in ScV is given by
∆(v1 · · · vn) =
σ∈Σi,n−i
vσ(1) · · · vσ(i) ⊗ vσ(i+1) · · · vσ(n).
2.4 DGLA’s and perturbations of the codifferential
Definition 1. For any chain complex (X, d), and odd ∂, with (d + ∂)2 =
0, we say that ∂ is a perturbation of the differential d. We call d + ∂ the
perturbed differential. Equivalently, we have [d, ∂] + ∂∂ = 0 in End(X). If ∂ is
also compatible with an existing coalgebra structure on X , we say that it is a
coalgebra perturbation.
Let (g, d) be a graded chain complex (d lowers degrees) with a bracket [ , ]
that is skew-symmetric (not necessarily Leibniz or a chain map). Consider the
differential graded symmetric coalgebra Sc[sg], the differential d being induced
by that on g. Also let ∂ be the coderivation on Sc[sg] of degree −1 induced by
the bracket.
Proposition 2. The bracket [ , ] turns (g, d) into a DGLA if and only if ∂ is
a coalgebra perturbation of d. Also, any DGLA structure on g is determined by
the coalgebra perturbation induced from the bracket.
When g is an ordinary (degree-zero) Lie algebra over a field, Sc[sg] =
Λcg with differential ∂ corresponding to the bracket is the ordinary Koszul
or Chevalley-Eilenberg complex computing the homology of g with coefficients
in the field.
2.5 Strongly homotopy Lie algebras
Definition 2. Let (g, d) be a chain complex and let d also denote the codiffer-
ential in Sc[sg] induced by d. A strongly homotopy Lie (sh-Lie, or L∞) structure
on g is a perturbation ∂ = ∂2 + · · · + ∂n + · · · of d, i.e. an odd coderivation
satisfying [d, ∂] + ∂∂ = 0 and ∂η = 0 (recall that η is the counit) so that the
sum d+ ∂ endows Sc[sg] with a new coaugmented DG coalgebra structure.
The corresponding mega-map ℓ2 + · · · + ℓn + · · · from S
c(sg) to g extends
the differential ℓ1 = d : sg → g, and the lower identities satisfied by
ℓ = ℓ1 + ℓ2 + · · ·+ ℓn + · · ·
read as follows:
ℓ21 = 0
ℓ1(ℓ2(a, b))± ℓ2(ℓ1(a), b)± ℓ2(ℓ1(b), a) = 0
ℓ1(ℓ3(a, b, c))± ℓ3(ℓ1(a), b, c)± ℓ3(ℓ1(b), a, c)± ℓ3(ℓ1(c), a, b)
±ℓ2(ℓ2(a, b), c)± ℓ2(ℓ2(a, c), b)± ℓ2(ℓ2(b, c), a) = 0.
An sh-Lie morphism between two sh-Lie (or DGL) algebras (g, d + · · · ) and
(g′, d′ + · · · ) is a collection of chain maps Fn : S
n[sg] → S
′], satisfying
∆′F (u) = (F ⊗ F )(∆u). Then F is uniquely determined by its projection onto
sg′, that is, we may assume Fn : S
n[sg] → sg
Definition 3. A quasi-isomorphism F between sh-Lie algebras g, g′ is an sh-Lie
morphism such that F1 : sg → sg
′ induces an isomorphism between H(g, d) and
H(g′, d′).
Remark 1. Quasi-isomorphisms between DGLA’s are especially important in
deformation theory. Such a map gives a one-to-one correspondence between
moduli spaces of solutions to MC equations in ~g[[~]] and ~g′[[~]] (see [5]):
given a quasi-isomorphism F : Sc[sg] → sg′, we define F̃ : ~g[[~]] → ~g′[[~]] by
F̃ (r) =
Fn(r, . . . , r)
(also see [6]).
2.6 The Hochschild chain complex and DGA’s
Let (A, µ) be a unital associative algebra (possibly graded), and T c[sA] denote
the tensor coalgebra on the suspension of A. We recall that
Coder(T c[sA]) ∼= Hom(T
c[sA], A).
In particular, the associative bilinear multiplication µ ∈ Hom(T c[sA], A) corre-
sponds to a square-zero coderivation ∂ : T c[sA] → T c[sA] defined by
∂(a1 ⊗ · · · ⊗ an)
(−1)i+1(a1 ⊗ · · · ⊗ µ(ai ⊗ ai+1)⊗ · · · ⊗ an)
+(−1)n+1(µ(an ⊗ a1)⊗ a2 ⊗ · · · ⊗ an−1).
The condition that ∂ is a codifferential is equivalent to the associativity con-
dition m ◦ m = 0 where ◦ is the Gerstenhaber composition on multilinear
maps (a right pre-Lie map). The complex (Hom(T c[sA], A), ∂) is known as
the Hochschild chain complex.
Now let (A, µ, d) be a DGA. Then d+µ ∈ Hom(T c[sA], A) corresponds to a
perturbed codifferential d+ ∂ satisfying (d+ ∂)2 = 0, which is equivalent to the
identities d2 = 0 and [d, ∂] + ∂∂ = 0. The latter can also be split into [d, ∂] = 0
and ∂∂ = 0.
Proposition 3. The multiplication µ turns (A, d) into a DGA if and only if ∂
is a coalgebra perturbation of d. Also, any DGA structure on A is determined
by the coalgebra perturbation induced from µ.
2.7 Strongly homotopy associative algebras
Definition 4. Let (A, d) be a chain complex and let d also denote the codifferen-
tial in T c[sA] induced by d. A strongly homotopy associative (or A∞) structure
on A is a perturbation ∂ = ∂2 + · · · + ∂n + · · · of d, i.e. an odd coderivation
satisfying [d, ∂] + ∂∂ = 0 and ∂η = 0 so that the sum d+ ∂ endows T c[sA] with
a new coaugmented DG coalgebra structure.
The corresponding mega-map m2+ · · ·+mn+ · · · from T
c[sA] to A extends
the differential m1 = d : sA → A, and the lower identities satisfied by
m = m1 +m2 + · · ·+mn + · · ·
read as follows:
m21 = 0
m1(m2(a, b))±m2(m1(a), b)±m2(m1(b), a) = 0
m1(m3(a, b, c))±m3(m1(a), b, c)±m3(a,m1(b), c)±m3(a, b,m1(c)) = 0.
The mega-identity is m ◦ m = 0, sometimes written in the braces notation
{m}{m} = 0.
3 Master equation
If (A, d) is a differential graded associative algebra (DGA), then the equation
dτ = ττ (1)
is called the Master Equation (ME) (or Maurer-Cartan equation (MCE), etc.).
Similarly if (g, d) is a DGLA, then the equation
[τ, τ ] (2)
is also called the Master Equation. Sometimes the sign convention is
dτ + ττ = 0 (3)
[τ, τ ] = 0. (4)
Clearly any solution of such an equation must be an odd element of the algebra.
Moreover, in case A is the graded universal enveloping algebra of g, or g is the
Lie algebra obtained from A by the usual bracket, then solutions of the DGLA
master equation are also solutions of the DGA master equation.
Remark 2. If τ is a solution of Eq. (2) or Eq. (4) in a DGLA g, then the
odd derivation dτ = d− adτ or dτ = d+ adτ respectively defines a new DGLA
structure on g with respect to the old bracket.
Example 1. If g is a DGLA or L∞ algebra, then the corresponding coal-
gebra perturbation ∂ in Coder(Sc(sg)) is a solution of the ME in the DGA
End(Sc(sg)), where the differential is D = add. ⋄
Example 2. Gauge Theory: Let ξ be a principal bundle with structure group
G and Lie algebra g. There is a graded Lie algebra structure on the ad(ξ)-valued
de Rham forms induced by g. Given a connection A and an ad(ξ)-valued 1-form
η, the sum A+ η is again a connection, and its curvature is
FA+η = FA + dAη +
[η, η].
In particular, FA = FA+η if and only if
dAη +
[η, η] = 0
(the Maurer-Cartan equation). Here dA is the covariant derivative of the con-
nection A. When A is a flat connection (zero curvature) then there exists a
DGLA structure on the ad(ξ)-valued differential forms (d2A = 0) and FA+η is
also flat iff the MCE is satisfied (then the covariant derivative for A+ η is dτ ).
4 Twisting cochain
The notion of a twisting cochain generalizes that of a connection in differential
geometry.
4.1 Differential on Hom
If (C, dC) and (A, dA) are chain complexes, the following differential D makes
Hom(C,A) into a chain complex: Dφ = dAφ± φdC .
4.2 Cup product and cup bracket
Proposition 4. For any differential graded coalgebra C and a differential graded
associative algebra (DGA) A, the chain complex (Hom(C,A), D) becomes a
DGA via the cup (convolution) product a ⌣ b defined by the composition
−→ C ⊗ C
−→ A⊗A
−→ A.
The coaugmentation and augmentation maps η and ǫ on C and A respectively
define an augmentation map on (Hom(C,A), D).
Proposition 5. For any differential graded coalgebra C and a DGLA g, the
chain complex (Hom(C, g), D) becomes a DGLA via the cup bracket [a, b] defined
by the composition
−→ C ⊗ C
−→ g ⊗ g
[ , ]
−→ g.
Example 3. If g is a Lie algebra, then the cup bracket on Hom(Sc[sg], g) is
defined as above. For example, if τ and κ are maps Sc1[g] → g, then [τ, κ] may
be nonzero only on Sc2[sg]. In this case, we compute
∆(xy) = 1⊗ xy + x⊗ y + y ⊗ x+ xy ⊗ 1 (x, y ∈ sg),
[τ, κ](xy) = [τ(x), κ(y)] + [τ(y), κ(x)]. (5)
Example 4. The Hochschild complex of an associative algebra (A, µ) (where
µ2 = 0): Let
C•(A) = Hom(T cA,A) = Hom(
A⊗n, A),
with differential D = adµ ∈ Der(C
•(A)). The cup product x ⌣ y = {µ}{x, y}
is the composition
−→ T cA⊗ T cA
−→ A⊗A
−→ A;
if x is an n-linear map and y is an m-linear map, then
(x ⌣ y)(a1 ⊗ · · · ⊗ an+m) = x(a1 ⊗ · · · ⊗ an) · y(an+1 ⊗ · · · ⊗ an+m).
Remark 3. The differential D above is an inner derivation and not derived
from differentials on A and T cA. Still, it is a derivation of the cup product.
4.3 Twisting cochain
Definition 5. Given a coaugmented DG coalgebra C and an augmented DGA
A, a twisting cochain is a homogeneous morphism t : C → A of degree −1 such
that ǫτ = 0 and τη = 0 , and which satisfies Dt = t ⌣ t.
In other words, a twisting cochain is a solution of the master equation on
Hom(C,A) with the usual differential D induced from those of C, A and the
product is the cup product.
Definition 6. Given a DG cocommutative coalgebra C and a DGLA g, a Lie
algebra twisting cochain t : C → g is a homogeneous map of degree −1 whose
composition with the coaugmentation map is zero, and which satisfies
[t, t] (6)
([ , ] being the cup bracket).
Recall that a DGLA structure on a graded chain complex (g, d) is given by
a perturbation ∂ of the corresponding codifferential on Sc[sg]. Moreover, the
piece d∂ + ∂d = 0 of (d+ ∂)2 = 0 says that the bracket is a chain map and the
piece ∂2 = 0 says that the bracket satisfies the Jacobi identity. Let us denote
the symmetric coalgebra with the codifferential ∂ by Sc
[ , ]
[sg]. Quillen’s notation
C[g] for the same DG coalgebra reminds us that this is the Koszul or Chevalley-
Eilenberg complex that computes the homology of g without any regard for the
additional differential d on g.
Example 5. For any DGLA g, its universal Lie algebra twisting cochain
τg : S
[ , ][sg] → g
is given by
τg(sx) = x for x ∈ g
τg(y) = 0 for y ∈ S
k[sg], k 6= 1.
That is, an element with tensor degree one goes to its desuspension and ev-
erything else goes to zero. Clearly, the composition τgη is zero, as τg = 0 on
constants. Next, we show that τg satisfies the equation (6), but we note that in
this construction the differential on g itself is taken to be zero. On the left-hand
side, we have
Dτg(x1 ∧ · · · ∧ xn) = τg∂(x1 ∧ · · · ∧ xn)
which is zero if n 6= 2 and is equal to [x1, x2] if n = 2. Meanwhile, on the
right-hand side, we have
[τg, τg](x1 ∧ · · · ∧ xn)
which is zero if n 6= 2 and is equal to
{[τg(x1), τg(x2)]± [τg(x2), τg(x1)]} = [x1, x2]
if n = 2. ⋄
Remark 4. The universal property of the universal LA twisting cochain is that
every Lie algebra twisting cochain factors through this one: that is, whenever
C is a coalgebra and τ : C → g is a twisting cochain, then τg ◦ c(τ) = τ where
c(τ) : C → Sc
[ , ]
[sg] is the unique coalgebra map induced by τ .
Using HPT, we will construct formal solutions
τ ∈ Hom(ScD[sH(g)], g)
of the master equation. Once we make the choice of a contraction, we will obtain
explicit inductive formulas for D and τ .
5 Homological perturbation theory (HPT)
“HPT is concerned with transferring various kinds of algebraic structure through
a homotopy equivalence”. Also: “HPT is a set of techniques for the transference
of structures from one object to another up to homotopy” (Real [14]).
5.1 Contraction
Definition 7. Let (M,dM ) and (N, dN ) be chain complexes, π : N → M
and ∇ : M → N be chain maps, and h ∈ End(N) be a morphism (possibly
preserving some extra structure) of degree 1. Then a contraction
N, h) (7)
of N onto M is a collection of the above data satisfying
π∇ = idM
D(h) = addN (h) = ∇π − idN
πh = 0, h∇ = 0, hh = 0.
Another way to describe this structure is to say that M is a strong deformation
retract (SDR) of N (also called Eilenberg-Zilber data). The properties on the
last line are referred to as the annihilation properties or side conditions. Note
that the first line makes π surjective (projection) and ∇, injective (inclusion).
The map h is also known as the homotopy operator between ∇π and idN :
∇π − idN = D(h) = dNh+ hdN
(D = addN is the induced differential on Hom(N,N)).
Often filtered contractions are considered.
Remark 5. Lambe and Stasheff [12] noticed that the side conditions on h are
not restrictive: if πh = 0 and h∇ = 0 are not satisfied, then we can replace
h by h′ = D(h)hD(h). Now if h2 = 0 is not satisfied either, we replace h′ by
h′′ = h′dNh
′, which finally gives us an operator h′′ satisfying the side conditions.
Lemma 2. Given a contraction (7), we have a (not necessarily direct) sum
N = Im(∇) + Im(h) + Im(dN ).
Proof. Each x ∈ N can be written as
x = ∇π(x) − hdN (x) − dNh(x). (8)
Lemma 3. [14] Given a contraction (7), we have
Im(∇) + Im(h) = Im(∇)⊕ Im(h) = Ker(h).
Proof. We have
Im(∇) ⊂ Ker(h) and Im(h) ⊂ Ker(h)
since h∇ = 0 and h2 = 0. Conversely, by (8), each x ∈ Ker(h) can be written
x = ∇π(x) − hdN (x) ∈ Im(∇) + Im(h).
That the sum is direct can be seen as follows: let x ∈ Im(∇) ∩ Im(h). Then we
have x = ∇(y) = h(z) for some y ∈ M and z ∈ N . Rewriting the decomposition
of x in Ker(h), we obtain
x = ∇π(x) − hdN (x)
= ∇πh(z)− hdN∇(y)
= 0− hdN∇(y) (πh = 0)
= h∇dM (y) (∇ chain map)
= 0 (h∇ = 0).
Corollary 1. For any contraction (7), we have
H(N, h) ∼= Im(∇) ∼= M.
Lemma 4. Given any contraction (7), we have
Im(h) + Im(dN ) = Im(h)⊕ Im(dN ).
Moreover, if dM ≡ 0, then
Im(h)⊕ Im(dN ) = Ker(π).
Proof. Say x ∈ Im(h) ∩ Im(dN ). Then x = h(y) = dN (z) for some y, z ∈ N ,
so that by (8) we obtain
x = ∇πh(y) − hdNdN (z)− dNhh(y) = 0.
It is always true that Im(h) ⊂ Ker(π) as πh = 0. If dM ≡ 0, we further have
the result
πdN (x) = −dMπx = 0,
so that altogether
Im(h)⊕ Im(dN ) ⊂ Ker(π).
Conversely, for x ∈ Ker(π), we see that (even without the condition dM = 0)
Ker(π) ⊂ Im(h) + Im(dN )
since
x = −hdN (x)− dNh(x) ∀x ∈ Ker(π)
due to (8). �
Lemma 5. For any contraction (7) with dM = 0 we have
Im(∇) + Im(dN ) = Im(∇) ⊕ Im(dN ) = Ker(dN ).
Proof. The given sum is direct: let x ∈ Im(∇) ∩ Im(dN ). Then x = ∇(y) =
dN (z) and by (8)
x = ∇πdN (z)− hdNdN (z)− dNh∇(y) = ∇πdN (z) = −∇dMπ(z) = 0.
Clearly, we have Im(dN ) ⊂ Ker(dN ). Also Im(∇) ⊂ Ker(dN ), because
dN∇(x) = −∇dM (x) = 0.
Conversely, by (8),
Ker(dN ) ⊂ Im(∇) + Im(dN )
since we can write
x = ∇π(x) − dNh(x)
(no condition on dM ) for x ∈ Ker(dN ). �
Corollary 2. For any contraction (7) with dM = 0 we have
H(N, dN ) ∼= Im(∇) ∼= M.
Proposition 6. For any contraction (7) with dM = 0 we have
N = Im(∇)⊕ Im(h)⊕ Im(dN )
where
Im(∇)⊕ Im(h) = Ker(h)
Im(h)⊕ Im(dN ) = Ker(π)
Im(∇)⊕ Im(dN ) = Ker(dN ),
2. Im(h)
→ Im(dN ) is an isomorphism with inverse Im(dN )
→ Im(h), and
3. Im(∇)
→ M is an isomorphism with inverse M
→ Im(∇); we also have
Im(∇) ∼= M = Im(π) ∼= H(N, dN ) ∼= H(N, h).
Remark 6. This is a Hodge-type decomposition reminiscent of the case of a
compact orientable Riemannian manifold M without boundary. If
∗ : Ωr(M) → Ωdim(M)−r
is the “Hodge star operator” (an isomorphism) and
d : Ωr−1(M) → Ωr(M)
is the de Rham differential, then we define a “partial inverse” d† (the adjoint
exterior derivative operator) to −d by d† = ± ∗ d∗. The commutator of d and
d† is called the “Laplace-Beltrami operator”: ∆ = dd† + d†d. Then there exists
a unique decomposition of the algebra of de Rham forms as follows:
Ωr(M) = Harmr(M)⊕ d(Ωr−1(M))⊕ d†(Ωr+1(M)),
where the “harmonic forms” are given by Harmr(M) = Ker(∆). In the case of
our general contraction with dM = 0, the operators h and dN replace d and d
respectively. What do we know about ∆ here? We have
∆ = D(h) = hdN + dNh = (h+ dN )
2 = ∇π − idN .
The kernel of this operator is equal to ∇(M), as we have
(∇π − idN )(x) = 0 ⇔ ∇π(x) = x ⇔ x ∈ ∇(M),
or Ker(∆) = Im(∇). So is there an analog of the Hodge star operator? If we
define an isomorphism ∗ = h+ dN + Id∇(M) (where the last operator is zero on
the remaining direct summands), then we have ∗−1 = −h− dN + Id∇(M), and
−1 = (h+dN+Id∇(M))dN (−h−dN+Id∇(M)) = −hdNh = −hIdIm(dN ) = h.
Remark 7. The operator d† is more like the BV operator than the (even)
Laplacian, which is not square-zero. Another similar case is Q (BRST operator)
and b0 (anti-ghost operator), for which we have Qb0 + b0Q = L0 (the degree
operator which is zero on the cohomology).
Proof. We only need to prove (2) and part of (3). First, we want to show that
−dNh = idIm(dN ) But then for x = dN (y), we have
− dNh(x) = [−dNh]dN (y)
= [hdN −∇π + idN ]dN (y)
= −∇[πdN ](y) + dN (y)
= −∇[−dMπ]dN (y) + x
Similarly, we would like to have −hdN = idIm(h). If x = h(y), then
− hdN (x) = [−hdN ]h(y)
= [dNh−∇π + idN ]h(y)
= −∇πh(y) + h(y)
Finally, we have π∇ = idM and ∇π∇ = ∇idM = ∇, which shows the isomor-
phism between ∇(M) and π(N). �
Example 6. Let (g, d) be a chain complex. Assume that the underlying ring
is a field. Then there exists a contraction
(H(g)
g, h) (9)
of chain complexes, where the differential on H(g) is zero: we can write g as a
linear sum
g = G⊕Ker(d) = G⊕ Im(d)⊕H(g)
by choosing arbitrary representatives of the homology classes etc.; let us show
the decomposition of an element x of g by
x = xG + xIm(d) + xH(g).
Then π is the projection of g onto H(g) and ∇ is the inclusion map of H(g) into
g. Note that as vector spaces G and Im(d) are isomorphic via d: let x, y ∈ G.
dx = dy ⇒ d(x − y) = 0 ⇒ x− y ∈ Ker(d) ∩G = {0}
and d : G → Im(d) is one-to-one as well as onto. We define h to be the inverse
of −d on Im(d) and zero on the rest of g. The linear map h is square-zero and
increases degree by one. Moreover,
(dh+ hd)(x)
= (dh+ hd)(xG + xIm(d) + xH(g))
= dh(xIm(d)) + hd(xG)
= −xIm(d) − xG
(∇π − idg)(xG + xIm(d) + xH(g))
= xH(g) − (xG + xIm(d) + xH(g))
= −xIm(d) − xG.
In comparison with the last corollary, we have
(N, dN ) = (g, d)
(M,dM ) = (H(g, d), 0)
Im(∇) = H(g, d)
Im(h) = G
Im(dN ) = Im(d).
Example 7. (The Tensor Trick) Any contraction (7) of chain complexes in-
duces a filtered contraction
(T c[M ]
T c[N ], T ch)
of coaugmented differential graded coalgebras. Here is how: the projection
PN : T
c[N ] → N
followed by the surjective chain map π : N → M gives us a linear map
π ◦ PN : T
c[N ] → M
(π ◦ PN )(x1 ⊗ · · · ⊗ xk) =
π(x1) if k = 1
0 otherwise.
which can then be made into a unique coalgebra map
T cπ : T c[N ] → T c[M ]
with the usual formula
T cπ(x1 ⊗ · · · ⊗ xk) =
x1 ⊗ · · · ⊗ π(xi)⊗ · · · ⊗ xk.
Next, the morphisms T cπ and T c∇ pass to the corresponding morphisms on
the coalgebras Sc[N ] and Sc[M ] respectively, and Sch is obtained from T ch by
symmetrization, to yield a contraction
(Sc[M ]
Sc[N ], Sch).
In particular, the contraction (9) induces
(Sc[sH(g)]
Sc[sg], Sch), (10)
which is a filtered contraction of coaugmented DG coalgebras. (Warning: Scπ
and Sc∇ are morphisms of coalgebras but Sch is not a coalgebra morphism,
although it is somewhat compatible with the coalgebra structure, being a ho-
motopy of coalgebra maps. One has to be careful when defining a homotopy of
cocommutative coalgebras.) ⋄
5.2 The first main theorem.
Assume that ∂ is the codifferential corresponding to an sh-Lie algebra struc-
ture on (g, d). Since the corresponding multilinear map on g has other compo-
nents than the binary bracket, we will denote the symmetric coalgebra on sg
with codifferential ∂ by Sc∂ [sg] and not by S
[ , ][sg]. Given two sh-Lie algebras
(g1, ∂1) and (g1, ∂2), an sh-morphism or sh-Lie map from g1 to g2 is a morphism
Sc∂1 [sg1] → S
[sg2] of DG coalgebras.
Theorem 1. Given a DGLA g and a contraction of chain complexes such as
(9), the data determine
(i) a differential D on Sc[sH(g)] (a coalgebra perturbation of the zero differen-
tial) turning the latter into a coaugmented DG coalgebra, hence endowing H(g)
with an sh-Lie algebra structure,
(ii) a Lie algebra twisting cochain
τ : ScD[sH(g)] → g
with adjoint τ̄ , written
τ̄ = (Sc∇)∂ : S
D[sH(g)] → C[g],
that induces an isomorphism on the homology, and
(iii) an extension of (Sc∇)∂ to a new contraction
(ScD[sH(g)]
(Scπ)∂
(Sc∇)∂
Sc∂ [sg], (S
cd)∂)
of filtered chain complexes (not necessarily of coalgebras).
Notes on Notation. While the induced bracket on H(g) is a strict graded
Lie bracket, the differential D may involve meaningful terms of higher order.
Let us introduce a table for the notation used in [8] for different types of chain
complexes and the corresponding symmetric coalgebras.
Chain complex Bracket(s) Sym. coalgebra Coderivation Property
(g, d) [ , ] (Sc[sg], d) ∂
graded generic DG coalgebra; coderivation
chain complex bilinear bracket induced diff. d induced by [ , ]
(g, d) [ , ]
Sc[ , ][sg], d
= C[g] ∂ (d+ ∂)
DGLA Lie bracket on g; generalized Koszul or coderivation
d derivation of it Chevalley-Eilenberg induced by [ , ]
complex
(g, 0) [ , ] (Sc[sg], 0) = Λ•(g) ∂ ∂2 = 0
Lie algebra Lie bracket Koszul complex codifferential
for homology induced by [ , ]
(g, d) ℓ2, ℓ3, . . . (S
∂ [sg], d) ∂ (d+ ∂)
2 = 0
L∞ algebra; higher brackets codiff. induced by
d = ℓ1 ℓ2, ℓ3, . . .
(H(g), 0) [ , ] (ScD[sH(g)], 0) D D
2 = 0
homology of induced Lie codiff. defined
DGLA (g, d) bracket on H(g) in the proof
with given
contraction
Sketch of Proof. We obtain the differential D and the twisting cochain τ on
Sc[sH(g)] as infinite series by induction: for b ≥ 1, write Scb for the homogeneous
degree-b component of Sc[sH(g)]. Then D, τ for b ≥ 2 are given by
τ = τ1 + τ2 + · · · , τ1 = ∇τH(g), τ
j : Scj → g, j ≥ 1, (11)
h([τ1, τb−1] + · · ·+ [τb−1, τ1])
D = D1 +D2 + · · · (12)
where Db−1 is the coderivation of Sc[sH(g)] determined by
τH(g)D
b−1 =
π([τ1, τb−1] + · · ·+ [τb−1, τ1]) : Scb → H(g).
That is, the coderivation followed by projection onto the degree-one subspace
sH(g) of Sc[sH(g)] is given by the above formula. In the notation of Subsec-
tion 2.3, we have fb = τH(g)D
b−1 and D = f̂ . For example (dropping the symbol
s for elements of sH(g)), we have τ1(x) = x ∈ H(g) ⊂ g, and τ2(xy) = h[x, y]
by (5). Let us also compute two terms of D:
τH(g)D
1(xy) = π[x, y] ∈ H(g),
τH(g)D
2(xyz) =
π( [x, h[y, z]] + [h[x, y], z] ),
etc. We can see why τ is a LA twisting cochain: since τ satisfies
τ = h
[τ, τ ]
we obtain
−dτ =
[τ, τ ]
in case of the particular SDR we constructed, and the last equation is the mas-
ter equation (the differential on H(g) being zero). The sums (11) and (12) are
infinite, but when either one is applied to a specific element in some subspace
of finite filtration degree, only finitely many terms will be nonzero. (The sum-
mand D1 is the ordinary Cartan-Chevalley-Eilenberg differential for the classi-
fying coalgebra of the graded Lie algebra H(g).) The proof that D is indeed
a coalgebra differential and τ is a twisting cochain “will be given elsewhere”.
A “spectral sequence argument” shows that τ̄ induces an isomorphism on the
homology.�
Remark 8. If ∇H(g) happens to be a Lie subalgebra of g, then [τ1, τ1] will
have values in H(g) and τ2 = (1/2)h[τ1, τ1], as well as the remaining τ j , will
be zero. Similarly, we will have D = D1.
Corollary 3. Under the hypotheses of Theorem 1,
τ : ScD[sH(g)] → g, (13)
viewed as an element of degree −1 of the DGLA Hom(ScD[sH(g)], g), satisfies
the master equation (2).
The twisting cochain (13) is our most general solution of the master equation.
The other solutions of the master equation can be derived from it.
6 Corollaries and the second main theorem
6.1 Other corollaries of Theorem 1.
Corollary 4. Under the hypotheses of Theorem 1, suppose in addition that
there is a differential D̃ on Sc[sH(g)] turning the latter into a coaugmented DG
coalgebra in such a way that (Scπ)∂ = D̃(Scπ). Then D = D̃ and (Scπ)∂ may
be taken to be Scπ. In particular, when (Scπ)∂ is zero, then the differential D
on Sc[sH(g)] is necessarily zero, that is, the new contraction in Theorem 1 has
the form
(Sc[sH(g)]
(Sc∇)∂
Sc∂ [sg], (S
ch)∂).
For example, this is the case when the composite
g ⊗ g
[ , ]
→ H(g)
is zero.
Corollary 5. Under the hypotheses of Theorem 1, suppose in addition that
there is a differential D̃ on Sc[sH(g)] turning the latter into a coaugmented DG
coalgebra in such a way that ∂(Sc∇) = (Sc∇)D̃. Then D = D̃ and (Sc∇)∂ =
S∇. In particular, when ∂(Sc∇) is zero, then the differential D on Sc[sH(g)]
is necessarily zero, that is, the new contraction in Theorem 1 has the form
(Sc[sH(g)]
(Scπ)∂
Sc∂ [sg], (S
ch)∂).
For example, this is the case when the composite
H(g)⊗H(g) →
→ g ⊗ g
[ , ]
is zero.
6.2 The second main theorem
Theorem 2. Given a DGLA g, a DGL subalgebra m of g, and a contraction
(H(g)
of chain complexes so that the composite
[ , ]
→ H(g)
is zero, then the induced bracket on H(g) is zero, that is, H(g) is abelian as a
graded Lie algebra, and the data determine a solution τ ∈ Hom(Sc[sH(g)], g) of
the master equation (2) in such a way that the following hold:
(i) The composite πτ coincides with the universal twisting cochain Sc[sH(g)] →
H(g) for the abelian Lie algebra H(g), and
(ii) the values of τ lie in m.
7 Differential Gerstenhaber and BV algebras
7.1 Differential Gerstenhaber algebras
Definition 8. A Gerstenhaber (or G-) algebra consists of
• A graded commutative and associative algebra (A, µ) (µ suppressed), and
• A graded Lie bracket (the Gerstenhaber or G− bracket) [ , ] : A⊗ A → A
of degree −1, such that
• For each homogeneous element a ∈ A, bracketing with a is a derivation of
the Lie bracket of degree |a| − 1.
That is, we want ad(a) to commute with the bracket for all a ∈ A.
Definition 9. A differential G-algebra is a Gerstenhaber algebra (A, [ , ]) with
a differential d of degree +1 on A which is a derivation of the multiplication on
We want [d, µ] = 0 and [d, d] = 0 in Gerstenhaber’s composition bracket
notation.
Definition 10. A differential G-algebra is called strict if the differential d is a
derivation of the G-bracket as well.
We want d to commute with the bracket.
Let (A, [ , ], d) be a strict differential G-algebra. We will for the moment
ignore all the extra structure on A except for the G-bracket and the differential.
As such, A is a DGLA, and we will change the notation to g to emphasize that.
We will use the grading
g1 = A
0, g0 = A
1, g−1 = A
2, . . . , g−n = A
n+1, . . .
so that the graded bracket and the differential on g are now “ordinary”: namely,
[ , ] : gj ⊗ gk → gj+k, d : gj → gj−1.
Consider a contraction of g onto H(g) as in (9). Let ∂ denote the operator
(“perturbation of d”) on Sc∂ [sg] corresponding to the Lie bracket on g. By the
Main Theorem (Theorem 1), we can transfer it to the symmetric coalgebra of
the homology: there exists a new contraction
(ScD[sH(g)]
(Scπ)∂
(Sc∇)∂
Sc∂ [sg], (S
cd)∂)
of not only filtered chain complexes but of filtered differential graded coalgebras.
The twisting cochain τ of (13) of Corollary 1 is now an element of
Hom(ScD[sH(g)], s
of degree −2, satisfying the master equation
[τ, τ ],
where D is the Hom-differential and the graded cup bracket on the right-hand
side refers to the one induced by the graded coalgebra structure on Sc[sH(g)]
and the graded Lie algebra structure on g.
7.2 Differential BV algebras
Definition 11. Let (A, [ , ]) be a G-algebra with an additional operator ∆ on
A of degree −1. If ∆ satisfies the condition
[a, b] = (−1)|a|
∆(ab)− (∆a)b − (−1)|a|a(∆b)
then it is said to be a generator of the G-algebra. In this case, (A,∆) is called
a weak Batalin-Vilkovisky (BV-) algebra. If, moreover, ∆ is exact (i.e. ∆2 = 0),
then (A,∆) is simply called a Batalin-Vilkovisky (BV-) algebra. Koszul has
shown that ∆ behaves as a derivation for the G-bracket:
∆[x, y] = [∆x, y]− (−1)|x|[x,∆y] ∀x, y ∈ A.
With respect to the original graded commutative and associative product on A,
we can only say that ∆ is a second order differential operator, or Φ3∆(a, b) = 0,
where Φr∆ are r-linear operators used to define higher order differential opera-
tors.
Now let us denote the bracket on a (weak) BV-algebra (A,∆) by [ , ].
Definition 12. If d is a differential of degree +1 that endows (A, [ , ]) with a
differential G-algebra structure, such that ∆d+d∆ = 0, then the triple (A,∆, d)
is called a (weak) differential BV-algebra.
Proposition 7. For any weak differential BV-algebra (A,∆, d), the differential
d behaves as a derivation of the G-bracket [ , ]:
d[x, y] = [dx, y]− (−1)|x|[x, dy].
That is, (A, [ , ], d) is a differential G-algebra.
Note: (A, [ , ],∆) is not a differential G-algebra unless ∆ is exact.
Example 8. [7] Let V be a Z2-graded finite dimensional vector space and sV
be its suspended-graded dual. If {xi} is a basis for V consisting of homogeneous
elements, then the dual basis {x∗i } has the property that xi and x
i always have
opposite parities. Then the algebra C[[x1, . . . , xn, x
1, . . . , x
n]] of formal power
series has the following BV operator (Laplacian):
Since the underlying algebra is graded commutative, the composition of two
derivations is a second order differential operator by any definition. Moreover
∆2 = 0, which makes a BV-algebra out of this data. ♦
7.3 Formality
7.3.1 Formality of differential graded P -algebras
Recall that our ground ring is a field of characteristic zero. Let P be a differential
graded operad and (A, d) be an algebra over P . We often want to know to
which extent the cohomology of a space reflects the underlying topological or
geometrical properties of that space.
Definition 13. The P -algebra A is called formal if there exists a strongly
homotopy P -algebra map (H(A), 0)
→ (A, d) which induces an isomorphism in
homology.
7.3.2 Examples
Example 9. (Commutative DG associative algebras.) A smooth mani-
fold M is called formal if the commutative associative DG algebra of de Rham
forms onM is formal in the sense of the above Definition. Examples are compact
Kähler manifolds, Lie groups, and complete intersections. Poisson manifolds
(proof by Sharygin and Talalaev). ⋄
Example 10. (DGLA’s.) The Hochschild complex for the algebra A =
C∞(M) of smooth functions on a Poisson manifold M (Kontsevich). ⋄
7.4 Differential BV algebras and formality
Definition 14. We will say a differential BV-algebra (A,∆, d) (A=g as a Lie
algebra) satisfies the statement of the Kählerian Formality Lemma (or the ∂∂̄
Lemma) if the maps
Ker(∆), d |Ker(∆)
→֒ (g, d),
Ker(∆), d |Ker(∆)
−→ H(g,∆)
are isomorphisms on the homology, where H(g,∆) is endowed with the zero
differential.
Remark 9. If the statement of the K.F.L. is satisfied, then proj can be extended
to a contraction
(H(g, d), 0)
Ker(∆), d |Ker(∆)
Since we have
H (Ker(∆), d) ∼= H(g, d)
H (Ker(∆), d) ∼= H(g,∆),
now we can have a contraction of Ker(∆) onto H(Ker(∆)) = H(g) as we did
with g and H(g).
Theorem 3. Let (A,∆, d) be a differential BV-algebra satisfying the statement
of the Kählerian Formality Lemma and extend the projection proj to a contrac-
(H(g, d), 0)
where
Ker(∆), d |Ker(∆)
and π =proj. Then H(g) is abelian as a graded Lie algebra and the data deter-
mine a solution
τ ∈ Hom(Sc[sH(g)], g)
of the master equation dτ =
[τ, τ ] in such a way that the following hold:
• The values of τ lie in m, that is, the composite
∆ ◦ τ : Sc[sH(g)] → g
is zero;
• The composite πτ coincides with the universal twisting cochain for the
abelian graded Lie algebra H(g); so that
• For k ≥ 2, the values of the component τk of τ on S
k[sH(g)] lie in Im(∆).
Proof. Follows from Theorem 2. �
Let us add the following condition to the ones in Theorem 3: suppose that
A consists of a single copy of the ground ring F (necessarily generated by the
unit 1 of A) and that ∆(1) = 0. Then 1 generates a central copy of the ground
ring in g (the ground ring commutes with all elements of A), and we may write
g = F ⊕ g̃
as a direct sum of differential graded Lie algebras. Here g̃ is the uniquely
determined complement of F . Why unique? We have g1 = A0 = F and we may
g−n =
An+1,
where g̃ will be closed under the (degree-zero) Lie bracket:
[ , ] : gj ⊗ gk → gj+k,
with j + k ≤ 0 if j, k ≤ 0.
Corollary 6. Assume that the hypotheses of Theorem 3, the abovementioned
conditions (A0 = F , ∆(1) = 0), and the condition H1(g) 6= 0 hold. Then the
contraction of the Theorem can be chosen in such a way that
Im(τk) ⊂ g̃ for k ≥ 2.
The statements of the main theorems have an interpretation in the context
of deformation theory, as explained below.
8 Deformation theory
Given a DGLA g, the construction of the universal solution τ from Theorems
2 and 3 relies on a chosen contraction. This provides a formal solution of the
master equation (MCE), with a perturbed differential D on Sc[sHg] in the
direction of (starting with) the Lie bracket induced on homology, endowing the
former with a dg-coalgebra structure and a twisting cochain:
τ : ScD[sHg] → g.
The moduli space interpretation of the set of solutions is along the lines of
Schlesinger-Stasheff [15]. Since our focus is on the construction of solutions of
the MCE, the reader is referred to the original text [8]. Additional details in
terms of deformation functors, tangent cohomology, and the Kuranishi functor
can be found in [13]. The relation between the latter functor and the construc-
tion of a twisting cochain corresponding to a contraction will be investigated
elsewhere.
References
[1] F. Akman and L.M. Ionescu, Higher derived brackets and deformation the-
ory I; arXiv:math.QA/0504541.
[2] F. Akman and L.M. Ionescu, Higher derived brackets and deformation the-
ory II, in preparation.
[3] F. Akman, L.M. Ionescu, and P. Sissokho, On deformation theory and
graph homology, J. Algebra 310 (2007), 730-741; arXiv:math.QA/0507077.
[4] S. Barannikov and M. Kontsevich, Frobenius manifolds and formality of
Lie algebras of polyvector fields, Internat. Math. Res. Notices 4 (1998),
201-215; arXiv:alg-geom/9710032.
[5] P. Chen, On the formality theorem for the DGLA of Drinfeld;
arXiv:math.QA/0601055.
[6] V. Dolgushev, A formality theorem for Hochschild chains;
arXiv:math.QA/ 0402248.
[7] D. Fiorenza, An introduction to the Batalin-Vilkovisky formalism; arXiv:
math.QA/0402057.
[8] J. Huebschmann and J. Stasheff, Formal solution of the master equa-
tion via HPT and deformation theory, Forum Math. 14 (2002), 847-868;
arXiv:math.AG/9906036.
[9] L.M. Ionescu, A combinatorial approach to coefficients in deformation
quantization; arXiv:math.QA/0404389.
[10] M. Kontsevich, Deformation quantization of Poisson manifolds I;
arXiv:q-alg/9709040.
[11] Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys. 69 (2004),
61-87.
[12] L. Lambe and J.D. Stasheff, Applications of perturbation theory to iterated
fibrations, Manuscripta Math. 58 (1987), 363-376.
[13] M. Manetti, Deformation theory via differential graded Lie algebras;
arXiv:math.AG/0507284.
[14] P. Real, Homological perturbation theory and associativity, Homology, Ho-
motopy, and Applications 2 (2000), 51-88.
[15] M. Schlessinger and J. Stasheff, Deformation theory and rational homotopy
type, Pub. Math. Sci. IHES (1998).
[16] T. Voronov, Higher derived brackets and homotopy algebras, J. Pure
Appl. Algebra 202 (2005), 133-153.
http://arxiv.org/abs/math/0504541
http://arxiv.org/abs/math/0507077
http://arxiv.org/abs/alg-geom/9710032
http://arxiv.org/abs/math/0601055
http://arxiv.org/abs/math/0402248
http://arxiv.org/abs/math/0402057
http://arxiv.org/abs/math/9906036
http://arxiv.org/abs/math/0404389
http://arxiv.org/abs/q-alg/9709040
http://arxiv.org/abs/math/0507284
	Introduction
	Perturbations of (co)differentials
	Derivations of the tensor algebra
	Coderivations of the tensor coalgebra
	Coderivations of the symmetric coalgebra
	DGLA's and perturbations of the codifferential
	Strongly homotopy Lie algebras
	The Hochschild chain complex and DGA's
	Strongly homotopy associative algebras
	Master equation
	Twisting cochain
	Differential on Hom
	Cup product and cup bracket
	Twisting cochain
	Homological perturbation theory (HPT)
	Contraction
	The first main theorem.
	Corollaries and the second main theorem
	Other corollaries of Theorem ??.
	The second main theorem
	Differential Gerstenhaber and BV algebras
	Differential Gerstenhaber algebras
	Differential BV algebras
	Formality
	Formality of differential graded P-algebras
	Examples
	Differential BV algebras and formality
	Deformation theory

  These notes, based on the paper "Formal Solution of the Master Equation via
HPT and Deformation Theory" by Huebschmann and Stasheff, were prepared for a
series of talks at Illinois State University with the intention of applying
Homological Perturbation Theory to the derived bracket constructions of
Kosmann-Schwarzbach and T. Voronov, and eventually writing Part II of the paper
"Higher Derived Brackets and Deformation Theory I" by the present authors.

Introduction
These notes, based on the paper [8] by Huebschmann and Stasheff, were pre-
pared for a series of talks at Illinois State University with the intention of ap-
plying Homological Perturbation Theory (HPT) to the construction of derived
brackets [11, 16], and eventually writing Part II of the paper [1].
Derived brackets are obtained by deforming the initial bracket via a deriva-
tion of the bracket. In [3] it was demonstrated that such deformations corre-
spond to solutions of the Maurer-Cartan equation, and the role of an “almost
contraction” was noted. This technique (see also [9]) is very similar to the itera-
tive procedure of [8] for finding the most general solution of the Maurer-Cartan
equation, i.e. the deformation of a given structure in a prescribed direction.
The present article, besides providing additional details of the condensed
article [8], forms a theoretical background for understanding and generalizing
the current techniques that give rise to derived brackets. The generalization,
which will be the subject matter of [2], will be achieved by using Stasheff and
Huebschmann’s universal solution. A second application of the universal solu-
tion will be in deformation quantization and will help us find the coefficients of
star products in a combinatorial manner, rather than as a byproduct of string
theory which underlies the original solution given by Kontsevich [10].
HPT is often used to replace given chain complexes by homotopic, smaller,
and more readily computable chain complexes (to explore “small” or “minimal”
models). This method may prove to be more efficient than “spectral sequences”
in computing (co)homology. One useful tool in HPT is
Lemma 1 (Basic Perturbation Lemma (BPL)). Given a contraction of N onto
M and a perturbation ∂ of dN , under suitable conditions there exists a pertur-
bation d∂ of dM such that H(M,dM + d∂) = H(N, dN + ∂).
The main question is: under what conditions does the BPL allow the preser-
vation of the data structures (DGA’s, DG coalgebras, DGLA’s etc.)? (We will
use the self-explanatory abbreviations such as DG for “differential graded”,
http://arxiv.org/abs/0704.0432v1
DGA for “differential graded (not necessarily associative) algebra”, and DGLA
for “differential graded Lie algebra”.)
Another prominent idea is that of a “(universal) twisting cochain” as a so-
lution of the “master equation”:
Proposition 1. Given a contraction of N onto M and a twisting cochain N →
A (A some DGA), there exists a unique twisting cochain M → A that factors
through the given one and which can be constructed inductively.
The explicit formulas are reminiscent of the Kuranishi map [13] (p.17), and
the relationship will be investigated elsewhere.
Note: we will assume that the ground ring is a field F of characteristic zero.
We will denote the end of an example with the symbol ♦ and the end of a proof
by �.
2 Perturbations of (co)differentials
2.1 Derivations of the tensor algebra
For any vector space V over F we have the isomorphismDer(TV ) ∼= Hom(V, TV )
where TV denotes the (augmented) tensor algebra on V . Namely, every linear
map f from V into TV extends uniquely into a derivation of the algebra TV
via the formula
f̂(v1 ⊗ · · · vn) =
v1 ⊗ · · · ⊗ f(vi)⊗ · · · vn.
Equivalently, every derivation of TV is determined by its restriction to V .
2.2 Coderivations of the tensor coalgebra
Similarly, we have the isomorphism Coder(T cV ) ∼= Hom(T cV, V ) where T cV is
the (coaugmented) coassociative tensor coalgebra of V , with counit η : T cV → F
(projection onto F ), and comultiplication
∆(v1 ⊗ · · · ⊗ vn) =
(v1 ⊗ · · · ⊗ vi)⊗ (vi+1 ⊗ · · · ⊗ vn).
Every linear map f = f1 + f2 + · · ·+ fn + · · · : T
cV → V (where fi : V
⊗i → V )
factors uniquely through a coderivation f̂ of T cV defined via the formula
f̂(v1 ⊗ · · · ⊗ vn) =
v1 ⊗ · · · ⊗ f1(vi)⊗ · · · vn
v1 ⊗ · · · ⊗ f2(vi ⊗ vi+1)⊗ · · · vn
+fn(v1 ⊗ · · · ⊗ vn).
That is, each coderivation on T cV is determined by itself followed by the projec-
tion onto V . Recall that the condition for f̂ to be a coderivation can be written
as ∆f̂ = (1⊗ f̂ + f̂ ⊗ 1)∆.
2.3 Coderivations of the symmetric coalgebra
Let us consider the cofree cocommutative counital coassociative algebra ST cV
on the vector space V as a subspace of T cV . The symmetric group Σn acts on
the left on V ⊗n via σ(v1 ⊗ · · · ⊗ vn) = vσ−1(1) ⊗ · · · ⊗ vσ−1(n). Then
ST cV =
(V ⊗n)Σn
is the space of invariants of this action. The action is compatible with the
coproduct on ST cV , so ST cV is a subcoalgebra of T cV which is cocommutative.
Note that ST (V ) is not a subalgebra with respect to the tensor multiplication
in T (V ); the product has to be symmetrized so that it projects back onto this
subspace (reminiscent of what T. Voronov does with derived brackets). The
projection (symmetrization) map P : T cV → ST cV is given by
P (v1 ⊗ · · · ⊗ vn) =
σ(v1 ⊗ · · · ⊗ vn).
This is not a coalgebra map, but is a retraction of the canonical inclusion
ST cV →֒ T cV . Now a coderivation D : T cV → T cV induces a coderivation
DS : ST
cV → ST cV by the composition ST cV →֒ T cV
→ T cV
→ ST cV .
In particular, a coderivation of T cV induces one of ΛcV = ST c[sV ] where V is
thought of as living in degree zero: we introduce the graded symmetric coalgebra
below. Once again, coderivations of ST cV are determined by their projections
onto ST c1V ; a map f = f1 + f2 + · · · : ST
c(V ) → V determines a coderivation
f̂ as in the tensor coalgebra case.
In the remaining part of this survey, we choose to identify ST cV with the
abstract symmetric coalgebra ScV under the isomorphism
v1 · · · vn 7→ P (v1 ⊗ · · · ⊗ vn).
The coproduct in ScV is given by
∆(v1 · · · vn) =
σ∈Σi,n−i
vσ(1) · · · vσ(i) ⊗ vσ(i+1) · · · vσ(n).
2.4 DGLA’s and perturbations of the codifferential
Definition 1. For any chain complex (X, d), and odd ∂, with (d + ∂)2 =
0, we say that ∂ is a perturbation of the differential d. We call d + ∂ the
perturbed differential. Equivalently, we have [d, ∂] + ∂∂ = 0 in End(X). If ∂ is
also compatible with an existing coalgebra structure on X , we say that it is a
coalgebra perturbation.
Let (g, d) be a graded chain complex (d lowers degrees) with a bracket [ , ]
that is skew-symmetric (not necessarily Leibniz or a chain map). Consider the
differential graded symmetric coalgebra Sc[sg], the differential d being induced
by that on g. Also let ∂ be the coderivation on Sc[sg] of degree −1 induced by
the bracket.
Proposition 2. The bracket [ , ] turns (g, d) into a DGLA if and only if ∂ is
a coalgebra perturbation of d. Also, any DGLA structure on g is determined by
the coalgebra perturbation induced from the bracket.
When g is an ordinary (degree-zero) Lie algebra over a field, Sc[sg] =
Λcg with differential ∂ corresponding to the bracket is the ordinary Koszul
or Chevalley-Eilenberg complex computing the homology of g with coefficients
in the field.
2.5 Strongly homotopy Lie algebras
Definition 2. Let (g, d) be a chain complex and let d also denote the codiffer-
ential in Sc[sg] induced by d. A strongly homotopy Lie (sh-Lie, or L∞) structure
on g is a perturbation ∂ = ∂2 + · · · + ∂n + · · · of d, i.e. an odd coderivation
satisfying [d, ∂] + ∂∂ = 0 and ∂η = 0 (recall that η is the counit) so that the
sum d+ ∂ endows Sc[sg] with a new coaugmented DG coalgebra structure.
The corresponding mega-map ℓ2 + · · · + ℓn + · · · from S
c(sg) to g extends
the differential ℓ1 = d : sg → g, and the lower identities satisfied by
ℓ = ℓ1 + ℓ2 + · · ·+ ℓn + · · ·
read as follows:
ℓ21 = 0
ℓ1(ℓ2(a, b))± ℓ2(ℓ1(a), b)± ℓ2(ℓ1(b), a) = 0
ℓ1(ℓ3(a, b, c))± ℓ3(ℓ1(a), b, c)± ℓ3(ℓ1(b), a, c)± ℓ3(ℓ1(c), a, b)
±ℓ2(ℓ2(a, b), c)± ℓ2(ℓ2(a, c), b)± ℓ2(ℓ2(b, c), a) = 0.
An sh-Lie morphism between two sh-Lie (or DGL) algebras (g, d + · · · ) and
(g′, d′ + · · · ) is a collection of chain maps Fn : S
n[sg] → S
′], satisfying
∆′F (u) = (F ⊗ F )(∆u). Then F is uniquely determined by its projection onto
sg′, that is, we may assume Fn : S
n[sg] → sg
Definition 3. A quasi-isomorphism F between sh-Lie algebras g, g′ is an sh-Lie
morphism such that F1 : sg → sg
′ induces an isomorphism between H(g, d) and
H(g′, d′).
Remark 1. Quasi-isomorphisms between DGLA’s are especially important in
deformation theory. Such a map gives a one-to-one correspondence between
moduli spaces of solutions to MC equations in ~g[[~]] and ~g′[[~]] (see [5]):
given a quasi-isomorphism F : Sc[sg] → sg′, we define F̃ : ~g[[~]] → ~g′[[~]] by
F̃ (r) =
Fn(r, . . . , r)
(also see [6]).
2.6 The Hochschild chain complex and DGA’s
Let (A, µ) be a unital associative algebra (possibly graded), and T c[sA] denote
the tensor coalgebra on the suspension of A. We recall that
Coder(T c[sA]) ∼= Hom(T
c[sA], A).
In particular, the associative bilinear multiplication µ ∈ Hom(T c[sA], A) corre-
sponds to a square-zero coderivation ∂ : T c[sA] → T c[sA] defined by
∂(a1 ⊗ · · · ⊗ an)
(−1)i+1(a1 ⊗ · · · ⊗ µ(ai ⊗ ai+1)⊗ · · · ⊗ an)
+(−1)n+1(µ(an ⊗ a1)⊗ a2 ⊗ · · · ⊗ an−1).
The condition that ∂ is a codifferential is equivalent to the associativity con-
dition m ◦ m = 0 where ◦ is the Gerstenhaber composition on multilinear
maps (a right pre-Lie map). The complex (Hom(T c[sA], A), ∂) is known as
the Hochschild chain complex.
Now let (A, µ, d) be a DGA. Then d+µ ∈ Hom(T c[sA], A) corresponds to a
perturbed codifferential d+ ∂ satisfying (d+ ∂)2 = 0, which is equivalent to the
identities d2 = 0 and [d, ∂] + ∂∂ = 0. The latter can also be split into [d, ∂] = 0
and ∂∂ = 0.
Proposition 3. The multiplication µ turns (A, d) into a DGA if and only if ∂
is a coalgebra perturbation of d. Also, any DGA structure on A is determined
by the coalgebra perturbation induced from µ.
2.7 Strongly homotopy associative algebras
Definition 4. Let (A, d) be a chain complex and let d also denote the codifferen-
tial in T c[sA] induced by d. A strongly homotopy associative (or A∞) structure
on A is a perturbation ∂ = ∂2 + · · · + ∂n + · · · of d, i.e. an odd coderivation
satisfying [d, ∂] + ∂∂ = 0 and ∂η = 0 so that the sum d+ ∂ endows T c[sA] with
a new coaugmented DG coalgebra structure.
The corresponding mega-map m2+ · · ·+mn+ · · · from T
c[sA] to A extends
the differential m1 = d : sA → A, and the lower identities satisfied by
m = m1 +m2 + · · ·+mn + · · ·
read as follows:
m21 = 0
m1(m2(a, b))±m2(m1(a), b)±m2(m1(b), a) = 0
m1(m3(a, b, c))±m3(m1(a), b, c)±m3(a,m1(b), c)±m3(a, b,m1(c)) = 0.
The mega-identity is m ◦ m = 0, sometimes written in the braces notation
{m}{m} = 0.
3 Master equation
If (A, d) is a differential graded associative algebra (DGA), then the equation
dτ = ττ (1)
is called the Master Equation (ME) (or Maurer-Cartan equation (MCE), etc.).
Similarly if (g, d) is a DGLA, then the equation
[τ, τ ] (2)
is also called the Master Equation. Sometimes the sign convention is
dτ + ττ = 0 (3)
[τ, τ ] = 0. (4)
Clearly any solution of such an equation must be an odd element of the algebra.
Moreover, in case A is the graded universal enveloping algebra of g, or g is the
Lie algebra obtained from A by the usual bracket, then solutions of the DGLA
master equation are also solutions of the DGA master equation.
Remark 2. If τ is a solution of Eq. (2) or Eq. (4) in a DGLA g, then the
odd derivation dτ = d− adτ or dτ = d+ adτ respectively defines a new DGLA
structure on g with respect to the old bracket.
Example 1. If g is a DGLA or L∞ algebra, then the corresponding coal-
gebra perturbation ∂ in Coder(Sc(sg)) is a solution of the ME in the DGA
End(Sc(sg)), where the differential is D = add. ⋄
Example 2. Gauge Theory: Let ξ be a principal bundle with structure group
G and Lie algebra g. There is a graded Lie algebra structure on the ad(ξ)-valued
de Rham forms induced by g. Given a connection A and an ad(ξ)-valued 1-form
η, the sum A+ η is again a connection, and its curvature is
FA+η = FA + dAη +
[η, η].
In particular, FA = FA+η if and only if
dAη +
[η, η] = 0
(the Maurer-Cartan equation). Here dA is the covariant derivative of the con-
nection A. When A is a flat connection (zero curvature) then there exists a
DGLA structure on the ad(ξ)-valued differential forms (d2A = 0) and FA+η is
also flat iff the MCE is satisfied (then the covariant derivative for A+ η is dτ ).
4 Twisting cochain
The notion of a twisting cochain generalizes that of a connection in differential
geometry.
4.1 Differential on Hom
If (C, dC) and (A, dA) are chain complexes, the following differential D makes
Hom(C,A) into a chain complex: Dφ = dAφ± φdC .
4.2 Cup product and cup bracket
Proposition 4. For any differential graded coalgebra C and a differential graded
associative algebra (DGA) A, the chain complex (Hom(C,A), D) becomes a
DGA via the cup (convolution) product a ⌣ b defined by the composition
−→ C ⊗ C
−→ A⊗A
−→ A.
The coaugmentation and augmentation maps η and ǫ on C and A respectively
define an augmentation map on (Hom(C,A), D).
Proposition 5. For any differential graded coalgebra C and a DGLA g, the
chain complex (Hom(C, g), D) becomes a DGLA via the cup bracket [a, b] defined
by the composition
−→ C ⊗ C
−→ g ⊗ g
[ , ]
−→ g.
Example 3. If g is a Lie algebra, then the cup bracket on Hom(Sc[sg], g) is
defined as above. For example, if τ and κ are maps Sc1[g] → g, then [τ, κ] may
be nonzero only on Sc2[sg]. In this case, we compute
∆(xy) = 1⊗ xy + x⊗ y + y ⊗ x+ xy ⊗ 1 (x, y ∈ sg),
[τ, κ](xy) = [τ(x), κ(y)] + [τ(y), κ(x)]. (5)
Example 4. The Hochschild complex of an associative algebra (A, µ) (where
µ2 = 0): Let
C•(A) = Hom(T cA,A) = Hom(
A⊗n, A),
with differential D = adµ ∈ Der(C
•(A)). The cup product x ⌣ y = {µ}{x, y}
is the composition
−→ T cA⊗ T cA
−→ A⊗A
−→ A;
if x is an n-linear map and y is an m-linear map, then
(x ⌣ y)(a1 ⊗ · · · ⊗ an+m) = x(a1 ⊗ · · · ⊗ an) · y(an+1 ⊗ · · · ⊗ an+m).
Remark 3. The differential D above is an inner derivation and not derived
from differentials on A and T cA. Still, it is a derivation of the cup product.
4.3 Twisting cochain
Definition 5. Given a coaugmented DG coalgebra C and an augmented DGA
A, a twisting cochain is a homogeneous morphism t : C → A of degree −1 such
that ǫτ = 0 and τη = 0 , and which satisfies Dt = t ⌣ t.
In other words, a twisting cochain is a solution of the master equation on
Hom(C,A) with the usual differential D induced from those of C, A and the
product is the cup product.
Definition 6. Given a DG cocommutative coalgebra C and a DGLA g, a Lie
algebra twisting cochain t : C → g is a homogeneous map of degree −1 whose
composition with the coaugmentation map is zero, and which satisfies
[t, t] (6)
([ , ] being the cup bracket).
Recall that a DGLA structure on a graded chain complex (g, d) is given by
a perturbation ∂ of the corresponding codifferential on Sc[sg]. Moreover, the
piece d∂ + ∂d = 0 of (d+ ∂)2 = 0 says that the bracket is a chain map and the
piece ∂2 = 0 says that the bracket satisfies the Jacobi identity. Let us denote
the symmetric coalgebra with the codifferential ∂ by Sc
[ , ]
[sg]. Quillen’s notation
C[g] for the same DG coalgebra reminds us that this is the Koszul or Chevalley-
Eilenberg complex that computes the homology of g without any regard for the
additional differential d on g.
Example 5. For any DGLA g, its universal Lie algebra twisting cochain
τg : S
[ , ][sg] → g
is given by
τg(sx) = x for x ∈ g
τg(y) = 0 for y ∈ S
k[sg], k 6= 1.
That is, an element with tensor degree one goes to its desuspension and ev-
erything else goes to zero. Clearly, the composition τgη is zero, as τg = 0 on
constants. Next, we show that τg satisfies the equation (6), but we note that in
this construction the differential on g itself is taken to be zero. On the left-hand
side, we have
Dτg(x1 ∧ · · · ∧ xn) = τg∂(x1 ∧ · · · ∧ xn)
which is zero if n 6= 2 and is equal to [x1, x2] if n = 2. Meanwhile, on the
right-hand side, we have
[τg, τg](x1 ∧ · · · ∧ xn)
which is zero if n 6= 2 and is equal to
{[τg(x1), τg(x2)]± [τg(x2), τg(x1)]} = [x1, x2]
if n = 2. ⋄
Remark 4. The universal property of the universal LA twisting cochain is that
every Lie algebra twisting cochain factors through this one: that is, whenever
C is a coalgebra and τ : C → g is a twisting cochain, then τg ◦ c(τ) = τ where
c(τ) : C → Sc
[ , ]
[sg] is the unique coalgebra map induced by τ .
Using HPT, we will construct formal solutions
τ ∈ Hom(ScD[sH(g)], g)
of the master equation. Once we make the choice of a contraction, we will obtain
explicit inductive formulas for D and τ .
5 Homological perturbation theory (HPT)
“HPT is concerned with transferring various kinds of algebraic structure through
a homotopy equivalence”. Also: “HPT is a set of techniques for the transference
of structures from one object to another up to homotopy” (Real [14]).
5.1 Contraction
Definition 7. Let (M,dM ) and (N, dN ) be chain complexes, π : N → M
and ∇ : M → N be chain maps, and h ∈ End(N) be a morphism (possibly
preserving some extra structure) of degree 1. Then a contraction
N, h) (7)
of N onto M is a collection of the above data satisfying
π∇ = idM
D(h) = addN (h) = ∇π − idN
πh = 0, h∇ = 0, hh = 0.
Another way to describe this structure is to say that M is a strong deformation
retract (SDR) of N (also called Eilenberg-Zilber data). The properties on the
last line are referred to as the annihilation properties or side conditions. Note
that the first line makes π surjective (projection) and ∇, injective (inclusion).
The map h is also known as the homotopy operator between ∇π and idN :
∇π − idN = D(h) = dNh+ hdN
(D = addN is the induced differential on Hom(N,N)).
Often filtered contractions are considered.
Remark 5. Lambe and Stasheff [12] noticed that the side conditions on h are
not restrictive: if πh = 0 and h∇ = 0 are not satisfied, then we can replace
h by h′ = D(h)hD(h). Now if h2 = 0 is not satisfied either, we replace h′ by
h′′ = h′dNh
′, which finally gives us an operator h′′ satisfying the side conditions.
Lemma 2. Given a contraction (7), we have a (not necessarily direct) sum
N = Im(∇) + Im(h) + Im(dN ).
Proof. Each x ∈ N can be written as
x = ∇π(x) − hdN (x) − dNh(x). (8)
Lemma 3. [14] Given a contraction (7), we have
Im(∇) + Im(h) = Im(∇)⊕ Im(h) = Ker(h).
Proof. We have
Im(∇) ⊂ Ker(h) and Im(h) ⊂ Ker(h)
since h∇ = 0 and h2 = 0. Conversely, by (8), each x ∈ Ker(h) can be written
x = ∇π(x) − hdN (x) ∈ Im(∇) + Im(h).
That the sum is direct can be seen as follows: let x ∈ Im(∇) ∩ Im(h). Then we
have x = ∇(y) = h(z) for some y ∈ M and z ∈ N . Rewriting the decomposition
of x in Ker(h), we obtain
x = ∇π(x) − hdN (x)
= ∇πh(z)− hdN∇(y)
= 0− hdN∇(y) (πh = 0)
= h∇dM (y) (∇ chain map)
= 0 (h∇ = 0).
Corollary 1. For any contraction (7), we have
H(N, h) ∼= Im(∇) ∼= M.
Lemma 4. Given any contraction (7), we have
Im(h) + Im(dN ) = Im(h)⊕ Im(dN ).
Moreover, if dM ≡ 0, then
Im(h)⊕ Im(dN ) = Ker(π).
Proof. Say x ∈ Im(h) ∩ Im(dN ). Then x = h(y) = dN (z) for some y, z ∈ N ,
so that by (8) we obtain
x = ∇πh(y) − hdNdN (z)− dNhh(y) = 0.
It is always true that Im(h) ⊂ Ker(π) as πh = 0. If dM ≡ 0, we further have
the result
πdN (x) = −dMπx = 0,
so that altogether
Im(h)⊕ Im(dN ) ⊂ Ker(π).
Conversely, for x ∈ Ker(π), we see that (even without the condition dM = 0)
Ker(π) ⊂ Im(h) + Im(dN )
since
x = −hdN (x)− dNh(x) ∀x ∈ Ker(π)
due to (8). �
Lemma 5. For any contraction (7) with dM = 0 we have
Im(∇) + Im(dN ) = Im(∇) ⊕ Im(dN ) = Ker(dN ).
Proof. The given sum is direct: let x ∈ Im(∇) ∩ Im(dN ). Then x = ∇(y) =
dN (z) and by (8)
x = ∇πdN (z)− hdNdN (z)− dNh∇(y) = ∇πdN (z) = −∇dMπ(z) = 0.
Clearly, we have Im(dN ) ⊂ Ker(dN ). Also Im(∇) ⊂ Ker(dN ), because
dN∇(x) = −∇dM (x) = 0.
Conversely, by (8),
Ker(dN ) ⊂ Im(∇) + Im(dN )
since we can write
x = ∇π(x) − dNh(x)
(no condition on dM ) for x ∈ Ker(dN ). �
Corollary 2. For any contraction (7) with dM = 0 we have
H(N, dN ) ∼= Im(∇) ∼= M.
Proposition 6. For any contraction (7) with dM = 0 we have
N = Im(∇)⊕ Im(h)⊕ Im(dN )
where
Im(∇)⊕ Im(h) = Ker(h)
Im(h)⊕ Im(dN ) = Ker(π)
Im(∇)⊕ Im(dN ) = Ker(dN ),
2. Im(h)
→ Im(dN ) is an isomorphism with inverse Im(dN )
→ Im(h), and
3. Im(∇)
→ M is an isomorphism with inverse M
→ Im(∇); we also have
Im(∇) ∼= M = Im(π) ∼= H(N, dN ) ∼= H(N, h).
Remark 6. This is a Hodge-type decomposition reminiscent of the case of a
compact orientable Riemannian manifold M without boundary. If
∗ : Ωr(M) → Ωdim(M)−r
is the “Hodge star operator” (an isomorphism) and
d : Ωr−1(M) → Ωr(M)
is the de Rham differential, then we define a “partial inverse” d† (the adjoint
exterior derivative operator) to −d by d† = ± ∗ d∗. The commutator of d and
d† is called the “Laplace-Beltrami operator”: ∆ = dd† + d†d. Then there exists
a unique decomposition of the algebra of de Rham forms as follows:
Ωr(M) = Harmr(M)⊕ d(Ωr−1(M))⊕ d†(Ωr+1(M)),
where the “harmonic forms” are given by Harmr(M) = Ker(∆). In the case of
our general contraction with dM = 0, the operators h and dN replace d and d
respectively. What do we know about ∆ here? We have
∆ = D(h) = hdN + dNh = (h+ dN )
2 = ∇π − idN .
The kernel of this operator is equal to ∇(M), as we have
(∇π − idN )(x) = 0 ⇔ ∇π(x) = x ⇔ x ∈ ∇(M),
or Ker(∆) = Im(∇). So is there an analog of the Hodge star operator? If we
define an isomorphism ∗ = h+ dN + Id∇(M) (where the last operator is zero on
the remaining direct summands), then we have ∗−1 = −h− dN + Id∇(M), and
−1 = (h+dN+Id∇(M))dN (−h−dN+Id∇(M)) = −hdNh = −hIdIm(dN ) = h.
Remark 7. The operator d† is more like the BV operator than the (even)
Laplacian, which is not square-zero. Another similar case is Q (BRST operator)
and b0 (anti-ghost operator), for which we have Qb0 + b0Q = L0 (the degree
operator which is zero on the cohomology).
Proof. We only need to prove (2) and part of (3). First, we want to show that
−dNh = idIm(dN ) But then for x = dN (y), we have
− dNh(x) = [−dNh]dN (y)
= [hdN −∇π + idN ]dN (y)
= −∇[πdN ](y) + dN (y)
= −∇[−dMπ]dN (y) + x
Similarly, we would like to have −hdN = idIm(h). If x = h(y), then
− hdN (x) = [−hdN ]h(y)
= [dNh−∇π + idN ]h(y)
= −∇πh(y) + h(y)
Finally, we have π∇ = idM and ∇π∇ = ∇idM = ∇, which shows the isomor-
phism between ∇(M) and π(N). �
Example 6. Let (g, d) be a chain complex. Assume that the underlying ring
is a field. Then there exists a contraction
(H(g)
g, h) (9)
of chain complexes, where the differential on H(g) is zero: we can write g as a
linear sum
g = G⊕Ker(d) = G⊕ Im(d)⊕H(g)
by choosing arbitrary representatives of the homology classes etc.; let us show
the decomposition of an element x of g by
x = xG + xIm(d) + xH(g).
Then π is the projection of g onto H(g) and ∇ is the inclusion map of H(g) into
g. Note that as vector spaces G and Im(d) are isomorphic via d: let x, y ∈ G.
dx = dy ⇒ d(x − y) = 0 ⇒ x− y ∈ Ker(d) ∩G = {0}
and d : G → Im(d) is one-to-one as well as onto. We define h to be the inverse
of −d on Im(d) and zero on the rest of g. The linear map h is square-zero and
increases degree by one. Moreover,
(dh+ hd)(x)
= (dh+ hd)(xG + xIm(d) + xH(g))
= dh(xIm(d)) + hd(xG)
= −xIm(d) − xG
(∇π − idg)(xG + xIm(d) + xH(g))
= xH(g) − (xG + xIm(d) + xH(g))
= −xIm(d) − xG.
In comparison with the last corollary, we have
(N, dN ) = (g, d)
(M,dM ) = (H(g, d), 0)
Im(∇) = H(g, d)
Im(h) = G
Im(dN ) = Im(d).
Example 7. (The Tensor Trick) Any contraction (7) of chain complexes in-
duces a filtered contraction
(T c[M ]
T c[N ], T ch)
of coaugmented differential graded coalgebras. Here is how: the projection
PN : T
c[N ] → N
followed by the surjective chain map π : N → M gives us a linear map
π ◦ PN : T
c[N ] → M
(π ◦ PN )(x1 ⊗ · · · ⊗ xk) =
π(x1) if k = 1
0 otherwise.
which can then be made into a unique coalgebra map
T cπ : T c[N ] → T c[M ]
with the usual formula
T cπ(x1 ⊗ · · · ⊗ xk) =
x1 ⊗ · · · ⊗ π(xi)⊗ · · · ⊗ xk.
Next, the morphisms T cπ and T c∇ pass to the corresponding morphisms on
the coalgebras Sc[N ] and Sc[M ] respectively, and Sch is obtained from T ch by
symmetrization, to yield a contraction
(Sc[M ]
Sc[N ], Sch).
In particular, the contraction (9) induces
(Sc[sH(g)]
Sc[sg], Sch), (10)
which is a filtered contraction of coaugmented DG coalgebras. (Warning: Scπ
and Sc∇ are morphisms of coalgebras but Sch is not a coalgebra morphism,
although it is somewhat compatible with the coalgebra structure, being a ho-
motopy of coalgebra maps. One has to be careful when defining a homotopy of
cocommutative coalgebras.) ⋄
5.2 The first main theorem.
Assume that ∂ is the codifferential corresponding to an sh-Lie algebra struc-
ture on (g, d). Since the corresponding multilinear map on g has other compo-
nents than the binary bracket, we will denote the symmetric coalgebra on sg
with codifferential ∂ by Sc∂ [sg] and not by S
[ , ][sg]. Given two sh-Lie algebras
(g1, ∂1) and (g1, ∂2), an sh-morphism or sh-Lie map from g1 to g2 is a morphism
Sc∂1 [sg1] → S
[sg2] of DG coalgebras.
Theorem 1. Given a DGLA g and a contraction of chain complexes such as
(9), the data determine
(i) a differential D on Sc[sH(g)] (a coalgebra perturbation of the zero differen-
tial) turning the latter into a coaugmented DG coalgebra, hence endowing H(g)
with an sh-Lie algebra structure,
(ii) a Lie algebra twisting cochain
τ : ScD[sH(g)] → g
with adjoint τ̄ , written
τ̄ = (Sc∇)∂ : S
D[sH(g)] → C[g],
that induces an isomorphism on the homology, and
(iii) an extension of (Sc∇)∂ to a new contraction
(ScD[sH(g)]
(Scπ)∂
(Sc∇)∂
Sc∂ [sg], (S
cd)∂)
of filtered chain complexes (not necessarily of coalgebras).
Notes on Notation. While the induced bracket on H(g) is a strict graded
Lie bracket, the differential D may involve meaningful terms of higher order.
Let us introduce a table for the notation used in [8] for different types of chain
complexes and the corresponding symmetric coalgebras.
Chain complex Bracket(s) Sym. coalgebra Coderivation Property
(g, d) [ , ] (Sc[sg], d) ∂
graded generic DG coalgebra; coderivation
chain complex bilinear bracket induced diff. d induced by [ , ]
(g, d) [ , ]
Sc[ , ][sg], d
= C[g] ∂ (d+ ∂)
DGLA Lie bracket on g; generalized Koszul or coderivation
d derivation of it Chevalley-Eilenberg induced by [ , ]
complex
(g, 0) [ , ] (Sc[sg], 0) = Λ•(g) ∂ ∂2 = 0
Lie algebra Lie bracket Koszul complex codifferential
for homology induced by [ , ]
(g, d) ℓ2, ℓ3, . . . (S
∂ [sg], d) ∂ (d+ ∂)
2 = 0
L∞ algebra; higher brackets codiff. induced by
d = ℓ1 ℓ2, ℓ3, . . .
(H(g), 0) [ , ] (ScD[sH(g)], 0) D D
2 = 0
homology of induced Lie codiff. defined
DGLA (g, d) bracket on H(g) in the proof
with given
contraction
Sketch of Proof. We obtain the differential D and the twisting cochain τ on
Sc[sH(g)] as infinite series by induction: for b ≥ 1, write Scb for the homogeneous
degree-b component of Sc[sH(g)]. Then D, τ for b ≥ 2 are given by
τ = τ1 + τ2 + · · · , τ1 = ∇τH(g), τ
j : Scj → g, j ≥ 1, (11)
h([τ1, τb−1] + · · ·+ [τb−1, τ1])
D = D1 +D2 + · · · (12)
where Db−1 is the coderivation of Sc[sH(g)] determined by
τH(g)D
b−1 =
π([τ1, τb−1] + · · ·+ [τb−1, τ1]) : Scb → H(g).
That is, the coderivation followed by projection onto the degree-one subspace
sH(g) of Sc[sH(g)] is given by the above formula. In the notation of Subsec-
tion 2.3, we have fb = τH(g)D
b−1 and D = f̂ . For example (dropping the symbol
s for elements of sH(g)), we have τ1(x) = x ∈ H(g) ⊂ g, and τ2(xy) = h[x, y]
by (5). Let us also compute two terms of D:
τH(g)D
1(xy) = π[x, y] ∈ H(g),
τH(g)D
2(xyz) =
π( [x, h[y, z]] + [h[x, y], z] ),
etc. We can see why τ is a LA twisting cochain: since τ satisfies
τ = h
[τ, τ ]
we obtain
−dτ =
[τ, τ ]
in case of the particular SDR we constructed, and the last equation is the mas-
ter equation (the differential on H(g) being zero). The sums (11) and (12) are
infinite, but when either one is applied to a specific element in some subspace
of finite filtration degree, only finitely many terms will be nonzero. (The sum-
mand D1 is the ordinary Cartan-Chevalley-Eilenberg differential for the classi-
fying coalgebra of the graded Lie algebra H(g).) The proof that D is indeed
a coalgebra differential and τ is a twisting cochain “will be given elsewhere”.
A “spectral sequence argument” shows that τ̄ induces an isomorphism on the
homology.�
Remark 8. If ∇H(g) happens to be a Lie subalgebra of g, then [τ1, τ1] will
have values in H(g) and τ2 = (1/2)h[τ1, τ1], as well as the remaining τ j , will
be zero. Similarly, we will have D = D1.
Corollary 3. Under the hypotheses of Theorem 1,
τ : ScD[sH(g)] → g, (13)
viewed as an element of degree −1 of the DGLA Hom(ScD[sH(g)], g), satisfies
the master equation (2).
The twisting cochain (13) is our most general solution of the master equation.
The other solutions of the master equation can be derived from it.
6 Corollaries and the second main theorem
6.1 Other corollaries of Theorem 1.
Corollary 4. Under the hypotheses of Theorem 1, suppose in addition that
there is a differential D̃ on Sc[sH(g)] turning the latter into a coaugmented DG
coalgebra in such a way that (Scπ)∂ = D̃(Scπ). Then D = D̃ and (Scπ)∂ may
be taken to be Scπ. In particular, when (Scπ)∂ is zero, then the differential D
on Sc[sH(g)] is necessarily zero, that is, the new contraction in Theorem 1 has
the form
(Sc[sH(g)]
(Sc∇)∂
Sc∂ [sg], (S
ch)∂).
For example, this is the case when the composite
g ⊗ g
[ , ]
→ H(g)
is zero.
Corollary 5. Under the hypotheses of Theorem 1, suppose in addition that
there is a differential D̃ on Sc[sH(g)] turning the latter into a coaugmented DG
coalgebra in such a way that ∂(Sc∇) = (Sc∇)D̃. Then D = D̃ and (Sc∇)∂ =
S∇. In particular, when ∂(Sc∇) is zero, then the differential D on Sc[sH(g)]
is necessarily zero, that is, the new contraction in Theorem 1 has the form
(Sc[sH(g)]
(Scπ)∂
Sc∂ [sg], (S
ch)∂).
For example, this is the case when the composite
H(g)⊗H(g) →
→ g ⊗ g
[ , ]
is zero.
6.2 The second main theorem
Theorem 2. Given a DGLA g, a DGL subalgebra m of g, and a contraction
(H(g)
of chain complexes so that the composite
[ , ]
→ H(g)
is zero, then the induced bracket on H(g) is zero, that is, H(g) is abelian as a
graded Lie algebra, and the data determine a solution τ ∈ Hom(Sc[sH(g)], g) of
the master equation (2) in such a way that the following hold:
(i) The composite πτ coincides with the universal twisting cochain Sc[sH(g)] →
H(g) for the abelian Lie algebra H(g), and
(ii) the values of τ lie in m.
7 Differential Gerstenhaber and BV algebras
7.1 Differential Gerstenhaber algebras
Definition 8. A Gerstenhaber (or G-) algebra consists of
• A graded commutative and associative algebra (A, µ) (µ suppressed), and
• A graded Lie bracket (the Gerstenhaber or G− bracket) [ , ] : A⊗ A → A
of degree −1, such that
• For each homogeneous element a ∈ A, bracketing with a is a derivation of
the Lie bracket of degree |a| − 1.
That is, we want ad(a) to commute with the bracket for all a ∈ A.
Definition 9. A differential G-algebra is a Gerstenhaber algebra (A, [ , ]) with
a differential d of degree +1 on A which is a derivation of the multiplication on
We want [d, µ] = 0 and [d, d] = 0 in Gerstenhaber’s composition bracket
notation.
Definition 10. A differential G-algebra is called strict if the differential d is a
derivation of the G-bracket as well.
We want d to commute with the bracket.
Let (A, [ , ], d) be a strict differential G-algebra. We will for the moment
ignore all the extra structure on A except for the G-bracket and the differential.
As such, A is a DGLA, and we will change the notation to g to emphasize that.
We will use the grading
g1 = A
0, g0 = A
1, g−1 = A
2, . . . , g−n = A
n+1, . . .
so that the graded bracket and the differential on g are now “ordinary”: namely,
[ , ] : gj ⊗ gk → gj+k, d : gj → gj−1.
Consider a contraction of g onto H(g) as in (9). Let ∂ denote the operator
(“perturbation of d”) on Sc∂ [sg] corresponding to the Lie bracket on g. By the
Main Theorem (Theorem 1), we can transfer it to the symmetric coalgebra of
the homology: there exists a new contraction
(ScD[sH(g)]
(Scπ)∂
(Sc∇)∂
Sc∂ [sg], (S
cd)∂)
of not only filtered chain complexes but of filtered differential graded coalgebras.
The twisting cochain τ of (13) of Corollary 1 is now an element of
Hom(ScD[sH(g)], s
of degree −2, satisfying the master equation
[τ, τ ],
where D is the Hom-differential and the graded cup bracket on the right-hand
side refers to the one induced by the graded coalgebra structure on Sc[sH(g)]
and the graded Lie algebra structure on g.
7.2 Differential BV algebras
Definition 11. Let (A, [ , ]) be a G-algebra with an additional operator ∆ on
A of degree −1. If ∆ satisfies the condition
[a, b] = (−1)|a|
∆(ab)− (∆a)b − (−1)|a|a(∆b)
then it is said to be a generator of the G-algebra. In this case, (A,∆) is called
a weak Batalin-Vilkovisky (BV-) algebra. If, moreover, ∆ is exact (i.e. ∆2 = 0),
then (A,∆) is simply called a Batalin-Vilkovisky (BV-) algebra. Koszul has
shown that ∆ behaves as a derivation for the G-bracket:
∆[x, y] = [∆x, y]− (−1)|x|[x,∆y] ∀x, y ∈ A.
With respect to the original graded commutative and associative product on A,
we can only say that ∆ is a second order differential operator, or Φ3∆(a, b) = 0,
where Φr∆ are r-linear operators used to define higher order differential opera-
tors.
Now let us denote the bracket on a (weak) BV-algebra (A,∆) by [ , ].
Definition 12. If d is a differential of degree +1 that endows (A, [ , ]) with a
differential G-algebra structure, such that ∆d+d∆ = 0, then the triple (A,∆, d)
is called a (weak) differential BV-algebra.
Proposition 7. For any weak differential BV-algebra (A,∆, d), the differential
d behaves as a derivation of the G-bracket [ , ]:
d[x, y] = [dx, y]− (−1)|x|[x, dy].
That is, (A, [ , ], d) is a differential G-algebra.
Note: (A, [ , ],∆) is not a differential G-algebra unless ∆ is exact.
Example 8. [7] Let V be a Z2-graded finite dimensional vector space and sV
be its suspended-graded dual. If {xi} is a basis for V consisting of homogeneous
elements, then the dual basis {x∗i } has the property that xi and x
i always have
opposite parities. Then the algebra C[[x1, . . . , xn, x
1, . . . , x
n]] of formal power
series has the following BV operator (Laplacian):
Since the underlying algebra is graded commutative, the composition of two
derivations is a second order differential operator by any definition. Moreover
∆2 = 0, which makes a BV-algebra out of this data. ♦
7.3 Formality
7.3.1 Formality of differential graded P -algebras
Recall that our ground ring is a field of characteristic zero. Let P be a differential
graded operad and (A, d) be an algebra over P . We often want to know to
which extent the cohomology of a space reflects the underlying topological or
geometrical properties of that space.
Definition 13. The P -algebra A is called formal if there exists a strongly
homotopy P -algebra map (H(A), 0)
→ (A, d) which induces an isomorphism in
homology.
7.3.2 Examples
Example 9. (Commutative DG associative algebras.) A smooth mani-
fold M is called formal if the commutative associative DG algebra of de Rham
forms onM is formal in the sense of the above Definition. Examples are compact
Kähler manifolds, Lie groups, and complete intersections. Poisson manifolds
(proof by Sharygin and Talalaev). ⋄
Example 10. (DGLA’s.) The Hochschild complex for the algebra A =
C∞(M) of smooth functions on a Poisson manifold M (Kontsevich). ⋄
7.4 Differential BV algebras and formality
Definition 14. We will say a differential BV-algebra (A,∆, d) (A=g as a Lie
algebra) satisfies the statement of the Kählerian Formality Lemma (or the ∂∂̄
Lemma) if the maps
Ker(∆), d |Ker(∆)
→֒ (g, d),
Ker(∆), d |Ker(∆)
−→ H(g,∆)
are isomorphisms on the homology, where H(g,∆) is endowed with the zero
differential.
Remark 9. If the statement of the K.F.L. is satisfied, then proj can be extended
to a contraction
(H(g, d), 0)
Ker(∆), d |Ker(∆)
Since we have
H (Ker(∆), d) ∼= H(g, d)
H (Ker(∆), d) ∼= H(g,∆),
now we can have a contraction of Ker(∆) onto H(Ker(∆)) = H(g) as we did
with g and H(g).
Theorem 3. Let (A,∆, d) be a differential BV-algebra satisfying the statement
of the Kählerian Formality Lemma and extend the projection proj to a contrac-
(H(g, d), 0)
where
Ker(∆), d |Ker(∆)
and π =proj. Then H(g) is abelian as a graded Lie algebra and the data deter-
mine a solution
τ ∈ Hom(Sc[sH(g)], g)
of the master equation dτ =
[τ, τ ] in such a way that the following hold:
• The values of τ lie in m, that is, the composite
∆ ◦ τ : Sc[sH(g)] → g
is zero;
• The composite πτ coincides with the universal twisting cochain for the
abelian graded Lie algebra H(g); so that
• For k ≥ 2, the values of the component τk of τ on S
k[sH(g)] lie in Im(∆).
Proof. Follows from Theorem 2. �
Let us add the following condition to the ones in Theorem 3: suppose that
A consists of a single copy of the ground ring F (necessarily generated by the
unit 1 of A) and that ∆(1) = 0. Then 1 generates a central copy of the ground
ring in g (the ground ring commutes with all elements of A), and we may write
g = F ⊕ g̃
as a direct sum of differential graded Lie algebras. Here g̃ is the uniquely
determined complement of F . Why unique? We have g1 = A0 = F and we may
g−n =
An+1,
where g̃ will be closed under the (degree-zero) Lie bracket:
[ , ] : gj ⊗ gk → gj+k,
with j + k ≤ 0 if j, k ≤ 0.
Corollary 6. Assume that the hypotheses of Theorem 3, the abovementioned
conditions (A0 = F , ∆(1) = 0), and the condition H1(g) 6= 0 hold. Then the
contraction of the Theorem can be chosen in such a way that
Im(τk) ⊂ g̃ for k ≥ 2.
The statements of the main theorems have an interpretation in the context
of deformation theory, as explained below.
8 Deformation theory
Given a DGLA g, the construction of the universal solution τ from Theorems
2 and 3 relies on a chosen contraction. This provides a formal solution of the
master equation (MCE), with a perturbed differential D on Sc[sHg] in the
direction of (starting with) the Lie bracket induced on homology, endowing the
former with a dg-coalgebra structure and a twisting cochain:
τ : ScD[sHg] → g.
The moduli space interpretation of the set of solutions is along the lines of
Schlesinger-Stasheff [15]. Since our focus is on the construction of solutions of
the MCE, the reader is referred to the original text [8]. Additional details in
terms of deformation functors, tangent cohomology, and the Kuranishi functor
can be found in [13]. The relation between the latter functor and the construc-
tion of a twisting cochain corresponding to a contraction will be investigated
elsewhere.
References
[1] F. Akman and L.M. Ionescu, Higher derived brackets and deformation the-
ory I; arXiv:math.QA/0504541.
[2] F. Akman and L.M. Ionescu, Higher derived brackets and deformation the-
ory II, in preparation.
[3] F. Akman, L.M. Ionescu, and P. Sissokho, On deformation theory and
graph homology, J. Algebra 310 (2007), 730-741; arXiv:math.QA/0507077.
[4] S. Barannikov and M. Kontsevich, Frobenius manifolds and formality of
Lie algebras of polyvector fields, Internat. Math. Res. Notices 4 (1998),
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[5] P. Chen, On the formality theorem for the DGLA of Drinfeld;
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[6] V. Dolgushev, A formality theorem for Hochschild chains;
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[9] L.M. Ionescu, A combinatorial approach to coefficients in deformation
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http://arxiv.org/abs/math/0504541
http://arxiv.org/abs/math/0507077
http://arxiv.org/abs/alg-geom/9710032
http://arxiv.org/abs/math/0601055
http://arxiv.org/abs/math/0402248
http://arxiv.org/abs/math/0402057
http://arxiv.org/abs/math/9906036
http://arxiv.org/abs/math/0404389
http://arxiv.org/abs/q-alg/9709040
http://arxiv.org/abs/math/0507284
	Introduction
	Perturbations of (co)differentials
	Derivations of the tensor algebra
	Coderivations of the tensor coalgebra
	Coderivations of the symmetric coalgebra
	DGLA's and perturbations of the codifferential
	Strongly homotopy Lie algebras
	The Hochschild chain complex and DGA's
	Strongly homotopy associative algebras
	Master equation
	Twisting cochain
	Differential on Hom
	Cup product and cup bracket
	Twisting cochain
	Homological perturbation theory (HPT)
	Contraction
	The first main theorem.
	Corollaries and the second main theorem
	Other corollaries of Theorem ??.
	The second main theorem
	Differential Gerstenhaber and BV algebras
	Differential Gerstenhaber algebras
	Differential BV algebras
	Formality
	Formality of differential graded P-algebras
	Examples
	Differential BV algebras and formality
	Deformation theory
