Medical, Pharma, Engineering, Science, Technology and Business

^{1}Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
**
E-mail:** [email protected]

^{2}Department of Mathematics, The Ohio State University at Newark,
1179 University Drive, Newark, OH 43055, USA**
E-mail:** [email protected]

**Received Date**: March 08, 2008; **Revised Date: **May 02, 2008

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

The cohomology of a finite-dimensional coalgebra over a finitely generated quadratic operad, with coefficients in itself, is defined and is shown to have the structure of a graded Lie algebra. The cohomology of such a coalgebra is isomorphic to the cohomology of its linear dual as graded Lie algebras.

Operads are mathematical devices that encode algebraic structures. They were originally introduced by May [24] to study iterated loop spaces in algebraic topology. Operads have since been used with great success in many other fields, including algebra, mathematical physics, and computer science. See [23] for a general survey of generalities and applications of operads. In particular, familiar structures such as associative algebras, Lie algebras, commutative algebras, and Gerstenhaber algebras are all algebras over some corresponding operads.

Let A be an associative algebra, and let C be a coassociative coalgebra. The structure
of the Hochschild cohomology [11] of A with self-coefficients is well known; it is
a Gerstenhaber algebra [6]. The same is true for the Hochschild coalgebra cohomology [12], which can be proved by basically the same arguments as in [6]. If, in addition, A is finitedimensional,
then and are isomorphic as graded vector spaces [26], where
A^{#} is the dual coalgebra of A.

In this paper, we generalize this duality isomorphism to finite-dimensional (co)algebras over
any finitely generated quadratic operad *P*. In particular, we will do the following:

(1) Define cohomology with self-coefficients for finite-dimensional P-coalgebras V .

(2) Observe that is a graded Lie algebra, whose graded Lie bracket is induced by one on the defining cochain complex .

(3) Show that and are isomorphic as graded Lie algebras, and vice versa,
where is the cohomology (as defined in [2]) of the dual P-algebra V ^{#} of V .

We note that the differential graded Lie algebra controls the deformations of V as a Pcoalgebra in the sense of Gerstenhaber [7].

When P is the associative algebra operad As, the cohomology coincides with. In this case, we recover the duality isomorphism in the Hochschild case discussed above, strengthened with the graded Lie brackets. Taking other finitely generated quadratic operads P, the above statements apply to other classical (co)algebras, including commutative [10], Lie [4], Poisson, and Gerstenhaber (co)algebras. They also cover, for example, the cases of Leibniz (co)algebras [14,15,16,18], Loday-type (co)algebras [1,5,17,19,20,21,27,28], and ennea-(co)algebras [13].

**1.1 Organization**

In the next section, we recall from [2,3] the constructions and some properties of the cochain complex that defines the cohomology with self-coefficients of a P-algebra A.

In Section 3, we study properties of P-coalgebras. In particular, it is observed (Theorem 3.3)
that a P-coalgebra structure is equivalent to a degree 1 differential derivation on the free graded
P^{!}-algebra generated by V , where P^{!} denotes the Koszul dual operad of P.

In Section 4, the cochain complex that defines the cohomology of a finitedimensional P-coalgebra V is defined. It is first shown to be a graded Lie algebra (Corollary 4.3). A P-coalgebra structure can be characterized as a square-zero element in this graded Lie algebra (Corollary 4.5). The differential on is then defined as, up to a nonzero scalar multiple, the inner derivation [−, π] with respect to the square-zero element π that defines the P-coalgebra structure on V (4.3). This makes into a differential graded Lie algebra (Corollary 4.6), which implies that is a graded Lie algebra (Corollary 4.7). Example 4.8 makes this explicit in the case of the Hochschild coalgebra cochain complex .

In Section 5, it is shown that, for a finite-dimensional P-coalgebra V , the differential graded Lie algebras and are isomorphic (Corollary 5.3). The duality isomorphism between and is then an immediate consequence after passing to cohomology (Corollary 5.4). The special case involving Hochschild (coalgebra) cohomology is explained in Example 5.5.

In Section 6, an explicit description of the differential and is given in terms of the
elementary operations o^{i} in the operad P^{!} (Theorems 6.1 and 6.2).

In this section, we recall some basic definitions and results about operads and the cohomology of an algebra over a quadratic operad with coefficients in itself.

**2.1 Conventions**

The symbol denotes the set of positive integers. Throughout this paper, we work over a fixed
field k of characteristic zero. Vector spaces, ⊗, Hom, and End (endomorphisms) are all meant
over k. For any positive integer n, Σ_{n} will denote the group of permutations on n elements. For σ ∈ Σ_{n}, (σ) ∈ {−1, 1} will stand for the sign of σ, and sgn_{n} will denote the sign representation
of Σ_{n}.

**2.2 Operads**

An operad [22,23,24,25] P consists of a right k[Σ_{n}]-module P(n), one for each n ∈
∗. For positive integers n, j_{1}, . . . , j_{n}, there is a structure map

γ : P(n)⊗P(j_{1})⊗· · ·⊗P(j_{n})→P(j_{1} + · · · + j_{n})

These structure maps satisfy some associativity, equivariance, and unity conditions, which can
be found in [24]. Using the operad structure maps, one defines the o_{i} operations as

f o_{i} g = γ(f; 1, . . . , 1, g, 1, . . . , 1) ∈ P(n + m − 1) (2.1)

for f ∈ P(n) and g ∈ P(m), where there are (i − 1) copies of 1’s in front of g. Conversely, the structure maps γ can be recovered from the ◦i operations as

(2.2)

for f ∈ P(n) and gi ∈ P(j_{i}) (1 ≤ i ≤ n). In the presence of the unit 1 ∈ P(1), the operad
structure maps γ are completely determined by the ◦i operations (see [22] or [23, Section 1.7.1,
p. 66]). In what follows, by using (2.1) and (2.2), we will use these two equivalent definitions of
an operad interchangeably.

**2.3 Non-Σ operads**

From the definition of an operad, if we omit the parts concerning the symmetric groups Σn (n ≥ 1), then we obtain the definition of a non-Σ operad.

**2.4 Operad morphisms**

Let P and Q be two operads. A morphism of operads from P to Q is a sequence a = {a(n), n ∈ } of k[Σn]-linear maps a(n): P(n) → Q(n) satisfying the conditions a(1)(1) = 1 and a(n + m − 1)(μ ◦i ν) = a(n)(μ) ◦i a(m)(ν) for n,m ∈ , 1 ≤ i ≤ n, μ ∈ P(n), and ν ∈ P(m).

**2.5 Endomorphism operad**

Let V be a vector space over k. For n ∈ , let End(V )(n) = Hom(V ^{⊗n}, V ). Then End(V) =
{End(V )(n), n ∈ } is naturally an operad under composition, called the endomorphism operad
of V . Indeed, f o_{i} g = f(1, . . . , g, . . . , 1), where g is in the ith place, and 1 is the identity map
on V . The Σ_{n}-action on End(V )(n) comes from composition with the left Σn-action on V ^{⊗n}.

**2.6 Algebras over an operad**

Let P be an operad. A P-algebra or an algebra over P is a vector space V over k along with a morphism of operads π : P → End(V ). Using adjunctions, a P-algebra structure on V can be expressed in terms of maps πn : P(n) ⊗Σn V ⊗n → V that satisfy the obvious associativity, equivariant, and unity conditions.

**Proposition 2.1** (see [9]). Let E be a right k[Σ_{2}]-module. Then there exists an operad F(E)
with F(E)(1) = k and F(E)(2) = E such that the following property holds: for any operad Q
and for any morphism of right k[Σ_{2}]-modules a: E → Q(2), there exists a unique morphism of
operads, : F(E)→Q, such that (2) = a.

**2.7 Free operad**

The operad F(E) is called the free operad generated by E. By the usual arguments, the free operad F(E) is unique up to operad isomorphisms.

**2.8 Free graded P-algebra**

For an operad P and a vector space V , define the free graded P-algebra generated by V as

where for σ ∈ Σn and v_{i} ∈ V . The homogeneous
degree n component of is denoted by. The P-algebra structure on is
the natural one defined by the operad structure on P and concatenation on V ^{⊗∗ }[3, 1.6.2].

**2.9 Graded derivations**

Let A = ⊕_{j≥0}A^{j} be a graded algebra over an operad P with structure maps πm: P(m) ⊗Σ_{m}
A^{⊗m} → A. A degree n derivation of A is a homogeneous degree n linear map d: A → A such
that

for m ∈ , μ ∈ P(m), and a_{i} ∈ A. Here s_{i} = |a_{1}| + · · · + |a_{i−1}| with s_{1} = 0, and |a| = j if
a ∈ A^{j} [3, Definition 2.3.1]. The set of degree n derivations of A is denoted by Der^{n}(A). Denote
by Der(A) the graded vector space ⊕_{n≥0} Der^{n}(A) of all derivations of A.

**2.10 Operadic ideals**

Let P be an operad. An ideal of P is a sequence I = {I(n), n ∈}, in which I(n) is a k[Σ_{n}]-
submodule of P(n), such that for μ ∈ P(n), ν ∈ P(m), x ∈ I(m), y ∈ I(n), and 1 ≤ i, j ≤ n,
one has that μ o_{i} x ∈ I(n + m − 1) and y o_{j} ν ∈ I(n + m − 1).

When I is an ideal of P, the quotient P/I = {(P/I)(n) = P(n)/I(n)} inherits an operad structure from P.

**2.11 Quadratic operads**

Let E be a right k[Σ_{2}]-module and let R be a right k[Σ_{3}]-submodule of F(E)(3). Let (R) be
the ideal generated by R. Then the quotient operad F(E)/(R) is called the quadratic operad
generated by E with relations R, denoted by P(k,E,R) [9]. A quadratic operad P(k,E,R) is
said to be finitely generated if E is a finite-dimensional vector space.

**Proposition 2.2** (see [3, Proposition 1.5.5]). Let P = P(k,E,R) be a quadratic operad. Then
a P-algebra structure on a vector space V is determined by a morphism of right k[Σ_{2}]-modules
π : P(2) = E → End(V )(2) such that (3)(R) = 0.

In this case, the morphism π : P(2) → End(V )(2), or equivalently its adjoint π : P(2) ⊗
Σ_{2}
V ^{⊗2 }→ V , is called the structural morphism of the P-algebra V .

**2.12 Quadratic duality**

Let F be a right k[Σ_{n}]-module. By F^{#} we mean the right k[Σ_{n}]-module F^{#} = Hom(F, k)⊗sgn_{n},
where the right Σ_{n}-action is given by for ∈ Hom(F, k) and x ∈ F.

Let E be a right k[Σ_{2}]-module. Then as right k[Σ_{3}]-modules, one has that [9] . Let R ⊂ F(E)(3) be a right k[Σ_{3}]-submodule, and let R⊥⊂ F(E^{#})(3) be the
annihilator of R in . The Koszul dual of the quadratic operad P =
P(k,E,R) is defined as the quadratic operad P^{!} = P(k,E^{#},R^{⊥}).

**2.13 The graded Lie algebra L_{P}(V )**

We briefly recall the cohomology of an algebra over a finitely generated quadratic operad, due to Balavoine [2,3].

For the rest of this section, let P = P(k,E,R) be a finitely generated quadratic operad,
and let V be a finite-dimensional vector space. To simplify notations, let (P^{!})^{#}(n) stand for
(P^{!}(n))^{#}. Define the vector spaces

where, with the natural structure of a left k[Σ_{n}]-module. Let
LP(V ) denote the graded vector space . Using the elementary operations
o_{i} in the operad P! and the non-Σ operad , one defines the operations
[2,3]

(2.3)

for and . These two operations are indeed well defined, i.e.,
independent of the choice of representing elements μ^{∗ }⊗ f and ν^{∗} ⊗ g. However, the individual
o_{i} operations are not well defined on L_{P}(V ) [3, Remark 2.4.4].

**Proposition 2.3** (see [3, Proposition 2.4.4]). The bracket [−,−] defined above makes L_{P}(V )
into a graded Lie algebra.

Indeed, once one establishes that o and [−,−] are well defined, this result follows from [8, (3)], since is a non-Σ operad.

**Proposition 2.4** (see [2, Proposition 3.1.4]). There is an isomorphism of vector spaces

With the obvious notations, the isomorphism Γ is given by

(2.4)

**2.14 The graded Lie algebra **

Using the isomorphism Γ, the graded Lie bracket on L_{P}(V ) can be transported to , which
makes into a graded Lie algebra of degree −1. More precisely, the graded Lie bracket on is defined as

The same can be said of the operation o, from which the graded Lie bracket is defined.

**Proposition 2.5** (see [3, Proposition 2.3.3]). There is an isomorphism of vector spaces

The space of derivations has a natural graded Lie bracket, namely, the commutator bracket. Using the isomorphism ω, this gives rise to another graded Lie bracket of degree −1 on . These two graded Lie brackets on are equal [3, 2.4.4].

**Theorem 2.6** (see [3, Theorem 2.4.1 and Corollary 2.4.2]). The following three sets are in
bijection with each other:

(1) The set of P-algebra structures on V .

(2) The set of degree 1 derivations that satisfy d^{2} = 0.

(3) The set of elements that satisfy [π, π] = 0.

When one of these equivalent conditions is satisfied, we say that (V, π) is a P-algebra.

Note that [π, π] = 0 is equivalent to the condition in Proposition 2.2.

**2.15 Cohomology of a P-algebra**

Let (A, π) be a P-algebra. Define the map by setting

(2.5)

By [2, Proposition 3.1.7], the map is a differential. The homology of the cochain complex is denoted by or , and it is called the cohomology of the P-algebra A with coefficients in itself or simply the operadic cohomology of A. With the induced Lie bracket, becomes a graded Lie algebra. Also, note that is a differential graded Lie algebra, which controls the deformations of V as a P-algebra [3, Section 4] in the sense of Gerstenhaber [7].

The purpose of this section is to give alternative characterizations of P-coalgebra structures on a finite-dimensional vector space in terms of differential derivations when P is a finitely generated quadratic operad.

**3.1 Coendomorphism operad**

Let V be a vector space. Let Coend(V) = {Hom(V, V ^{⊗n})} be the coendomorphism operad of
V with the obvious structure maps, dual to those in End(V ). For an operad P, a P-coalgebra
structure on V is a morphism P →Coend(V ) of operads.

For example, a coassociative coalgebra structure is equivalent to an As-coalgebra structure, where As is the associative algebra operad.

**Proposition 3.1. **Let P = P(k,E,R) be a finitely generated quadratic operad, and let V be
a finite-dimensional vector space. Then a P-coalgebra structure on V is determined by a k[Σ_{2}]-
equivariant morphism π : E = P(2) → Coend(V )(2), such that , : F(E) →
Coend(V ) is the unique operad morphism associated to π.

Proof. The same proof as Proposition 2.2 works here. Indeed, if π is as stated, then the morphism must factor through the quotient P = F(E)/(R). This is because commutes with
the o_{i} operations and every element in the ideal (R) is a sum of elements that are iterated o_{i}
products with at least one entry in R.

The condition can be expressed as [π, π] = 0 (Corollary 4.5).

**3.2 Duality isomorphism**

Let V be a finite-dimensional vector space. Denote by V ^{#} its linear dual Hom(V, k). Then for
each n ≥ 1, there is a linear isomorphism (see [26, Proposition 2.8])

(3.1)

given by ζ^{n}(f) = f^{#}, where

for α_{i} ∈ V ^{#} and a ∈ V . The notations on the right-hand side of the previous line is given by

**Theorem 3.2.** Let V be a finite-dimensional vector space. Then the maps ζ^{n} (n ≥ 1) assemble
to form an isomorphism of operads.

**Proof.** The maps ζ^{n} are linear isomorphisms, and it is clear that It remains to
check that they are compatible with the operad structure maps γ and that ζ^{n} is Σ_{n}-equivariant.
For f ∈ Coend(V )(k) and g_{i} ∈ Coend(V )(n_{i}) (1 ≤ i ≤ k), we have

where o denotes composition of functions. This shows that ζ is compatible with the operad structure maps.

One can see that the map ζ^{n} is Σ_{n}-equivariant because the Σ_{n}-action on Coend(V )(n) (resp.,
End(V^{ #})(n)) comes from composition with the right (resp., left) Σ_{n}-action on V ^{⊗n }(resp.,
(V ^{#})^{⊗n}). Indeed, pick f ∈ Coend(V )(n), σ ∈ Σ_{n}, α_{i} ∈ V ^{#} (1 ≤ i ≤ n), and a ∈ V . Then we
have

The equality (∗) comes from the fact that the two sets of elements in k, and , are equal. This shows that ζ^{n} is Σ_{n}-equivariant.

This result leads to the following alternative characterizations of a P-coalgebra structure.

**Theorem 3.3.** Let P = P(k,E,R) be a finitely generated quadratic operad, and let V be a finitedimensional
vector space. Then the following three sets are in bijection with each other:

(1) The set of P-coalgebra structures on V .

(2) The set of k[Σ_{2}]-equivariant morphisms π : E → Coend(V )(2) such that (3)(r) = 0 for r ∈ R.

(3) The set of degree 1 derivations that satisfy d^{2} = 0.

**Proof.** The bijection between (1) and (2) is Proposition 3.1. By Theorem 2.6, a derivation d as
stated corresponds to a P-algebra structure on V ^{#}, i.e., an operad morphism : P →End(V ^{#}).
Therefore, it follows from Theorem 3.2 that the composition ζ^{−1}: P → Coend(V ) is also
a morphism of operads. This is by definition a P-coalgebra structure on V . The argument can
be reversed to prove the converse, thereby giving a bijection between (1) and (3).

In this section, we give another characterization of a P-coalgebra structure in terms of a graded Lie bracket and define cohomology of P-coalgebras.

Throughout this section, let V be a finite-dimensional vector space, and let P = P(k,E,R) be a finitely generated quadratic operad.

**4.1 The graded Lie algebra**

Define the vector spaces

Here , which is the same as Hom(V, V ^{⊗n}) as a vector space
and has the natural left Σ_{n}-action.

**Theorem 4.1.** For each n ≥ 1, there is an isomorphism of vector spaces.

**Proof.** The isomorphism is the composition of the following isomorphisms:

The isomorphisms (1) and (4) use the fact that |Σ_{n}| = n! is invertible in k, which implies that
there is a canonical isomorphism whenever X is a right Σ_{n}-module (see, e.g., [23,
(3.60)]). The isomorphisms (2) and (3) rely on the fact that P^{!}(n) is finite-dimensional.

Tracing through the various isomorphisms above, can be described more explicitly as

(4.1)

where , and v ∈ V .

Using the o_{i} operations on the operad P^{!} and the non-Σ operad , one defines the
operations o and [−,−] on exactly as in (2.3).

**Proposition 4.2.** (, [−,−]) is a graded Lie algebra.

Proof. The same argument as in [3, Proposition 2.4.4] shows that the operations o and [−,−] are well defined on . Since is a non-Σ operad, the result follows from [8, (3)].

**4.2 The graded Lie algebra**

Define the operations o and [−,−] on via . Namely, define

(4.2)

The following result is an immediate consequence of Theorem 4.1 and Proposition 4.2.

**Corollary 4.3.** (, [−,−]) is a graded Lie algebra of degree −1.

There is another graded Lie bracket on defined in terms of differential derivations using the following result.

**Proposition 4.4.** There is an isomorphism of vector spaces.

**Proof.** Applying Proposition 2.5 to V instead of V ^{#}, we obtain the first isomorphism:

The second isomorphism is simply dualization.

The natural graded Lie algebra structure on the space of derivations induces a graded Lie bracket of degree −1 on via ω (Proposition 4.4). Exactly as in the case of P-algebra [3, p. 221], this graded Lie bracket is equal to the one defined before (Corollary 4.3).

**4.3 Square-zero characterization of P-coalgebras**

Using the graded Lie bracket on , we can give another characterization of a P-coalgebra structure, adding to the list in Theorem 3.3.

**Corollary 4.5.** There is a one-to-one correspondence between the P-coalgebra structures on V
and elements satisfying.

**Proof.** As in [3, Corollary 2.4.2], under the isomorphism ω in Proposition 4.4, an element corresponds to an element such that. The result now
follows from Theorem 3.3.

**4.4 Coboundary in**

Now let V be a finite-dimensional P-coalgebra with structural morphism , i.e., [π, π] = 0. Following Balavoine [3], define a map by setting

(4.3)

for , Note that this has the exact same formula as the differential in (2.5).

**Corollary 4.6.** The map is a differential on . In particular, (, , [−,−]) is a differential graded Lie algebra.

**Proof.** The map is a differential because [−,−] is a graded Lie bracket and
[π, π] = 0.

**4.5 Cohomology of P-coalgebras**

The cohomology of the cochain complex (, ) is denoted by or and is called the cohomology of V with coefficients in itself.

Essentially the same discussion as in [3, Section 4] also applies here, showing that the differential graded Lie algebra (, , [−,−]) controls the deformations of the P-coalgebra (V, π).

The following result is an immediate consequence of Corollary 4.6.

**Corollary 4.7.** The graded vector space inherits the structure of a graded Lie algebra
from .

**Example 4.8** (Hochschild coalgebra cohomology). Let V be a finite-dimensional As-coalgebra
(i.e., a finite-dimensional coassociative coalgebra) with comultiplication Δ: V → V ^{⊗2}. Using
the fact [9] As = As^{!} = {k[Σ_{n}]}, we deduce that and (n ≥ 2), where denotes the Hochschild coalgebra cohomology of V with
self-coefficients [12,26]. The last equality comes from the fact (see Theorems 6.1 and 6.2)
that , where bc is the Hochschild coalgebra coboundary on the cochain complex
{Hom(V, V ^{⊗n})}. More precisely, it is defined as the alternating sum

(4.4)

for f ∈ Hom(V, V ^{⊗n}), where o denotes composition of functions.

Using (2.3), one observes that the graded Lie bracket on (and hence also on= ) is given by

(4.5)

for f ∈ and g ∈

Throughout this section, let V be a finite-dimensional vector space, and let P = P(k,E,R) be
a finitely generated quadratic operad. Recall that V ^{#} = Hom(V, k) denotes the linear dual of
V . The purpose of this section is to show that and are isomorphic as graded
Lie algebras.

**5.1 Cochain level isomorphism**

Consider the map

for n ≥ 1. It is clear that ξ^{n} is a linear isomorphism, since each of the three maps that define it
is an isomorphism (Proposition 2.4, (3.1), and Theorem 4.1).

Using the formulas for Γ (2.4), (4.1), and ζ^{n} (3.1), one infers that ξ^{n} is, in fact, the
dualization isomorphism

More explicitly, given , and x ∈ V , we have

(5.1)

where

**Theorem 5.1.** The maps ξ^{n} assemble to form an isomorphism of graded
Lie algebras.

**Proof.** It remains to show that ξ commutes with the graded Lie brackets. It suffices to show that
each of the three maps that define ξ commutes with the graded Lie brackets. Both isomorphisms Γ and commute with the graded Lie brackets by definition.

On the other hand, the operations o and [−,−] on L_{P}(V ) and are defined in terms of
the o_{i} operations in P^{!}, , and . It follows from the fact that the ◦i operations can
be written in terms of the operad structure map γ (2.1) and Theorem 3.2 that the isomorphism
in the middle, , commutes with the operation ◦ and hence also the graded Lie
bracket.

**Corollary 5.2.** Let π be an element in. Then π defines a P-coalgebra structure on V if and only if defines a P-algebra structure on V^{ #}.

**Proof.** From Theorem 5.1, it follows that . The result now follows from
the fact that ξ^{3} is an isomorphism, Theorem 2.6, and Corollary 4.5.

**Corollary 5.3.** Let (V, π) be a finite-dimensional P-coalgebra. Then the map

is an isomorphism of differential graded Lie algebras.

**Proof.** In view of Theorem 5.1 and Corollary 5.2, it remains to show that ξ commutes with the
differentials. Pick an element . Then we have that

Passing to cohomology, we obtain the following result.

**Corollary 5.4.** The map ξ induces an isomorphism of graded
Lie algebras.

**Example 5.5** (= Example 4.8, continued). Let V be a finite-dimensional coassociative coalgebra
(i.e., a finite-dimensional As-coalgebra) with comultiplication . Then is the usual multiplication of the linear dual V^{ #}. In
this case, Corollary 5.4 says that there is a duality isomorphism of graded Lie algebras, where denotes the Hochschild cohomology [11] of the associative
algebra V^{ #} with self-coefficients. The graded Lie bracket in is as described in
(4.5). The graded Lie bracket in is the Gerstenhaber bracket [6].

**5.2 Duality isomorphism: From P-algebra to P-coalgebra**

There are also the obvious counterparts of the results above that relate the cohomology of a finitedimensional P-algebra (V, π) with that of the finite-dimensional P-coalgebra .

**Theorem 5.6.** Let (V, π) be a finite-dimensional P-algebra. Then the map

is an isomorphism of differential graded Lie algebras. Passing to cohomology, it induces an isomorphism

of graded Lie algebras.

Since the arguments are essentially the same as the ones given above, we will omit them.

In this section, we give an explicit description of the differential (4.3) in for a finitedimensional P-coalgebra (V, π), where P = P(k,E,R) is a finitely generated quadratic operad.

**6.1 The component maps** and

Pick elements and v ∈ V . Write

and

Using (4.1), (4.2), (4.3), and the definition of the graded Lie bracket in (2.3), one infers that

(6.1)

where

(6.2)

and

In view of (6.1), in order to understand the differential , it suffices to describe the component maps and .

**6.2 Explicit formula for**

In order to describe (f)(v) more explicitly using f itself, we use the notations

(6.3)

The following result is the P-coalgebra analogue of [2, Theorem 3.2.3], and it corresponds to the term

in (4.4).

**Theorem 6.1.** We have that

where the notations are as in (6.3).

The proof will be given below. Notice that the second component in the sum in Theorem 6.1 has an alternative description as follows. If , then has the form

It follows that

**6.3 Explicit formula for**

To deal with (f)(v), we use the notations

(6.4)

The following result is the P-coalgebra analogue of [2, Theorem 3.2.4], and it corresponds to the sum

in (4.4).

**Theorem 6.2.** We have that

where the notations are as in (6.4).

**Proof of Theorem 6.1.** We follow the notations and parts of the arguments in [2, 3.2.2]. For
a fixed element τ ∈ Σ_{n}, we set

for . According to [3, (7)], the signature of τ_{i} is given by

It follows that

(6.5)

Likewise, we have

(6.6)

Now set j = τ^{−1}(i) and . Substituting (6.5) and (6.6) back into (6.2), it follows that

(6.7)

Since (6.7) holds for any τ ∈ Σ_{n}, we can average it over Σ_{n}, which gives rise to

(6.8)

Since the second o_{j} is in , the last component in (6.8) can be rewritten as

(6.9)

To finish the proof, note that by using the explicit formula for (4.1), we can also write f(v) as

(6.10)

The proof of Theorem 6.1 now finishes by combining (6.8), (6.9), and (6.10).

Theorem 6.2 can be proved by modifying the proof of Theorem 6.1 slightly.

**6.4 Example:**

As an example, we compute (f)(v) explicitly using Theorems 6.1 and 6.2. Given f ∈ and v ∈ V, we have that , which can be considered as an element in V because, and . Since , it follows that

(6.11)

The last equality follows from the formula for (4.1).

Likewise, since for i ∈ {1, 2}, a similar analysis as above shows that

(6.12)

Combining (6.11) and (6.12), we have that

(6.13)

in which o denotes composition of functions.

**6.5** **as coderivations**

Recall that for a coassociative coalgebra (C, Δ), a coderivation on C is a linear map f : C → C such that

(6.14)

for x ∈ C, where . The condition (6.14) can also be stated in the element-free form as

We generalize this to P-coalgebras. For a finite-dimensional P-coalgebra (V, π), we define a coderivation on (V, π) to be a linear map f : V → V such that

Here ◦ denotes composition of functions, and we regard π as an element of

Denote by CoderP(V ) the vector space of all coderivations on (V, π).

The formula (6.13) for shows that is annihilated by if and only if f is a coderivation on (V, π). This leads to the following result.

**Corollary 6.3.** There is an equality of vector spaces.

The first author is supported by NBHM Post doctoral fellowship.

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