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Journal of Generalized Lie Theory and Applications
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Cohomology and duality for coalgebras over a quadratic operad

Anita MAJUMDAR1 and Donald YAU2

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

2Department 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

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Abstract

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.

1 Introduction

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] Image of A with self-coefficients is well known; it is a Gerstenhaber algebra [6]. The same is true for the Hochschild coalgebra cohomology Image [12], which can be proved by basically the same arguments as in [6]. If, in addition, A is finitedimensional, then Image andImage 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 Image with self-coefficients for finite-dimensional P-coalgebras V .

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

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

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

When P is the associative algebra operad As, the cohomology Image coincides withImage. 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 Image that defines the cohomologyImage 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 Image that defines the cohomology Image 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 Image on Image 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 Image into a differential graded Lie algebra (Corollary 4.6), which implies that Image is a graded Lie algebra (Corollary 4.7). Example 4.8 makes this explicit in the case of the Hochschild coalgebra cochain complex Image.

In Section 5, it is shown that, for a finite-dimensional P-coalgebra V , the differential graded Lie algebras Image and Image are isomorphic (Corollary 5.3). The duality isomorphism between Image and Image 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 Image and Image is given in terms of the elementary operations oi in the operad P! (Theorems 6.1 and 6.2).

2 Algebras over an operad

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 Image 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, Σn will denote the group of permutations on n elements. For σ ∈ Σn, (σ) ∈ {−1, 1} will stand for the sign of σ, and sgnn will denote the sign representation of Σn.

2.2 Operads

An operad [22,23,24,25] P consists of a right k[Σn]-module P(n), one for each n ∈Image ∗. For positive integers n, j1, . . . , jn, there is a structure map

γ : P(n)⊗P(j1)⊗· · ·⊗P(jn)→P(j1 + · · · + jn)

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

f oi 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

Image (2.2)

for f ∈ P(n) and gi ∈ P(ji) (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 ∈ Image} 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 ∈Image , 1 ≤ i ≤ n, μ ∈ P(n), and ν ∈ P(m).

2.5 Endomorphism operad

Let V be a vector space over k. For n ∈ Image, let End(V )(n) = Hom(V ⊗n, V ). Then End(V) = {End(V )(n), n ∈ Image} is naturally an operad under composition, called the endomorphism operad of V . Indeed, f oi g = f(1, . . . , g, . . . , 1), where g is in the ith place, and 1 is the identity map on V . The Σn-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[Σ2]-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[Σ2]-modules a: E → Q(2), there exists a unique morphism of operads, Image: F(E)→Q, such that Image(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

Image

where Image for σ ∈ Σn and vi ∈ V . The homogeneous degree n component of Image is denoted byImage. The P-algebra structure on Image 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≥0Aj be a graded algebra over an operad P with structure maps πm: P(m) ⊗Σm A⊗m → A. A degree n derivation of A is a homogeneous degree n linear map d: A → A such that

Image

for m ∈ Image, μ ∈ P(m), and ai ∈ A. Here si = |a1| + · · · + |ai−1| with s1 = 0, and |a| = j if a ∈ Aj [3, Definition 2.3.1]. The set of degree n derivations of A is denoted by Dern(A). Denote by Der(A) the graded vector space ⊕n≥0 Dern(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 ∈Image}, in which I(n) is a k[Σn]- 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 oi x ∈ I(n + m − 1) and y oj 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[Σ2]-module and let R be a right k[Σ3]-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[Σ2]-modules π : P(2) = E → End(V )(2) such that Image(3)(R) = 0.

In this case, the morphism π : P(2) → End(V )(2), or equivalently its adjoint π : P(2) ⊗ Σ2 V ⊗2 → V , is called the structural morphism of the P-algebra V .

2.12 Quadratic duality

Let F be a right k[Σn]-module. By F# we mean the right k[Σn]-module F# = Hom(F, k)⊗sgnn, where the right Σn-action is given by Image for Image ∈ Hom(F, k) and x ∈ F.

Let E be a right k[Σ2]-module. Then as right k[Σ3]-modules, one has that [9] ImageImage . Let R ⊂ F(E)(3) be a right k[Σ3]-submodule, and let R⊥⊂ F(E#)(3) be the annihilator of R in Image. 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 LP(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

Image

whereImage, with the natural structure of a left k[Σn]-module. Let LP(V ) denote the graded vector space Image. Using the elementary operations oi in the operad P! and the non-Σ operad Image, one defines the operations [2,3]

Image (2.3)

for Image and Image. These two operations are indeed well defined, i.e., independent of the choice of representing elements μ⊗ f and ν ⊗ g. However, the individual oi operations are not well defined on LP(V ) [3, Remark 2.4.4].

Proposition 2.3 (see [3, Proposition 2.4.4]). The bracket [−,−] defined above makes LP(V ) into a graded Lie algebra.

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

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

Image

With the obvious notations, the isomorphism Γ is given by

Image (2.4)

2.14 The graded Lie algebra Image

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

Image

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

Image

The space Image 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 Image. These two graded Lie brackets on Image 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 Image that satisfy d2 = 0.

(3) The set of elements Image 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 Image in Proposition 2.2.

2.15 Cohomology of a P-algebra

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

Image (2.5)

By [2, Proposition 3.1.7], the map Image is a differential. The homology of the cochain complexImage is denoted byImage or Image, 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, Image becomes a graded Lie algebra. Also, note thatImage 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].

3 Coalgebras over an operad

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[Σ2]- equivariant morphism π : E = P(2) → Coend(V )(2), such that Image, Image : 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 Image must factor through the quotient P = F(E)/(R). This is because Image commutes with the oi operations and every element in the ideal (R) is a sum of elements that are iterated oi products with at least one entry in R.

The condition Image 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])

Image (3.1)

given by ζn(f) = f#, where

Image

for αi ∈ V # and a ∈ V . The notations on the right-hand side of the previous line is given by

Image

Theorem 3.2. Let V be a finite-dimensional vector space. Then the maps ζn (n ≥ 1) assemble to form an isomorphism Image of operads.

Proof. The maps ζn are linear isomorphisms, and it is clear that Image It remains to check that they are compatible with the operad structure maps γ and that ζn is Σn-equivariant. For f ∈ Coend(V )(k) and gi ∈ Coend(V )(ni) (1 ≤ i ≤ k), we have

Image

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

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

Image

The equality (∗) comes from the fact that the two sets of elements in k, Image and Image, are equal. This shows that ζn is Σn-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[Σ2]-equivariant morphisms π : E → Coend(V )(2) such that Image(3)(r) = 0 for r ∈ R.

(3) The set of degree 1 derivations Image that satisfy d2 = 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 Image: P →End(V #). Therefore, it follows from Theorem 3.2 that the composition ζ−1Image: 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).

4 Cohomology of P-coalgebras

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 algebraImage

Define the vector spaces

Image

Here Image, which is the same as Hom(V, V ⊗n) as a vector space and has the natural left Σn-action.

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

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

Image

The isomorphisms (1) and (4) use the fact that |Σn| = n! is invertible in k, which implies that there is a canonical isomorphism Image whenever X is a right Σn-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, Image can be described more explicitly as

Image (4.1)

where Image, and v ∈ V .

Using the oi operations on the operad P! and the non-Σ operad Image, one defines the operations o and [−,−] on Image exactly as in (2.3).

Proposition 4.2. (Image, [−,−]) 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 Image. SinceImage is a non-Σ operad, the result follows from [8, (3)].

4.2 The graded Lie algebraImage

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

Image (4.2)

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

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

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

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

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

Image

The second isomorphism is simply dualization.

The natural graded Lie algebra structure on the space Image of derivations induces a graded Lie bracket of degree −1 on Image 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 Image, 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 Image satisfyingImage.

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

4.4 Coboundary in Image

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

Image (4.3)

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

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

Proof. The map Image 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 (Image, Image) is denoted byImage or Image 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 (Image, Image, [−,−]) 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 Image inherits the structure of a graded Lie algebra from Image.

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[Σn]}, we deduce that Image and ImageImage (n ≥ 2), where Image 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 Image, where bc is the Hochschild coalgebra coboundary on the cochain complex {Hom(V, V ⊗n)}. More precisely, it is defined as the alternating sum

Image (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 Image (and hence also onImage= Image) is given by

Image (4.5)

for f ∈Image and g ∈ Image

5 Cohomological duality

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 Image and Image are isomorphic as graded Lie algebras.

5.1 Cochain level isomorphism

Consider the map

Image

for n ≥ 1. It is clear that ξn 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), Image (4.1), and ζn (3.1), one infers that ξn is, in fact, the dualization isomorphism

Image

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

Image (5.1)

where

Image

Theorem 5.1. The maps ξn assemble to form an isomorphism Image 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 Image commute with the graded Lie brackets by definition.

On the other hand, the operations o and [−,−] on LP(V ) and Image are defined in terms of the oi operations in P!, Image, and Image . 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, Image, commutes with the operation ◦ and hence also the graded Lie bracket.

Corollary 5.2. Let π be an element inImage. Then π defines a P-coalgebra structure on V if and only if Image defines a P-algebra structure on V #.

Proof. From Theorem 5.1, it follows that Image. The result now follows from the fact that ξ3 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

Image

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 Image. Then we have that

Image

Passing to cohomology, we obtain the following result.

Corollary 5.4. The map ξ induces an isomorphism Image 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 Image. Then Image is the usual multiplication of the linear dual V #. In this case, Corollary 5.4 says that there is a duality isomorphism Image of graded Lie algebras, whereImage denotes the Hochschild cohomology [11] of the associative algebra V # with self-coefficients. The graded Lie bracket in Imageis as described in (4.5). The graded Lie bracket in Image 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 Image.

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

Image

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

Image

of graded Lie algebras.

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

6 The differential Image

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

6.1 The component mapsImage andImage

Pick elements Image and v ∈ V . Write

Image

and

Image

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

Image (6.1)

where

Image (6.2)

and

Image

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

6.2 Explicit formula for Image

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

Image (6.3)

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

Image

in (4.4).

Theorem 6.1. We have that

Image

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 Image, thenImage has the form

Image

It follows that

Image

6.3 Explicit formula for Image

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

Image (6.4)

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

Image

in (4.4).

Theorem 6.2. We have that

Image

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 ∈ Σn, we set

Image

for Image . According to [3, (7)], the signature of τi is given by

Image

It follows that

Image (6.5)

Likewise, we have

Image (6.6)

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

Image (6.7)

Since (6.7) holds for any τ ∈ Σn, we can average it over Σn, which gives rise to

Image (6.8)

Since the second oj is in Image, the last component in (6.8) can be rewritten as

Image (6.9)

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

Image (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:Image

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

Image (6.11)

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

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

Image (6.12)

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

Image (6.13)

in which o denotes composition of functions.

6.5Image as coderivations

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

Image (6.14)

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

Image

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

Image

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

Image

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

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

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

Acknowledgement

The first author is supported by NBHM Post doctoral fellowship.

References

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