Cohomology and duality for coalgebras over a quadratic operad

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. 2000 MSC: 18D50, 20G10, 16W30


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] H * h (A, A) of A with self-coefficients is well known; it is a Gerstenhaber algebra [6]. The same is true for the Hochschild coalgebra cohomology H * c (C, C) [12], which can be proved by basically the same arguments as in [6]. If, in addition, A is finitedimensional, then H * h (A, A) and H * c (A # , A # ) 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 H * P (V ) with self-coefficients for finite-dimensional P-coalgebras V . (2) Observe that H * P (V ) is a graded Lie algebra, whose graded Lie bracket is induced by one on the defining cochain complex C * P (V ).
(3) Show that H * P (V ) and H * P (V # ) are isomorphic as graded Lie algebras, and vice versa, where H * P (V # ) is the cohomology (as defined in [2]) of the dual P-algebra V # of V .
We note that the differential graded Lie algebra C * P (V ) controls the deformations of V as a Pcoalgebra in the sense of Gerstenhaber [7].

Organization
In the next section, we recall from [2,3] the constructions and some properties of the cochain complex C * P (A) that defines the cohomology H * P (A) 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 C * P (V ) that defines the cohomology H * P (V ) 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 δ π on C * P (V ) 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 C * P (V ) into a differential graded Lie algebra (Corollary 4.6), which implies that H * P (V ) is a graded Lie algebra (Corollary 4.7). Example 4.8 makes this explicit in the case of the Hochschild coalgebra cochain complex C * In Section 5, it is shown that, for a finite-dimensional P-coalgebra V , the differential graded Lie algebras C * P (V ) and C * P (V # ) are isomorphic (Corollary 5.3). The duality isomorphism between H * P (V ) and H * P (V # ) 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 δ π in C * P (V ) is given in terms of the elementary operations • i in the operad P ! (Theorems 6.1 and 6.2).

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.

Conventions
The symbol N * 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 .

Operads
An operad [22,23,24,25] P consists of a right k[Σ n ]-module P(n), one for each n ∈ 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 • i operations as 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 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.

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

Endomorphism operad
Let V be a vector space over k. For n ∈ N * , let End(V )(n) = Hom(V ⊗n , V ). Then End(V ) = {End(V )(n), n ∈ N * } is naturally an operad under composition, called the endomorphism operad of V . Indeed, f • 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 Σ n -action on V ⊗n .

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 π n : P(n) ⊗ Σn V ⊗n → V that satisfy the obvious associativity, equivariant, and unity conditions. Proposition 2.1 (see [9]

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.

Free graded P-algebra
For an operad P and a vector space V , define the free graded P-algebra generated by V as The P-algebra structure on F gr P (V ) is the natural one defined by the operad structure on P and concatenation on V ⊗ * [3, 1.6.2].

Graded derivations
Let A = ⊕ j≥0 A j be a graded algebra over an operad P with structure maps π m :

Operadic ideals
Let P be an operad. An ideal of P is a sequence I = {I(n), 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 • i x ∈ I(n + m − 1) and y • 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.

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.

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

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 for μ * ⊗ f ∈ L n P (V ) and ν * ⊗ g ∈ L m P (V ). These two operations are indeed well defined, i.e., independent of the choice of representing elements μ * ⊗ f and ν * ⊗ g. However, the individual

]). There is an isomorphism of vector spaces
With the obvious notations, the isomorphism Γ is given by

The graded Lie algebra C * P (V )
Using the isomorphism Γ, the graded Lie bracket on L P (V ) can be transported to C * P (V ), which makes C * P (V ) into a graded Lie algebra of degree −1. More precisely, the graded Lie bracket on The same can be said of the operation •, from which the graded Lie bracket is defined.
There is an isomorphism of vector spaces The space Der(F gr P ! (V # )) 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 C * P (V ). These two graded Lie brackets on C * P (V ) are equal [ When one of these equivalent conditions is satisfied, we say that (V, π) is a P-algebra.

Cohomology of a P-algebra
Let (A, π) be a P-algebra. Define the map δ n π : C n P (A) → C n+1 P (A) by setting By [2, Proposition 3.1.7], the map δ * π is a differential. The homology of the cochain complex , 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, H * P (A) becomes a graded Lie algebra. Also, note that (C * P (V ), δ π , [−, −]) 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].

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.

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

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]) for α i ∈ V # and a ∈ V . The notations on the right-hand side of the previous line is given by Proof. The maps ζ n are linear isomorphisms, and it is clear that ζ 1 (Id V ) = Id V # . It remains to check that they are compatible with the operad structure maps γ and that ζ n is Σ n -equivariant.
where • denotes composition of functions. This shows that ζ is compatible with the operad structure maps.
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 . (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 ϕ : P → End(V # ). Therefore, it follows from Theorem 3.2 that the composition ζ −1 ϕ : 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).

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.

The graded Lie algebra C * P (V )
Define the operations • and [−, −] on C * P (V ) via Γ. Namely, define The following result is an immediate consequence of Theorem 4.1 and Proposition 4.2. There is another graded Lie bracket on C * P (V ) defined in terms of differential derivations using the following result. 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 Der(F gr P ! (V )) of derivations induces a graded Lie bracket of degree −1 on C * P (V ) 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).

Square-zero characterization of P-coalgebras
Using the graded Lie bracket on C * P (V ), 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 ϕ ∈ C 2 P (V ) satisfying [ϕ, ϕ] = 0.

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

Coboundary in C * P (V )
Now let V be a finite-dimensional P-coalgebra with structural morphism π ∈ C 2 P (V ), i.e., [π, π] = 0. Following Balavoine [3], define a map δ for f ∈ C n P (V ). Note that this has the exact same formula as the differential in C * P (V ) (2.5).

Cohomology of P-coalgebras
The cohomology of the cochain complex (C * P (V ), δ π ) is denoted by H n P (V ) or H n P (V, π) and is called the cohomology of V with coefficients in itself.
The following result is an immediate consequence of Corollary 4.6.   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 C n As (V ) = Hom(V, V ⊗n ) and H n As (V ) = H n c (V, V ) (n ≥ 2), where H * c (V, V ) 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 2 n+1 δ n π = b n c , where b c is the Hochschild coalgebra coboundary on the cochain complex {Hom(V, V ⊗n )}. More precisely, it is defined as the alternating sum for f ∈ Hom(V, V ⊗n ), where • denotes composition of functions. Using (2.3), one observes that the graded Lie bracket on C * As (V ) (and hence also on H * for f ∈ C m As (V ) and g ∈ C n As (V ).

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 H * P (V ) and H * P (V # ) are isomorphic as graded Lie algebras.

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), Γ (4.1), and ζ n (3.1), one infers that ξ n is, in fact, the dualization isomorphism More explicitly, given ϕ ∈ C n P (V ), μ ∈ (P ! ) # (n), α ∈ (V # ) ⊗n ∼ = (V ⊗n ) # , and x ∈ V , we have Theorem 5.1. The maps ξ n assemble to form an isomorphism ξ : C * 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 Γ −1 commute with the graded Lie brackets by definition.
On the other hand, the operations • and [−, −] on L P (V ) and L P (V ) are defined in terms of the • i operations in P ! , End(V ), and Coend(V ). 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, (Id P ! (n) ⊗ζ n ), commutes with the operation • and hence also the graded Lie bracket.

Corollary 5.2.
Let π be an element in C 2 P (V ). Then π defines a P-coalgebra structure on V if and only if ξ 2 π ∈ C 2 P (V # ) defines a P-algebra structure on V # .
Proof. In view of Theorem 5.1 and Corollary 5.2, it remains to show that ξ commutes with the differentials. Pick an element f ∈ C n P (V ). Then we have that Passing to cohomology, we obtain the following result.

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 (V # , (ξ 2 ) −1 π).
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.

The differential δ π
In this section, we give an explicit description of the differential δ π (4.3) in C * P (V ) for a finitedimensional P-coalgebra (V, π), where P = P(k, E, R) is a finitely generated quadratic operad.
Using (4.1), (4.2), (4.3), and the definition of the graded Lie bracket in L P (V ) (2.3), one infers that In view of (6.1), in order to understand the differential δ n π , it suffices to describe the component maps δ 1,n π and δ 2,n π .

Explicit formula for δ 1,n π
In order to describe δ 1,n π (f )(v) more explicitly using f itself, we use the notations The following result is the P-coalgebra analogue of [2, Theorem 3.2.3], and it corresponds to the term Theorem 6.1. We have that where the notations are as in (6.3).
Theorem 6.2 can be proved by modifying the proof of Theorem 6.1 slightly.
The formula (6.13) for δ 1 π (f ) shows that f ∈ C 1 P (V ) is annihilated by δ 1 π if and only if f is a coderivation on (V, π). This leads to the following result.