# Duality in Gerstenhaber algebras

###### Abstract

Let be a differential graded coalgebra, the Adams cobar construction and the dual algebra. We prove that for a large class of coalgebras there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies and . This result permits to describe a Hodge decomposition of the loop space homology of a closed oriented manifold, in the sense of Chas-Sullivan, when the field of coefficients is of characteristic zero.

AMS Classification : 16E40, 17A30, 17B55, 81T30, 55P35.

Key words : Hochschild cohomology, Gerstenhaber algebra, free loop space, loop homology.

## 1 Introduction

In the last two decades there has been a great deal of interest in Gerstenhaber algebras since they arise, as BV-algebras, in the BRST theory of topological field theory and in operad setting as well as in string theory. Let us denote by (resp. ) the Hochschild complex (resp. cohomology) of the differential graded algebra with coefficients in . It is well known that is a Gerstenhaber algebra. A new geometrical impact to Hochschild cohomology has been given recently by the work of Chas and Sullivan [6].

Let be an integral domain and let denotes the graded dual of the graded module . For a supplemented graded coalgebra , is a supplemented graded algebra (without any finiteness restriction), and the reduced coproduct can be iterated unambiguously to produce . If for each and some , , the coalgebra is called locally conilpotent.

Our first result reads

Theorem 1. Let (C,d) be a -free locally conilpotent differential graded coalgebra and the normalized cobar construction on . There exists a homomorphism of differential graded algebras

which induces a homomorphism of Gerstenhaber algebras

a) The homomorphism is an isomorphism whenever is finitely generated in each degree, and either i) or else ii) .

b) If is a field then is natural with respect to quasi-isomorphisms of differential graded coalgebras and is an isomorphism whenever is finitely generated in each degree, and either i) and , or else ii) .

The proof of Theorem 1 relies heavily on properties of free models and on differential graded Lie algebras of derivations. In order to minimize the periquisites, we do not introduce the twisting cochain formalism. Nonetheless it underlies most of the computational peculiarities we perform by hands.

Let us precise that a free model is a tensor algebra together with a differential such that is the union of an increassing familly such that and . Observe ([9]-Proposition 3.1) that for any differential graded algebra there is a quasi-isomorphism of differential graded algebras . If is a locally conilpotent coalgebra, the normalized cobar construction on is a free model (5.2).

We denote by the differential graded Lie algebra of derivations on with the commutator bracket and differential . We introduce the following extension, denoted , of this differential graded Lie algebra:

where . The next result is a corner stone in the proof of Theorem 1.

Theorem 2. Let be a free model. There exist two injective quasi-isomorphisms of differential graded Lie algebras

where is the differential graded algebra with and , .

If we introduce, following Getzler and Jones [13, 14], the Hochschild complex, of an -coalgebra , Theorem 2 can be read: There exist injective quasi-isomorphisms of differential graded Lie algebras

Since the graded Lie algebra is natural with respect to quasi-isomorphisms (3.4) we deduce from Theorems 1 and 2:

Corollary 1. Consider as in Theorem 1. Let and be free models, then there exists isomorphisms of graded Lie algebras:

The geometric meaning of Theorem 1 is in terms of loop homology. Recall that the loop homology of a closed orientable manifold of dimension is the ordinary homology of the free loop space with degrees shifted by , i.e. . In [6], Chas and Sullivan have defined a Batalin-Vilkovisky algebra structure on . Thus a loop product and a loop bracket are defined such that is a Gerstenhaber algebra. More recently, in his thesis [20], Thomas Traddler has introduced the notion of -bimodules and -inner product for - algebras such that

a) the Hochschild homology is defined and is naturally a BV-algebra for an - algebra with -inner product

b) the geometric chains on a manifold naturally possesses via intersection the structure of an -algebras with -inner product

c) for simply connected closed manifolds a) and b) are compatible with the ”Chas-Sullivan” BV-structure.

Denote by (resp. ) the normalized singular chain coalgebra (resp. cochain algebra) on with coefficients in . The cap product with a fixed fundamental cycle, , is not a map of -bimodules. Nonetheless, ([10] Theorem 2), Poincaré duality induces an isomorphism The Jones isomorphism [16] composed with , is an isomorphism of graded modules. R. Cohen and J. Jones ([7]-corollary 10) have announced that this isomorphism also identify the loop product on with the Gerstenhaber product on . Then Theorem 1 and naturality of (3.4) would imply a “Gerstenhaber algebra analog” of a result obtained by Burghelea-Fiederowicz [4] and Goodwillie [15]: Let be a simply-connected closed oriented manifold. The loop algebra is isomorphic to the Gerstenhaber algebra . Here denotes the based loops on a pointed space . Indeed for any pointed 1-connected space there is a natural equivalence , [8].

Corollary 1 is particulary interesting when is an Adams-Hilton model [1] for , since the Adams-Hilton model of a space is completely determined by a cellular decomposition of . If is a field of characteristic zero we can go further. The Adams-Hilton model of is the universal enveloping algebra of a differential graded Lie algebra [2]. Then,

where is considered as an -module via the adjoint representation. Denote by the vector space generated by the elements , with . Then the vector spaces are stable under the adjoint action of and . This gives the “Hodge decomposition”:

Corollary 2. Under the above hypothesis there exists isomorphisms of graded vector spaces:

The remaining of the paper is organized as follows:

2) Sketch of the proof of Theorem 1.

3) Hochschild cohomology of a differential graded algebra.

4) Proof of Theorem 2.

5) The Hochschild cochain complex of a differential graded coalgebra.

6) Proof of Propositions B and C (see below).

## 2 Sketch of the Proof of Theorem 1.

2.1. Notation. In the rest of the paper, except in 2.6 and 2.7, will be a principal ideal domain. If is a (lower) graded -module (when we need upper graded -module we put as usual) then:

a) ,

b) denotes the tensor algebra on , while we denote by the free supplemented coalgebra generated by .

Since we work with graded differential objets, we will make a special attention to signs. Recall that if and are differential graded -modules then

a) is a differential graded -module : , ,

b) is a differential graded -module : , ,

c) the commutator, , gives to the differential graded -module a structure of differential graded Lie algebra,

d) if is a differential graded coalgebra with diagonal and is a differential graded algebra with product then the cup product, , gives to the differential graded -module a structure of differential graded algebra.

2.2. A (graded) Gerstenhaber algebra is a commutative graded algebra with a degree 1 linear map

such that:

a) the suspension of is a graded Lie algebra with bracket

b) the product is compatible with the bracket, .

This last condition means that for any the adjunction map is a -derivation: ie. for , , , .

A homomorphism of Gerstenhaber algebras is a homomorphism of graded algebras such that is a homomorphism of graded Lie algebras.

For our purpose it is convenient to introduce the notion of pre-Gerstenhaber algebra. This a differential graded algebra together with a degree 1 homomorphism of differential graded modules where denotes a differential graded Lie algebra. Observe that

a) .

b) induces on a structure of graded a Lie algebra compatible with the differential while no compatibility condition with the product is required.

A homomorphism of pre-Gerstenhaber algebras is a commutative diagram of homomorphisms of differential graded modules

such that is a homomorphism of differential graded algebras and is a homomorphism of differential graded Lie algebras. Clearly, induces a homomorphism of graded algebras and a homomorphism of differential graded Lie algebras .

If is an isomorphism of graded modules then is called a quasi-isomorphism of pre-Gerstenhaber algebras.

If the structure of pre-Gerstenhaber algebra on (resp. ) induces a structure of Gerstenhaber algebra on (resp. on ) then is a homomorphism of Gerstenhaber algebras.

2.3. Let be a differential graded algebra. We consider the degree 1 isomorphism that extends a linear map to a coderivation

where

i) is the non-unital bar construction (3.1),

ii) is the differential graded Lie algebra of coderivations of .

The map defines the Hochschild complex as a pre-Gerstenhaber algebra. (It induces on the usual Gerstenhaber algebra structure [19].)

2.4. Dually, if is a supplemented differential graded coalgebra the degree 1 isomorphism that extends a linear map to a derivation

where is the non-counital cobar construction (5.1), makes the Hochschild complex of into a pre-Gerstenhaber algebra.

We also consider the pre-Gerstenhaber algebra together with a degree 1 linear isomorphism:

where

i) is the normalized cobar construction

ii) is the differential graded Lie algebra considered in the introduction.

2.5. Intermediate results.

Proposition A. Let be a -free locally conilpotent differential graded coalgebra. The inclusion induces, by naturality, a quasi-isomorphism of pre-Gerstenhaber algebras

Proposition B. Let be a differential graded coalgebra. The usual linear duality induces an homomorphism of pre-Gerstenhaber algebras which is

i) an isomorphism whenever is a free graded -module of finite type,

ii) a quasi-isomorphism if is a free graded -module of finite type.

Proposition C. Let be a -free locally conilpotent differential graded coalgebra. The bar-cobar adjunction induces a quasi-isomorphism of differential graded algebras

which admits a linear section such that is a homomorphism of differential graded Lie algebras.

2.6. Let be a -free locally conilpotent differential graded coalgebra. The composite

is a homomorphism of differential graded algebras but not a homomorphism of pre-Gerstenhaber algebra. Nonetheless,

is a homomorphism of Gerstenhaber algebras. It results from the constructions involved in Propositions A, B and C that is natural with respect to quasi-isomorphisms of differential graded coalgebras: Given a quasi-isomorphism of locally conilpotent supplemented -free differential graded coalgebras such that either i) and or else ii) and , then by Remark 2.3 of [8], is a quasi-isomorphism of -free differential graded algebras. Moreover, if is a field, is also a quasi-isomorphism between -free differential graded algebras. By 3.4, there exist isomorphisms of Gerstenhaber algebras and such that the following diagram commutes

2.7. The reduction process described below permits to deduce part b) of Theorem 1 from part a).

In case ii), let be a -free supplemented differential graded coalgebra such that is finitely generated in each degree, and is -free. By Proposition 4.2 of [8], there exists a free model and a quasi-isomorphism of augmented differential graded algebras , where is -free of finite type. We consider then the previous diagram with . By Propositions A, B and C, the lower line in the above diagram is an isomorphism thus so is the upper line.

The case i) is similar.

Required definitions and proofs are detailed in the following sections.

## 3 The Hochschild complex of a differential graded algebra.

3.1. Let be a differential graded supplemented algebra, , and (resp. ) be a right (resp. left) differential graded -bimodule. The two-sided bar constructions, and are defined as follows:

A generic element is written with degree The differential is defined by

with and:

Hereafter we will consider the normalized and the non-unital bar constructions on :

(In the latter is considered as a non unital algebra ([18]-p. 142))). The differentials and are given by the same formula:

We will also frequently use the twisting cochain of

and the following result:

Proposition. [9, Lemma 4.3] If is a differential graded algebra such that is a -free module then the canonical map

is a semi-free resolution of -modules (Here denotes the envelopping algebra).

3.2. Let be a supplemented differential graded algebra and a differential graded -bimodule. The canonical isomorphism of graded modules

carries a differential on . More explicitly, if , we have:

and

where .

The Hochschild cochain complex of with coefficients in the -bimodule is the differential module

(It is important here to remark that the differential graded module does not coincide with none of the differential graded modules and .) The Hochschild cohomology of with coefficients in is

Let be a homomorphism of differential graded algebras. Then is a differential graded bimodule via and is a differential graded algebra.

3.3 Let be a homomorphism of differential graded coalgebras. We denote by the subcomplex of consisting of -coderivations and by the linear isomorphism extending each linear map into a coderivation.

a) The degree 1 linear isomorphism :

satisfies ,

b) the structure of pre-Gerstenhaber algebra defined by induces the usual structure of Gerstenhaber algebra on the Hochschild cohomology .

Observe that and that for ,

3.4. Naturality. Let and be homomorphisms of differential graded algebras and . Then the two natural maps

are homomorphisms of differential graded algebras. Let and be the obvious maps which make commutative the following diagram of homomorphisms of differential graded modules:

It follows from ([9]-Proposition 2.3) that if , , are -free modules and if and are quasi-isomorphisms then and are quasi-isomorphisms. Thus, so are and .

Proposition. If is a quasi-isomorphism of differential graded algebras and if and are -free modules then the composite, denoted

is an isomorphism of Gerstenhaber algebras.

Proof. As observed above the maps

are quasi-isomorphisms of differential graded algebras. Let and be the natural maps which make commutative the following diagram of homomorphisms of differential graded modules:

We have to prove that the composite is a homomorphism of graded Lie algebras. By ([9], Proposition 3.1) the quasi-isomorphism factors as the composite of two quasi-isomorphisms of differential graded algebras

where admits a -linear retraction , and admits a -linear section . By functoriality, we have . It suffices therefore to prove that and are homomorphisms of graded Lie algebras.

In the case , admits the -linear section and so is surjective. Let , , be cycles in . Since is a surjective quasi-isomorphism of complexes, there exists cycles, , in such that

Thus

and if (resp. ) denotes the class of (resp. ), then

In the case , is surjective and the same argument works, mutatis mutandis.

## 4 Proof of Theorem 2.

In this section will denote a differential graded algebra (not necessarly a free model). The proof of Theorem 2 is a direct consequence of Lemmas 4.2, 4.3 and 4.7 below.

4.1. Let be as defined in the introduction. We define the injective degree linear map

as follow. Consider the counity and the twisting cochain . Given , we denote by the -linear map such that . We put

Let be the differential graded algebra where the differential is related to by

We define

where is the unique derivation of defined by