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Journal of Generalized Lie Theory and Applications
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An operadic approach to deformation quantization of compatible Poisson brackets, I

Vladimir Dotsenko*

 

Independent University of Moscow, Bolshoj Vlasievsky per. 11, 119002 Moscow, Russia

*Corresponding Author:
Vladimir Dotsenko
Independent University of Moscow
Bolshoj Vlasievsky per. 11
119002 Moscow, Russia
E-mail: [email protected]

Received date November 8, 2006 Revised date March 14, 2007

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Abstract

An analogue of the Livernet–Loday operad for two compatible brackets, which is a flat deformation of the bi-Hamiltonian operad is constructed. The Livernet–Loday operad can be used to define ?-products and deformation quantization for Poisson structures. The constructed operad is used in the same way, introducing a definition of operadic deformation quantization of compatible Poisson structures.

1 Introduction

Throughout the text, algebras and operads are defined over an arbitrary field image of characteristic zero (unless the ground field or ring is specified explicitly).

The general deformation quantization problem for Poisson structures is set, for example, by Bayen–Flato–Fronsdal–Lichnerowicz–Sternheimer [1]. Let us give a pure algebraic definition.

Definition 1. A *-product on a vector space A is a image linear associative product onimage such that the algebraimage is commutative. Thus, forimage image

It is immediate to check that A with the operations image becomes a Poisson algebra, i.e. the product a· b is commutative and associative, and the bracket {a, b} is a Lie bracket satisfying the Leibniz rule.

Definition 2. For a Poisson algebra A, its deformation quantization is a *-product on A such that the associated Poisson algebra structure coincides with the original structure on A.

Some constructions of deformation quantization are known now for the case which was the most important for [1], namely the algebra of functions on a smooth Poisson manifold; see, for example, the work of Kontsevich [7]. It seems to be much more difficult to deal with deformation quantization of two compatible Poisson brackets – even on the level of introducing the problem and giving the necessary definitions. While for a Poisson bracket the quantum object should possess a simple natural structure (namely, an associative algebra structure), for two compatible brackets it is not clear at a glance which structures in principle one should expect. For example, an obvious idea that a pencil of Poisson structures can be quantized into a pencil of associative products is wrong. Note that although one can apply the Kontsevich’s deformation quantization techniques to get a 2-parametric family of associative products, these products do not form a pencil, i.e. are not obtained by taking linear combinations of two associative products, and in general the algebraic relations between the different products from this family are unknown.

Using our previous results on the bi-Hamiltonian operad from [2, 3], we generalise the Livernet–Loday idea of an operadic definition of deformation quantization from the Poisson structures to the bi–Hamiltonian structures. According to our computation, the following algebraic structure should be related to the corresponding deformation quantization problem.

Definition 3. The operad image is a quadratic operad generated by two binary operationsimage such that the following identities hold for each elements of any algebra over this operad:

image

Thus the first operation is an associative product, and the second one is a Lie bracket which is compatible with the skew-symmetrization of the product and satisfies the Leibniz rule with the symmetrization of the product.

The paper is organized as follows. We begin with some known facts on the operad defined by Livernet and Loday which can be used to define ?-products operadically. We prove that this operad is related to the algebras of locally constant functions on the complements of hyperplane arrangements [4]. Then we define a family of associative algebras (depending on a point of k2 which provides a deformation for dual spaces of components of the bi-Hamiltonian operad (which are double even Orlik–Solomon algebras from [2]); we call them double Gelfand–Varchenko algebras. The collection of these algebras possesses a cooperad structure, and it turns out that the dual operad is isomorphic to image everywhere except for the origin. We prove that this operad is Koszul and that our deformation of the bi-Hamiltonian operad is flat. (The proof uses the theorem on filtered operads proved in [6].) Finally, we discuss some aspects of the operadic problem of deformation quantization for compatible Poisson brackets. We hope to discuss examples of such a quantization and further results on it elsewhere.

Throughout the paper we use quadratic operads and the Koszul duality for operads. The corresponding definitions and notation can be found in [3, 5, 10].

2 The Livernet–Loday operad and related algebras

Definition 4. The double Livernet–Loday operad image is a quadratic operad overimage generated by a commutative operation image and a skew-commutative operationimage satisfying the identities

image

Proposition 1 ([9]). For each image the operad image is isomorphic to the associative operad. If image, then the operad image is isomorphic to the Poisson operad.

The relevance of this operad for deformation quantization is explained by the following proposition.

Proposition 2 ([9]). A *-product on a vector space V is the same as an image -algebra structureimage

The following is clear from the proof of the previous statement in [9].

Proposition 3. A deformation quantization of a Poisson algebra A is a *-product on this algebra such that the operations on the vector spaceimage induced from the image -structure coincide with the original Poisson operations on A.

Markl and Remm [9] proved that the operad image is a Hopf operad. It follows that the dual spaces to its components are associative algebras. The following fact is quite surprising though.

Theorem 1. The associative algebra dual to the nth component of the Livernet–Loday operad can be presented by generatorsimage and relations

image

Thus for image it is isomorphic to the Gelfand–Varchenko algebra of locally constant functions on the complement of the real hyperplane arrangement An−1 (studied in [4]).

Proof. A straightforward calculation shows that the above relations are satisfied in the duals of the components. Thus it rema ins to prove that the mappings from the corresponding abstract algebras to the duals image are isomorphisms. It is easy to see that these abstract algebras are isomorphic to the Gelfand–Varchenko algebras. Thus the dimensions of these algebras are indeed equal to the dimensions of the duals (the number of regions for the arrangement An−1 is equal to n!, which is also the dimension of the nth component of the associative operad), and it is enough to check the surjectivity, which is clear.

3 Double Gelfand–Varchenko algebras

Although the operad imagecan be (and, in fact, is) studied separately from the double Gelfand– Varchenko algebras, the construction of these algebras was crucial in our work and so we discuss them here. Moreover, one can easily check that — under reasonable restrictions — these give the essentially unique deformation of double even Orlik–Solomon algebras and thus lead to a distinguished approach to deformation quantization.

Definition 5. The double Gelfand–Varchenko algebra image is an associative commutative algebra with generators image andimage and relations

image

It turns out that the collection of double Gelfand–Varchenko algebras possesses a cooperad structure. The corresponding formulae coincide with the formulae from [2].1 (One of the restrictions fixing the deformation which were mentioned above, is that the cocomposition is given by the same formulae. It looks too restrictive at a glance, but it is true in the case of the Poisson operad.) Namely, we have the following

Definition 6. Let I, J be finite sets andimage Let us define an algebra homomorphism

image

by

image

where  stands for either of the letters x, y.

Lemma 1. For all I, J the homomorphism ρIJ is well-defined.

It is easy to check that these mappings satisfy all the cocomposition axioms, and so define a cooperadic structure. To make the computations in the dual operad more simple and transparent, we used the following technical result.

Proposition 4. The algebra image is isomorphic to the associative commutative algebra with generators image and the relations

image

4 The double Livernet–Loday operad

Definition 7. The double Livernet–Loday operad image is generated overimage by a commutative operation image and two skew-commutative operationsimage andimage satisfying the identities.

image

Proposition 5. For each point imageexcept for the origin the operad imageis isomorphic to the operad image then the operadimage is isomorphic to the bi- Hamiltonian operad.

Proof. The case of the origin is clear. Let image Consider a new operation

image

It is clear that the operad generated by this operation and · is isomorphic to image (and thus, according to Proposition 1, to the associative operad). The remaining bracket gives the second operation of image

For the sake of completeness, we describe here the quadratic dual operad. A straightforward calculation proves the following

Proposition 6. The quadratic dual operad image is generated overimage by a skewcommutative operation image and two commutative operationsimage satisfying the identities

image

5 Filtrations and Koszulness

Here we recall the results of A. Khoroshkin that are crucial for proving the Koszulness of operads in several important cases.

Let image be a set-theoretic operad. It is called an ordered operad if each set image(n) is ordered and this order is compatible with operadic compositions: for image j = 1, . . . , n, we have

image

Suppose that for each image and eachimage we have a subspaceimage The collection of these subspace is called a image-valued filtration on image, if

• for all image we haveimage

• for the maximal element image

• fompositions are compatible with the filtration

image

The associated graded operad image is, by the definition, the sum

image

A image-valued filtration on image is generated by binary operations if for each α the space image coincides with the span of compositions of all binary operations that belong to someimage withimage

The following theorem is proved in [6]; both theorems with proofs generalise Theorem 7.1 of [11] from the quadratic algebras to quadratic operads.

Theorem 2. Suppose that a image-valued filtration on a quadratic operad image (with generatorsimage and relationsimage ) is generated by binary operations and satisfies the following conditions:

(i) the operad image oes not have nontrivial relations of degree 3 (between the elements belonging to image

(ii) the quadratic operad image with generatorsimage and relationsimage

Then the operad image is quadratic (and thus isomorphic to image ,and the operad image is Koszul.

As an example we give a proof of the theorem on distributive laws between Koszul operads [8]. The proof of this statement in [8] is partially based on an erroneous formula for the composition in the symmetric case [8, Prop. 1.7] and is therefore incomplete. The proof below is essentially due to A. Khoroshkin [6].

Let image andimage be two quadratic operads. Denote byimage the corresponding sets of generators and by image the spaces of relations. Denote also byimage the subspace in the free operad image generated by all elementsimage (and 1 stands for the identity unary operation). The notation image has the analogous meaning. Assume that there is an S3-equivariant mapping

image

Let image be the quadratic operad with generatorsimage and relationsimage where

image

Theorem 3 ([8, 6]). Assume that the natural mapping

image

is an isomorphism. Then the operad image is Koszul, andimage

Proof. The main ingredient of the proof is some particular set-theoretic operad image and a image- filtration on image for which the associated graded operad is isomorphic toimage (where the composition in the reverse order is equal to zero).

Consider the free set-theoretic operad generated by two binary operations a and b. Let T be an arbitrary element of this operad, i.e. a binary tree whose internal vertices are labeled by a and b. By the definition, the degree deg(T) is equal to the number of a’s among the labels, and the number of inversions inv(T) is equal to the number of pairs (v1, v2) of internal vertices of T where v2 belongs to the set of descendants of v1, the label of v2 is equal to a, and the label of v1 is equal to b. We have

image

Let mn,k be the maximal number of inversions for the elements of image

Let us define the operad of inversions image.Its componentimage is a disjoint union of sets

the ordering of this component is lexicographic (to compare image we use the lexicographic ordering for the pairs (j, i) and (l, k)), and the compositions are given by

image

where maximum is taken over all collections image with

image

There exists a natural image -valued filtration F on image: it assigns the label a to the generators belonging to image , and the label b to the generators belonging to image. The associated graded operadimage is isomorphic to image (with zero reverse composition). Indeed, the associate graded relations include image remaining as is, while the relations x − d(x) are transformed into x, since d(x) belongs to the previous level of the filtration with respect to the number of inversions.

The operad image is Koszul, since for the corresponding Koszul complexes we have the following isomorphisms and quasi-isomorphisms:

image

Moreover, the operad image has no nontrivial relations of degree 3, since the corresponding components of image andimage are isomorphic. It follows that image is Koszul, andimage Taking the associated graded operad does not affect the image-module structure, so image

This proof can be easily generalised for the following problem.

Let again image and image be two quadratic operads (all notation remains the same). Assume that there are S3-equivariant mappings

and

image

Let image be the quadratic operad with generatorsimage and relationsimage where

image

Theorem 4. Assume that the natural mapping

image

is an isomorphism. Then the operad image is Koszul, andimage modulesimage are isomorphic.

Proof. The same filtration can be used here as well. It turns out again that image (the associated graded relations includedimage remaining as is, the relations x − d(x) are transformed into x since d(x) belongs to the previous level of filtration with respect to the number of inversions, while the relations x−s(x) are transformed into x since s(x) belongs to the previous level of filtration with respect to the degree), and the remaining proof is the same.

Corollary 1. The operad image is isomorphic toimage

Proof. We apply the theorem to image A straightforward calculation shows that there are no nontrivial relations of degree 3 in image

This corollary together with the results of [3] gives the following

Corollary 2. image are isomorphic, so we have constructed a flat deformation of the bi-Hamiltonian operad.

Similarly to [2], we can derive some information about double Gelfand–Varchenko algebras from the results about the double Livernet–Loday operad.

The following proposition is straightforward.

Proposition 7. Consider the operadimage dual to the cooperad of double Gelfand–Varchenko algebras. Then

(i) the relations ofimage

(ii) quadratic relations of image are exactly the relations ofimage

It follows that there exists a natural mapping of the operads

image

Theorem 5. The operad image is quadratic and Π is an isomorphism.

Proof. The proof is completely analogous to the proof in the case of double even Orlik–Solomon algebras in [2]. We already proved that the operad image is a flat deformation of the bi- Hamiltonian operad. In fact, it is clear that the monomials introduced in [2] provide a basis for this operad as well. These monomials can be used to prove that the pairing between image and the collection of double Gelfand–Varchenko algebras is nondegenerate (where we have to check the triangularity of a certain matrix), which proves the theorem.

6 Deformation quantization: an operadic definition

We introduce the following definitions which are motivated by Propositions 2 and 3.

Definition 8. An operadic image-product on a vector space V is a image -algebra structure on theimage

Definition 9. An operadic deformation quantization of a bi-Hamiltonian algebra A is aimage product on this algebra such that the operations on the vector spaceimage induced from theimage -structure coincide with the original (bi-Hamiltonian) operations on A.

Similarly to Proposition 2, one can check that for any image -algebra V overimage the product on V given by

image

is associative. Unfortunately, this definition does not possess the most nice property of the definition for the Poisson algebras. Namely, one can prove

Proposition 8. There exist obstructions for the existence of an operadic deformation quantization; they are nontrivial for a generic bi-Hamiltonian algebra, and even for a generic bi- Hamiltonian structure on image

Remark 1. We expect that the deformation quantization theorem holds in a weaker form for this operadic version: for a bi-Hamiltonian algebra A, there should exist a structure of a strongly homotopy image -algebra [5] on image

Acknowledgements

The author is grateful to B. Feigin, A. Khoroshkin and L. Rybnikov for useful discussions. This work is partially supported by the LIEGRITS fellowship MRTN-CT-2003-505078.

1As usual, we define the algebra or an arbitrary finite set I (indexing the generators); thus, and cocompositions are some mappings from to

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