Calculations on Lie Algebra of the Group of Affine Symplectomorphisms

Abstract

We find the image of the affine symplectic Lie algebra from the Leibniz homology to the Lie algebra homology . The result shows that the image is the exterior algebra generated by the forms . Given the relevance of Hochschild homology to string topology and to get more interesting applications, we show that such a map is of potential interest in string topology and homological algebra by taking into account that the Hochschild homology is isomorphic to . Explicitly, we use the alternation of multilinear map, in our elements, to do certain calculations.

1. Introduction

Recall that the group of affine symplectomorphisms, which is the affine symplectic group , is given by all transformations of the form , where is a symplectic matrix and a fixed element of [1]. The Lie algebra of is called the affine symplectic Lie algebra. By using some details and facts from [2, 3], Lodder [4] has proved that the structure of the Leibniz homology of is determined by the exterior algebra of the forms , as follows: , where and are the unit vector fields parallel to and axes, respectively, and the Lie algebra homology has been proved to have an isomorphic vector space as follows: , where is the singular homology of the real symplectic Lie algebra and .

Here, we find the image of the affine symplectic Lie algebra from the Leibniz homology to the Lie algebra homology . The result shows that the image is the tensor of a real number with the exterior algebra . We use the alternation of multilinear map, in our elements, to do certain calculations.

Any advances for computations in Hochschild homology are fundamental in string topology because of the high connection between both Hochschild homology and free loop spaces [5]. In this paper, we show that the image of in is important to find the image in Hochschild homology . In particular, we find the image via the map where the maps and are induced by the chain maps, on the chain level, and , respectively, and is the adjoint universal enveloping algebra of . Since is isomorphic to the Hochschild homology [6], we get the image in .

In symmetric geometry, the study of symplectic algebras is important in manifolds because of the structure of the symplectic group preserving the transformations of the symplectic vector space at any point of symplectic manifolds, and the reader is kindly requested to refer to [7] to get more applications and interactions with classical mechanics. Moreover, considering position and momentum in the frame of quantum state in physics, symplectic group can be considered as an important tool in the phase space. Thus, in the paper, both the source about symplectic algebras and the target related to Hochschild (co)homology make the paper in the intersection field from mathematics to physics.

By referring to [6], we recall that Leibniz homology is a noncommutative theory for Lie algebras, while Hochschild homology is a noncommutative theory for algebras, in the sense that Leibniz homology does not require the skew-symmetry of the bracket for a Lie algebra, while Hochschild homology does not require commutativity of the product in an algebra.

2. Preliminaries

For any Lie algebra over a ring , the Lie algebra homology of , written , is the homology of the chain complex which was introduced by Chevalley and Eilenberg in [8]; namely,wherewhere the notation means that element has been deleted. In this paper denotes homology with real coefficients, where . Lie homology, with coefficients in the adjoint representation of the universal enveloping algebra , is the homology of the chain complex :where

The canonical projectiongiven by is a map of chain complexes and induces a -linear map on homology

To see more details, the reader is kindly requested to look at [9].

3. Leibniz and Hochschild Homology

Returning to the general setting of any Lie algebra over a ring , we recall that the Leibniz homology [10] of , written , is the homology of the chain complexwhere

Definition 1. Let be a commutative ring and be a -bimodule of an associative (not necessarily commutative) -algebra . We define the Hochschild complex as the sequence of maps , where the module is in degree . The Hochschild boundary map is given by for and for all . The homology groups of the Hochschild complex are called the Hochschild homology groups . For , we write .

4. Affine Symplectic Lie Algebra

We begin by , where are the unit vector fields parallel to and axes, respectively. Then the real symplectic Lie algebra has a basis

Let be the abelian Lie algebra with the basis . The affine symplectic Lie algebra has the basis . Thus, there is a short exact sequence of Lie algebras

In the following example, we find the Lie brackets of the elements in by taking into account the basic elements illustrated above.

Example 2. The basis of the real symplectic Lie algebra contains exactly these elements which can be denoted by , respectively. It is known that and for all . By taking the Lie brackets of the others, it follows thatNow we take , which means that is the Eigenvector of . Similarly, if we continue the computations, we get that is the Eigenvector not only for the bracket but also for for all .

The above example shows that the Cartan subalgebra of is which is the tangent of the maximal torus subset in the Lie group .

5. The Image of in

By convention, we denote the affine symplectic Lie algebra by . There is a canonical projection , where is the tensor algebra of and is the exterior algebra of , which is naturally defined by for Thus, the map induces a -linear map on homology

From [4], there are these two vector spaces isomorphisms and . Let us start with the element .

By using the alternation multilinear form, we can rewrite the elements from the wedge notation into tensor product by taking into account the signs of the permutations, so

For more general setting, let us take , so we getThus

The result makes sense because and .

6. The Image in the Hochschild Homology

Hochschild homology plays a significant role in string topology, so any progress on computations about this kind of homology will be interesting for mathematicians and for those who are working in theoretical physics. First, we find the nonzero images of Leibniz homology in the Lie algebra homology of the adjoint universal enveloping algebra . In particular, we find the image via the map where the maps and are induced by the chain maps and on the chain level. Naturally and can be defined as follows: and the inclusion . It is not difficult to prove that and are chain maps. Now if we are trying to find , where , we get similar procedure steps as we have done above, by taking into account that is different a little bit from the map and we will get the same result. I mean . The image result is in because in [4] it was proven that , where . After taking the induced map , we get in . If we put the mentioned homological algebras in more general setting as operadic theory and generalize the above result in category theory, it will be more and more applicable in many different fields of study. To see how the homological algebra meets operad, we can read [11].

Definition 3 (the antisymmetrization map ). Suppose that and as they were in the previous definition. We define [6] the antisymmetrization map as one that sends the element (for and for all ) to by , where is a permutation in the symmetric group on the set of indices , and acts on (the left of) by .

Theorem 4. From page 98 of [6], we know that if is a Lie -algebra and is a -bimodule, then we have the following isomorphism: . For , we get .

By applying the above theorem, we get the image of in the Hochschild homology as ( for any .

Corollary 5. For the affine symplectic Lie algebra , the image of in the Hochschild homology can be identified injectively as the exterior algebra .

7. Relation to String Topology and Hochschild Cohomology

Recall that string topology is the study of the algebraic and differential topology of the spaces of paths and loops in compact and oriented manifolds. In this paper, consider a symplectic manifold , so is canonically oriented by its symplectic forms and it is closed manifold because the forms are closed. Actually, the operations of the loop homology algebra of a manifold are very difficult to compute, but there are several conjectures connecting the string topology with algebraic structures on the Hochschild cohomology of algebras related to the manifold. Thus it is worthy to find the nonzero image in the Hochschild cohomology of the associative algebra .

Although it is not that easy to compute Hochschild cohomology in general, still there are some ways to do it. In this paper, we know from the previous section that the elements in are mapped injectively to . In other words, contains as a direct summand. Now, we know that

Taking into account that is the dual space of , where and is dual of and is dual of , we end up with this following result about the image in Hochschild cohomology of the given algebra.

Corollary 6. The Hochschild cohomology contains as a direct summand.

As an algebraic point of departure and theoretical physics point of view, the Hochschild cohomology of an associative algebra has natural product with a Lie type bracket of degree , satisfying Jacobi identity and graded anticommutativity such that both natural product and Lie type bracket are compatible to make a Gerstenhaber algebra. Furthermore, the Gerstenhaber algebra structure can be viewed as algebraic properties of the loop homology algebra of a manifold. Here, we concentrate our work by setting .

Competing Interests

The author declares no competing interests.