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Calculations on Lie Algebra of the Group of Affine Symplectomorphisms.

1. Introduction

Recall that the group of affine symplectomorphisms, which is the affine symplectic group AS[p.sub.n], is given by all transformations [mathematical expression not reproducible], where A is a 2n x 2n symplectic matrix and [z.sub.0] a fixed element of [R.sup.2n] [1]. The Lie algebra [g.sub.n] of AS[p.sub.n] 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 [g.sub.n] is determined by the exterior algebra of the forms [mathematical expression not reproducible] are the unit vector fields parallel to [x.sub.i] and [y.sub.i] axes, respectively, and the Lie algebra homology [H.sup.Lie.sub.*]([g.sub.n]) has been proved to have an isomorphic vector space as follows: [mathematical expression not reproducible] is the singular homology of the real symplectic Lie algebra s[p.sub.n] and [mathematical expression not reproducible].

Here, we find the image of the affine symplectic Lie algebra [g.sub.n] from the Leibniz homology H[L.sub.*]([g.sub.n]) to the Lie algebra homology [H.sup.Lie.sub.*] ([g.sub.n]). The result shows that the image is the tensor of a real number with the exterior algebra [[conjunction].sup.*] ([w.sub.n]). 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 H[L.sub.*]([g.sub.n]) in [H.sup.Lie.sub.*-1]([g.sub.n]) is important to find the image in Hochschild homology [H.sup.Lie.sub.*-1](U([g.sub.n])). In particular, we find the image via the map

[mathematical expression not reproducible], (1)

where the maps [j.sub.*] and [[pi].sub.*] are induced by the chain maps, on the chain level, [mathematical expression not reproducible], respectively, and U[([g.sub.n]).sup.ad] is the adjoint universal enveloping algebra of [g.sub.n]. Since [H.sup.Lie.sub.*-1] ([g.sub.n] > U[([g.sub.n]).sup.ad]) is isomorphic to the Hochschild homology [H.sup.Lie.sub.*-1](U([g.sub.n])) [6], we get the image in [H.sup.Lie.sub.*-1] (U([g.sub.n])).

In symmetric geometry, the study of symplectic algebras is important in manifolds because of the structure of the symplectic group SP(2n, R) 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 skewsymmetry 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 g over a ring k, the Lie algebra homology of g, written [H.sup.Lie.sub.*](g, k), is the homology of the chain complex [[conjunction].sup.*](g) which was introduced by Chevalley and Eilenberg in [8]; namely,

[mathematical expression not reproducible], (2)

where

[mathematical expression not reproducible], (3)

where the notation [[??].sub.i] means that element has been deleted. In this paper [H.sup.Lie.sub.*](g) denotes homology with real coefficients, where k = R. Lie homology, with coefficients in the adjoint representation of the universal enveloping algebra U[(g).sup.ad], is the homology of the chain complex U(g) [cross product] [[conjunction].sup.*] (g):

[mathematical expression not reproducible], (4)

where

[mathematical expression not reproducible]. (5)

The canonical projection

[mathematical expression not reproducible] (6)

given by g [cross product] [g.sup.[conjunction]n] [right arrow] ][g.sup..[conjunction](n+1)] is a map of chain complexes and induces a k-linear map on homology

[[rho].sub.*] : [H.sup.Lie.sub.n] (g, g) [right arrow] [H.sup.Lie.sub.n+1] (g,k). (7)

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 g over a ring fc, we recall that the Leibniz homology [10] of g, written H[L.sub.*] (g), is the homology of the chain complex

[mathematical expression not reproducible], (8)

where

[mathematical expression not reproducible]. (9)

Definition 1. Let k be a commutative ring and M be a g-bimodule of an associative (not necessarily commutative) k-algebra g. We define the Hochschild complex [mathematical expression not reproducible], where the module M [cross product] [g.sup.[cross product]n] is in degree n. The Hochschild boundary map [mathematical expression not reproducible] is given by

[mathematical expression not reproducible]. (10)

for m [member of] M and [g.sub.i] [member of] g for all i = 1, ..., n. The homology groups of the Hochschild complex C[H.sub.n](g, M) are called the Hochschild homology groups H[H.sub.n](g, M). For g = M, we write H[H.sub.n](g).

4. Affine Symplectic Lie Algebra

We begin by [mathematical expression not reproducible] are the unit vector fields parallel to [x.sub.i] and [y.sub.i] axes, respectively. Then the real symplectic Lie algebra s[p.sub.n] has a basis

[mathematical expression not reproducible]. (11)

Let [I.sub.n] be the abelian Lie algebra with the basis [mathematical expression not reproducible]. The affine symplectic Lie algebra [g.sub.n] has the basis [B.sub.1] U B2. Thus, there is a short exact sequence of Lie algebras

[mathematical expression not reproducible]. (12)

In the following example, we find the Lie brackets of the elements in [sp.sub.2] by taking into account the basic elements illustrated above.

Example 2. The basis [B.sub.1] of the real symplectic Lie algebra [sp.sub.2] contains exactly these elements

[mathematical expression not reproducible], (13)

which can be denoted by [e.sub.1] > [e.sub.2] > ... > [e.sub.10], respectively. It is known that [mathematical expression not reproducible]. By taking the Lie brackets of the others, it follows that

[mathematical expression not reproducible]. (14)

Now we take [[e.sub.1] > [e.sub.7]] = 2 [x.sub.1] ([partial derivative]/[partial derivative][y.sup.1]) = 2[e.sub.1], which means that [e.sub.1] is the Eigenvector of [[e.sub.1] > [e.sub.7]]. Similarly, if we continue the computations, we get that [e.sub.i] is the Eigenvector not only for the bracket [[e.sub.7] > [e.sub.i]] but also for [[e.sub.10] > [e.sub.i]] for all i = 1, ... n.

The above example shows that the Cartan subalgebra of [sp.sub.2] is {[e.sub.7] > [e.sub.10]} which is the tangent of the maximal torus subset in the Lie group SP(2, R).

5. The Image of H[L.sub.*]([g.sub.n]) in [H.sup.Lie.sub.*]([g.sub.n])

By convention, we denote the affine symplectic Lie algebra by [g.sub.n]. There is a canonical projection T([g.sub.n]) [right arrow] [[conjunction].sup.*]([g.sub.n]), where T([g.sub.n]) is the tensor algebra of [g.sub.n] and [[conjunction].sup.*]([g.sub.n]) is the exterior algebra of [g.sub.n], which is naturally defined by [mathematical expression not reproducible] Thus, the map n induces a k-linear map on homology

[mathematical expression not reproducible]. (15)

From [4], there are these two vector spaces isomorphisms [mathematical expression not reproducible]. Let us start with the element [mathematical expression not reproducible].

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

[mathematical expression not reproducible]. (16)

For more general setting, let us take [[conjunction].sup.2][w.sub.n] [member of] H[L.sub.*]([g.sub.n]), so we get

[mathematical expression not reproducible]. (17)

Thus

[mathematical expression not reproducible]. (18)

The result makes sense because [mathematical expression not reproducible].

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 H[L.sub.*] ([g.sub.n]) in the Lie algebra homology [H.sup.Lie.sub.*-1]([g.sub.n], U[([g.sub.n]).sup.ad]) of the adjoint universal enveloping algebra U[([g.sub.n]).sup.ad]. In particular, we find the image via the map

[mathematical expression not reproducible], (19)

where the maps [j.sub.*] and [[pi].sub.*] are induced by the chain maps [pi] and j on the chain level. Naturally [pi] and j can be defined as follows: [mathematical expression not reproducible] and the inclusion [mathematical expression not reproducible]. It is not difficult to prove that [pi] and j are chain maps. Now if we are trying to find [[pi].sub.*] ([w.sub.n]), where [w.sub.n] [member of] H[L.sub.*]([g.sub.n]), we get similar procedure steps as we have done above, by taking into account that n is different a little bit from the map [pi]' and we will get the same result. I mean [[pi].sub.*] ([[conjunction].sup.l][w.sub.n]) = [[conjunction].sup.l][w.sub.n]. The image result [mathematical expression not reproducible]. After taking the induced map [mathematical expression not reproducible]. 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 [[epsilon].sub.n]n). Suppose that g and M as they were in the previous definition. We define [6] the antisymmetrization map [mathematical expression not reproducible], where [sigma] is a permutation in the symmetric group [S.sub.n] on the set of indices [1, ..., n}, and [sigma] acts on (the left of) [mathematical expression not reproducible].

Theorem 4. From page 98 of [6], we know that if g is a Lie K-algebra and M is a U(g)-bimodule, then we have the following isomorphism: [mathematical expression not reproducible].

By applying the above theorem, we get the image of H[L.sub.*]([g.sub.n]) in the Hochschild homology [mathematical expression not reproducible].

Corollary 5. For the affine symplectic Lie algebra [g.sub.n], the image of H[L.sub.*]([g.sub.n]) in the Hochschild homology [H.sup.Lie.sub.*- 1](U([g.sub.n])) can be identified injectively as the exterior algebra [[conjunction].sup.*]([w.sub.n]).

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 M, so M 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 [H.sup.Lie.sub.*-1] (U([g.sub.n])) of the associative algebra U([g.sub.n]).

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 H[L.sub.*]([g.sub.n]) are mapped injectively to [H.sup.Lie.sub.*- 1](U([g.sub.n])). In other words, H[H.sub.*-1](U([g.sub.n])) contains [[conjunction].sup.*]([w.sub.n]) as a direct summand. Now, we know that

[H.sup.Lie.sub.*-1] (U ([g.sub.n])) [equivalent] Hom (H[H.sub.*-1] (U ([g.sub.n])), R). (20)

Taking into account that [[conjunction].sup.*]([w.sup.*.sub.n]) is the dual space of [mathematical expression not reproducible], we end up with this following result about the image in Hochschild cohomology of the given algebra.

Corollary 6. The Hochschild cohomology H[H.sup.*-1] (U([g.sub.n])) contains [[conjunction].sup.*]([w.sup.*.sub.n]) as a direct summand.

As an algebraic point of departure and theoretical physics point of view, the Hochschild cohomology H[H.sup.*](A) of an associative algebra has natural product with a Lie type bracket of degree -1, satisfying Jacobi identity and graded anticommutativity such that both natural product and Lie type bracket are compatible to make H[H.sup.*] (A) 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 A = U([g.sub.n]).

http://dx.doi.org/10.1155/2017/9513237

Competing Interests

The author declares no competing interests.

References

[1] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford University Press, 2nd edition, 1998.

[2] P. J. Hilton and U. Stammbach, A Course in Homological Algebra, Springer, Berlin, Germany, 1971.

[3] G. Hochschild and J.-P. Serre, "Cohomology of Lie algebras," Annals of Mathematics, vol. 57, pp. 591-603, 1953.

[4] J. M. Lodder, "Lie algebras of Hamiltonian vector fields and symplectic manifolds," Journal of Lie Theory, vol. 18, no. 4, pp. 897-914, 2008.

[5] R. L. Cohen, K. Hess, and A. A. Voronov, String Topology and Cyclic Homology, Advanced Courses in Mathematics--CRM Barcelona, 2006.

[6] J.-L. Loday, Cyclic Homology, vol. 301 of Fundamental Principles of Mathematical Sciences, Springer, Berlin, Germany, 2nd edition, 1998.

[7] V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60, Springer, Berlin, Germany, 1989.

[8] C. Chevalley and S. Eilenberg, "Cohomology theory of Lie groups and Lie algebras," Transactions of the American Mathematical Society, vol. 63, pp. 85-124, 1948.

[9] J. M. Lodder, "From Leibniz homology to cyclic homology," K-Theory, vol. 27, no. 4, pp. 359-370, 2002.

[10] T. Pirashvili, "On Leibniz homology," Annales de l'Institut Fourier, vol. 44, no. 2, pp. 401-411, 1994.

[11] J.-L. Loday and B. Vallette, Algebraic Operads, vol. 346 of Fundamental Principles of Mathematical Sciences, Springer, Heidelberg, Germany, 2012.

Zuhier Altawallbeh

Department of Mathematics, Tafila Technical University, Tafila 66110, Jordan

Correspondence should be addressed to Zuhier Altawallbeh; zuhier1980@gmail.com

Received 13 November 2016; Accepted 4 January 2017; Published 23 January 2017

Academic Editor: Manuel De Leon
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Title Annotation:Research Article
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Publication:Advances in Mathematical Physics
Date:Jan 1, 2017
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