Calculations on Lie Algebra of the Group of Affine Symplectomorphisms

We find the image of the affine symplectic Lie algebra gn from the Leibniz homology HL∗(gn) to the Lie algebra homology HLie ∗ (gn). The result shows that the image is the exterior algebra ∧∗(wn) generated by the formswn = ∑ni=1(∂/∂xi ∧∂/∂yi). Given the relevance of Hochschild homology to string topology and to get more interesting applications, we show that such amap is of potential interest in string topology and homological algebra by taking into account that the Hochschild homology HH∗−1(U(gn)) is isomorphic to HLie ∗−1(gn, U(gn)ad). Explicitly, we use the alternation of multilinear map, in our elements, to do certain calculations.


Introduction
Recall that the group of affine symplectomorphisms, which is the affine symplectic group AS  , is given by all transformations Ψ : R 2 → R 2 of the form Ψ() =  0 + , where  is a 2 × 2 symplectic matrix and  0 a fixed element of R 2 [1].The Lie algebra g  of AS  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  is determined by the exterior algebra of the forms   = ∑  =1 (/  ∧ /  ), as follows: HL * (g  ) ≃ ∧ * (  ), where /  and /  are the unit vector fields parallel to   and   axes, respectively, and the Lie algebra homology H Lie * (g  ) has been proved to have an isomorphic vector space as follows: H Lie * (g  ) ≃ H * (  , R) ⊗ ∧ * (  ), where H * (  , R) is the singular homology of the real symplectic Lie algebra   and ∧ * (  ) = ∑ ≥0 ∧  (  ).
Here, we find the image of the affine symplectic Lie algebra g  from the Leibniz homology HL * (g  ) to the Lie algebra homology H Lie * (g  ).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 HL * (g  ) in H Lie * −1 (g  ) is important to find the image in Hochschild homology HH * −1 ((g  )).In particular, we find the image via the map where the maps  * and  * are induced by the chain maps, on the chain level, CL * (g  ) ), respectively, and (g  ) ad is the adjoint universal enveloping algebra of g  .Since H Lie * −1 (g  , (g  ) ad ) is isomorphic to the Hochschild homology HH * −1 ((g  )) [6], we get the image in HH * −1 ((g  )).
In symmetric geometry, the study of symplectic algebras is important in manifolds because of the structure of the symplectic group SP(2, 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 2 Advances in Mathematical Physics 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.

Preliminaries
For any Lie algebra g over a ring , the Lie algebra homology of g, written H Lie * (g, ), is the homology of the chain complex ∧ * (g) which was introduced by Chevalley and Eilenberg in [8]; namely, where where the notation ∧   means that element has been deleted.In this paper H Lie * (g) denotes homology with real coefficients, where  = R. Lie homology, with coefficients in the adjoint representation of the universal enveloping algebra (g) ad , is the homology of the chain complex (g) ⊗ ∧ * (g): where The canonical projection given by g ⊗ g ∧ → g ∧(+1) 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].

Leibniz and Hochschild Homology
Returning to the general setting of any Lie algebra g over a ring , we recall that the Leibniz homology [10] of g, written HL * (g), is the homology of the chain complex where Definition 1.Let  be a commutative ring and  be a gbimodule of an associative (not necessarily commutative) algebra g.We define the Hochschild complex CH * (g, ) as the sequence of maps CH * (g, ) : where the module  ⊗ g ⊗ is in degree .The Hochschild boundary map  :  ⊗ g ⊗ →  ⊗ g ⊗−1 is given by (10) for  ∈  and   ∈ g for all  = 1, . . ., .The homology groups of the Hochschild complex CH  (g, ) are called the Hochschild homology groups HH  (g, ).For g = , we write HH  (g).

Affine Symplectic Lie Algebra
We begin by ( 1 ,  2 , . . .,   ,  1 ,  2 , . . .,   ) ∈ R 2 , where /  , /  are the unit vector fields parallel to   and   axes, respectively.Then the real symplectic Lie algebra   has a basis  (12) In the following example, we find the Lie brackets of the elements in  2 by taking into account the basic elements illustrated above.
Example 2. The basis B 1 of the real symplectic Lie algebra  2 contains exactly these elements which can be denoted by  1 ,  2 , . . .,  10 , respectively.It is known that [  ,   ] = 0 and [  ,   ] = −[  ,   ] for all ,  = 1, 2, . . ., .By taking the Lie brackets of the others, it follows that Similarly, if we continue the computations, we get that   is the Eigenvector not only for the bracket [ 7 ,   ] but also for [ 10 ,   ] for all  = 1, . . ..
The above example shows that the Cartan subalgebra of  2 is { 7 ,  10 } which is the tangent of the maximal torus subset in the Lie group SP(2, R).

The Image of HL
By convention, we denote the affine symplectic Lie algebra by g  .There is a canonical projection (g  ) → ∧ * (g  ), where (g  ) is the tensor algebra of g  and ∧ * (g  ) is the exterior algebra of g  , which is naturally defined by   : g ⊗  → g ∧  for  ≥ 0 Thus, the map   induces a -linear map on homology From [4], there are these two vector spaces isomorphisms HL * (g  ) ≃ ∧ * (  ) and H Lie * (g  ) ≃ H * (  , R) ⊗ ∧ * (  ).Let us start with the element   = ∑  =1 (/  ∧/  ) ∈ HL * (g  ).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 ∧ 2   ∈ HL * (g  ), so we get The result makes sense because

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 HL * (g  ) in the Lie algebra homology H Lie * −1 (g  , (g  ) ad ) of the adjoint universal enveloping algebra (g  ) ad .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  :

𝑛
. It is not difficult to prove that  and  are chain maps.Now if we are trying to find  * (  ), where   ∈ HL * (g  ), 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 ).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].
Corollary 5.For the affine symplectic Lie algebra g  , the image of  * (g  ) in the Hochschild homology  * −1 ((g  )) can be identified injectively as the exterior algebra ∧ * (  ).

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 HH * −1 ((g  )) of the associative algebra (g  ).
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 HL * (g  ) are mapped injectively to HH * −1 ((g  )).In other words, HH * −1 ((g  )) contains ∧ * (  ) as a direct summand.Now, we know that HH * −1 ( (g  )) ≃ Hom (HH * −1 ( (g  )) , R) . (20) 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 HH * −1 ((g  )) contains ∧ * ( *  ) as a direct summand.
As an algebraic point of departure and theoretical physics point of view, the Hochschild cohomology HH * () 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 HH * () 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  = (g  ).