# Calculations on Lie Algebra of the Group of Affine Symplectomorphisms

Authors: Zuhier Altawallbeh
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

We find the image of the affine symplectic Lie algebragn from the Leibniz homology HL⁎(gn) to the Lie algebra homology H⁎Lie(gn). The result shows that the image is the exterior algebra ∧⁎(wn) generated by the forms wn=∑i=1n(∂/∂xi∧∂/∂yi). 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 HH⁎-1(U(gn)) is isomorphic to H⁎-1Lie(gn,U(gn)ad). Explicitly, we use the alternation of multilinear map, in our elements, to do certain calculations.


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## Body

## 1. Introduction

Recall that the group of affine symplectomorphisms, which is the affine symplectic groupASpn, is given by all transformations Ψ:R2n→R2n of the form Ψ(z)=z0+Az, where A is a 2n×2n symplectic matrix and z0 a fixed element of R2n [1]. The Lie algebra gn of ASpn 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 gn is determined by the exterior algebra of the forms wn=∑i=1n(∂/∂xi∧∂/∂yi), as follows: HL∗(gn)≃∧∗(wn), where ∂/∂xi and ∂/∂yi are the unit vector fields parallel to xi and yi axes, respectively, and the Lie algebra homology H∗Lie(gn) has been proved to have an isomorphic vector space as follows: H∗Lie(gn)≃H∗(spn,R)⊗∧∗(wn), where H∗(spn,R) is the singular homology of the real symplectic Lie algebra spn and ∧∗(wn)=∑k≥0∧k(wn).Here, we find the image of the affine symplectic Lie algebragn from the Leibniz homology HL∗(gn) to the Lie algebra homology H∗Lie(gn). The result shows that the image is the tensor of a real number with the exterior algebra ∧∗(wn). 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∗(gn) in H∗-1Lie(gn) is important to find the image in Hochschild homology HH∗-1(U(gn)). In particular, we find the image via the map (1)j∗∘π∗:HL∗gn→π∗H∗-1Liegn,gn→j∗H∗-1Liegn,Ugnad,where the maps j∗ and π∗ are induced by the chain maps, on the chain level, CL∗(gn)→πC∗-1Lie(gn,gn) and C∗-1Lie(gn,gn)→jC∗-1Lie(gn,U(gn)ad), respectively, and U(gn)ad is the adjoint universal enveloping algebra of gn. Since H∗-1Lie(gn,U(gn)ad) is isomorphic to the Hochschild homology HH∗-1(Ugn) [6], we get the image in HH∗-1(Ugn).In symmetric geometry, the study of symplectic algebras is important in manifolds because of the structure of the symplectic groupSP(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 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 algebrag over a ring k, the Lie algebra homology of g, written H∗Lie(g,k), is the homology of the chain complex ∧∗(g) which was introduced by Chevalley and Eilenberg in [8]; namely,(2)k⟵0g⟵,g∧2⟵d⋯⟵dg∧n-1⟵dg∧n⟵d⋯,where(3)dg1∧g2∧⋯∧gn=∑1≤i<j≤n-1i+jgi,gj∧g1∧⋯∧g∧i∧⋯∧g∧j∧⋯∧gn,where the notation g∧i means that element has been deleted. In this paper H∗Lie(g) denotes homology with real coefficients, where k=R. Lie homology, with coefficients in the adjoint representation of the universal enveloping algebra U(g)ad, is the homology of the chain complex U(g)⊗∧∗(g):(4)Ug⟵Ug⊗g⟵Ug⊗g∧2⟵⋯⟵Ug⊗g∧n-1⟵dUg⊗g∧n⟵⋯,where(5)dg0⊗g1∧g2∧⋯∧gn=∑1≤j≤n-1j+1g0,gj⊗g1∧g2∧⋯g∧j∧⋯∧gn+∑1≤i<j≤n-1i+jg0⊗gi,gj∧g1∧⋯∧g∧i∧⋯∧g∧j∧⋯∧gn.The canonical projection(6)ρ:g⊗∧∗g⟶∧∗+1ggiven by g⊗g∧n→g∧(n+1) is a map of chain complexes and induces a k-linear map on homology(7)ρ∗:HnLieg,g⟶Hn+1Lieg,k.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 algebrag over a ring k, we recall that the Leibniz homology [10] of g, written HL∗(g), is the homology of the chain complex(8)k⟵0g⟵,g⊗2⟵∂⋯⟵∂g⊗n-1⟵∂g⊗n⟵∂⋯,where(9)∂g1⊗g2⊗⋯⊗gn=∑1≤i<j≤n-1jg1⊗g2⊗⋯⊗gi-1⊗gi,gj⊗gi+1⊗⋯⊗g∧j⊗⋯⊗gn.Definition 1.
Letk be a commutative ring and M be a g-bimodule of an associative (not necessarily commutative) k-algebra g. We define the Hochschild complex CH∗(g,M) as the sequence of maps CH∗(g,M):⋯→bM⊗g⊗n→bM⊗g⊗n-1→b⋯→bM⊗g→bM, where the module M⊗g⊗n is in degree n. The Hochschild boundary map b:M⊗g⊗n→M⊗g⊗n-1 is given by (10)bm⊗g1⊗g2⊗⋯gn≔mg1⊗g2⊗⋯⊗gn+∑i=1n-1-1im⊗g1⊗⋯⊗gigi+1⊗⋯⊗gn+-1ngnm⊗g1⊗⋯⊗gn-1for m∈M and gi∈g for all i=1,…,n. The homology groups of the Hochschild complex CHn(g,M) are called the Hochschild homology groups HHn(g,M). For g=M, we write HHn(g).


## 4. Affine Symplectic Lie Algebra

We begin by(x1,x2,…,xn,y1,y2,…,yn)∈R2n, where ∂/∂xi,∂/∂yi are the unit vector fields parallel to xi and yi axes, respectively. Then the real symplectic Lie algebra spn has a basis (11)B1=xk∂∂yk,yk∂∂xkwherek=1,2,…,n,xi∂∂yj+xj∂∂yi,yi∂∂xj+yj∂∂xiwhere1≤i<j≤n,yj∂∂yi-xi∂∂xjwherei=1,2,…n,j=1,2,…,n.LetIn be the abelian Lie algebra with the basis B2=∂/∂x1,∂/∂x2,…,∂/∂xn,∂/∂y1,∂/∂y2,…,∂/∂yn. The affine symplectic Lie algebra gn has the basis B1∪B2. Thus, there is a short exact sequence of Lie algebras(12)0⟶In⟶ign⟶πspn⟶0.In the following example, we find the Lie brackets of the elements insp2 by taking into account the basic elements illustrated above.Example 2.
The basisB1 of the real symplectic Lie algebra sp2 contains exactly these elements (13)x1∂∂y1,x2∂∂y2,y1∂∂x1,y2∂∂x2,x1∂∂y2+x2∂∂y1,y1∂∂x2+y2∂∂x1,y1∂∂y1-x1∂∂x1,y2∂∂y1-x1∂∂x2,y1∂∂y2-x2∂∂x1,y2∂∂y2-x2∂∂x2,which can be denoted by e1,e2,…,e10, respectively. It is known that ei,ei=0 and ei,ej=-ej,ei for all i,j=1,2,…,n. By taking the Lie brackets of the others, it follows that(14)e1,e2=0,e1,e3=x1∂∂x1-y1∂∂y1=-e7,e1,e4=0,e1,e5=0,e1,e6=x1∂∂x2-y2∂∂y1=-e8.
Now we takee1,e7=2x1(∂/∂y1)=2e1, which means that e1 is the Eigenvector of e1,e7. Similarly, if we continue the computations, we get that ei is the Eigenvector not only for the bracket e7,ei but also for e10,ei for all i=1,…n.The above example shows that the Cartan subalgebra ofsp2 is e7,e10 which is the tangent of the maximal torus subset in the Lie group SP(2,R).


## 5. The Image ofHL∗(gn) in H∗Lie(gn)

By convention, we denote the affine symplectic Lie algebra bygn. There is a canonical projection T(gn)→∧∗(gn), where T(gn) is the tensor algebra of gn and ∧∗(gn) is the exterior algebra of gn, which is naturally defined by π′:gn⊗l→gn∧l for l≥0 Thus, the map π′ induces a k-linear map on homology(15)π∗′:HL∗gn→H∗Liegn.From [4], there are these two vector spaces isomorphisms HL∗(gn)≃∧∗(wn) and H∗Lie(gn)≃H∗(spn,R)⊗∧∗(wn). Let us start with the element wn=∑i=1n(∂/∂xi∧∂/∂yi)∈HL∗(gn).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(16)π∗′wn=π∗′∑∂∂xi∧∂∂yi=π∗′12∑i=1n∂∂xi⊗∂∂yi-∂∂yi⊗∂∂xi=12∑i=1n∂∂xi∧∂∂yi-∂∂yi∧∂∂xi=12∑i=1n2∂∂xi∧∂∂yi=wn.For more general setting, let us take∧2wn∈HL∗(gn), so we get(17)π∗′∧2wn=π∗′∑i=1n∂∂xi∧∂∂yi∧∑i=1n∂∂xi∧∂∂yi=π∗′∂∂x1∧∂∂y1∧∑i=1n∂∂xi∧∂∂yi+⋯+∂∂xn∧∂∂yn∧∑i=1n∂∂xi∧∂∂yi=π∗′2∑1≤i<j≤n∂∂xi∧∂∂yi∧∂∂xj∧∂∂yj=π∗′2124∑1≤i<j≤nσ∈S4sgn⁡σ∂∂xiσ1⊗∂∂yiσ2⊗∂∂xjσ3⊗∂∂yjσ4=2∑1≤i<j≤n∂∂xi∧∂∂yi∧∂∂xj∧∂∂yj=∧2wn.Thus(18)π∗′∧2wn=∧2wn∈H∗Liegn.The result makes sense becauseH∗Lie(gn)≃H∗(spn,R)⊗∧∗(wn) and H0(spn,R)⊗∧∗(wn)=R⊗∧∗(wn)≃∧∗(wn).


## 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 homologyHL∗(gn) in the Lie algebra homology H∗-1Lie(gn,U(gn)ad) of the adjoint universal enveloping algebra U(gn)ad. In particular, we find the image via the map (19)j∗∘π∗:HL∗gn→π∗H∗-1Liegn,gn→j∗H∗-1Liegn,Ugnad,where the maps j∗ and π∗ are induced by the chain maps π and j on the chain level. Naturally π and j can be defined as follows: π:gn⊗l→gn⊗gn∧(l-1) and the inclusion j:gn⊗gn∧(l-1)↪U(gn)⊗gn∧(l-1). It is not difficult to prove that π and j are chain maps. Now if we are trying to find π∗(wn), where wn∈HL∗(gn), we get similar procedure steps as we have done above, by taking into account that π is different a little bit from the map π′ and we will get the same result. I mean π∗(∧lwn)=∧lwn. The image result ∧lwn is in H∗-1Lie(gn,gn) because in [4] it was proven that H∗-1Lie(gn,gn)≃H∗(spn,R)⊗∧∗-(wn), where ∧∗-(wn)=∑l≥1∧lwn. After taking the induced map j∗, we get j∗(π∗(∧lwn))=∧lwn in H∗-1Lie(gn,U(gn)ad). 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 mapϵn).
Suppose thatg and M as they were in the previous definition. We define [6] the antisymmetrization map ϵn:M⊗∧ng→M⊗g⊗n as one that sends the element m⊗g1∧⋯∧gn (for m∈M and gi∈g for all i=1,…,n) to ϵn(m⊗g1⊗⋯⊗gn) by ϵn≔∑σ∈Snsgn⁡(σ)σ, where σ is a permutation in the symmetric group Sn on the set of indices {1,…,n}, and σ acts on (the left of) m⊗g1⊗⋯⊗gn by σ·(m⊗g1⊗g2⊗⋯gn)=m⊗gσ-1(1)⊗⋯⊗gσ-1(n).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: HH∗(U(g),M)≃ϵ∗H∗Lie(g,Mad). For M=U(g), we get HH∗(U(g))≃ϵ∗H∗Lie(g,U(g)ad).By applying the above theorem, we get the image ofHL∗(gn) in the Hochschild homology HH∗-1(U(gn)) as (ϵ∗∘j∗∘π∗)(wn)=wn for any wn∈HL∗(gn).Corollary 5.
For the affine symplectic Lie algebragn, the image of HL∗(gn) in the Hochschild homology HH∗-1(U(gn)) can be identified injectively as the exterior algebra ∧∗(wn).


## 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 manifoldM, 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 HH∗-1(U(gn)) of the associative algebra U(gn).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 inHL∗(gn) are mapped injectively to HH∗-1(U(gn)). In other words, HH∗-1(U(gn)) contains ∧∗(wn) as a direct summand. Now, we know that(20)HH∗-1Ugn≃HomHH∗-1Ugn,R.Taking into account that∧∗(wn∗) is the dual space of ∧∗(wn), where wn∗=dxi∧dyi and dxi is dual of ∂/∂xi and dyi is dual of ∂/∂yj, we end up with this following result about the image in Hochschild cohomology of the given algebra.Corollary 6.
The Hochschild cohomologyHH∗-1(U(gn)) contains ∧∗(wn∗) as a direct summand.As an algebraic point of departure and theoretical physics point of view, the Hochschild cohomologyHH∗(A) of an associative algebra A 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) 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(gn).



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