Polarized vectorial Poisson structures

Abstract

We study various properties of polarized vectorial Poisson structures subordinate to polarized k-symplectic manifolds, and also, we study the notion of polarized vectorial Poisson manifold. Some properties and examples are given.

Full text

arXiv:1812.08461v1 [math.SG] 20 Dec 2018
Polarized vectorial Poisson structures
Azzouz AWANE. Ismail BENALI. Souhaila EL AMINE
LAMS. Ben M’sik’s Faculty of Sciences. B.P.7955. Bd Driss Harti. Casablanca.
Hassan II University of Casablanca.
azzouz.awane@univh2c.ma; ismail.benali-etu@etu.univh2c.ma;
souhaila.elamine-etu@etu.univh2c.ma
Abstract
We study various properties of polarized vectorial Poisson structures subordinate
to polarized k-symplectic manifolds, and also, we study the notion of polarized
vectorial Poisson manifold. Some properties and examples are given.
M.S.C2010: 53D05, 37K05, 53D17.
Keywords: k-symplectic structures. Hamiltonian. polarized vectorial Poisson
manifold.
1Introduction
Mathematical and physics considerations have led to introduce the polarized k−symplectic
structures ([1], [3] and [12]). The Poisson aspect of polarized k-symplectic manifolds al-
lows us to introduce and study the notion of polarized vectorial Poisson structures.
Recall that a Polarized k-symplectic structure on an n(k+ 1)-dimensional foliated man-
ifold Mis a pair (θ, F) in which Fis an n−codimensional foliation and θis a closed and
nondegenerate Rk-valued differential 2−form vanishing on vector fields tangent to the
leaves of F.
The polarized k-symplectic Darboux’s theorem show that around each point x0of M
there is a local coordinate system (xpi, yi)1≤p≤k,1≤i≤nsuch that
θ=
k
X
p=1 n
X
i=1
dxpi ∧dyi!⊗vp
and, Fis defined by the equations : dy1= 0,...,dyn= 0.Where (vp)1≤p≤kis the
canonical basis of Rk.
A polarized Hamiltonian vector field is a foliate vector field Xsuch that i(X)θis exact.
An associated polarized Hamiltonian to Xis an Rk-valued function H∈C∞M, Rk
1

such that i(X)θ=−dH. Locally the polarized Hamiltonians have the following form
H=
k
X
p=1 
X
j
aj(y1, ..., yn)xpj +bp(y1, ..., yn)
⊗vp
where ajand bpare basic functions.
The set of all polarized Hamiltonians is a proper vector subspace of C∞(M , Rk), that we
denote by H(M, F). This subspace admits a natural Lie algebra law {,}, called polarized
vectorial Poisson structure subordinate to the polarized k−symplectic structure.
In this work, we study various aspects of polarized vectorial Poisson manifolds and we
give some properties and examples of polarized Hamiltonians. This leads us to introduce
the notion of polarized vectorial Poisson structure on a foliated manifold. We give in this
paper, a natural polarized k−symplectic structure (θ, F) on the space hom G,Rk+1,
for a given real Lie algebra G; and also, the associated linear polarized Poisson structure
on Hhom G,Rk+1,Fhaving for support the space Hhom G,Rk+1 ,Fdepending
on the Lie algebra law of G.
2 Polarized k-symplectic manifolds
Let Mbe an n(k+ 1)-dimensional smooth manifold endowed with an n-codimensional
foliation F.Let θ=θp⊗vp∈A2(M)⊗Rkbe an Rk-valued differential 2−form. We
denote by Ethe sub-bundle of T M defined by the tangent vectors of the leaves of the
foliation F. And also, we denote by Γ(E), the set of all cross-sections of the M-bundle
E−→ M, and by Ap(M) the set of all differential p-forms on M. (vp)1≤p≤kbeing the
canonical basis of the real vector space Rk.
We recall that ([1], [3]), (θ, E) is a polarized k-symplectic structure on Mif: (i) θclosed
i.e. dθ = 0 ; (ii) θnondegenerate, i.e., for all X∈X(M), i(X)θ= 0 =⇒X= 0 and
(iii) θ(X, Y ) = 0 for all X, Y ∈Γ (E).
We recall also the following theorem ([1], [3]), which gives the local model of a polarized
k-symplectic structure in the Darboux’s sense.
Theorem 1. If (θ , E)is a polarized k-symplectic structure on M, then for every point
x0of M, there exist an open neighborhood Uof Mcontaining x0equipped with a local
coordinate system (xpi, yi)1≤p≤k,1≤i≤ncalled an adapted coordinate system, such that the
Rk-valued differential 2−form θis represented on Uby
θ|U=
k
X
p=1
θp⊗vp=
k
X
p=1 n
X
i=1
dxpi ∧dyi!⊗vp
and F|Uis defined by the equations dy1=... =dyn= 0.
2

The theorem’s expressions imply the following local transition formulas of the canonical
coordinates
yi=yiy1,...,yn,xpi =
n
X
j=1
xpj ∂yj
∂yi+ϕpi y1, . . . , yn
Indeed, these expressions are affine with respect to xpj .
Recall that, [11], a real function f∈C∞(M) is called basic, if for any vector field Y
tangent to F, the function Y(f) is identically zero. We denote by A0
b(M, F) the subring
of C∞(M) of basic functions.
Let f∈C∞(M), the following properties are equivalent : (i) fis basic ; (ii) fis constant
on each leaf of F.
We recall also, that a vector field X∈X(M) is said to be foliate, or that it is an
infinitesimal automorphism of Fif in a neighborhood of any point of M, the local one
parameter group associated to Xleaves the foliation Finvariant.
We have the following equivalence : (i) Xis foliate. (ii) [X, Y ]∈Γ(E) for all Y∈Γ(E).
(iii) In a local coordinate system (xpi, yi)1≤p≤k,1≤i≤n, the vector field Xhas the following
form
X=X
p,i
ξpi xqj q,j , y1,...,yn)∂
∂xpi +
n
X
j=1
ηj(y1, . . . , yn)∂
∂yj
.
We denote byI(M, F) the space of foliate vector fields for F.The following properties
are satisfied :
1. I(M, F) is a Lie algebra.
2. I(M, F) is a module over the ring A0
b(M, F) of basic functions.
A smooth r-form αa on Mis said to be basic ([11]) if: i(Y)α= 0 et i(Y)dα = 0 ∀Y∈
Γ(E).
The following properties are equivalent : (i) αis basic. (ii) In every simple distinguished
open set, equipped with Darboux’s local coordinate system, αtake the form
α=X
1≤i1<...<ir≤n
αi1...irdyi1∧...∧dyir
where the coefficients αi1...irare basic functions.
3 Polarized Hamiltonian vector fields
The notations being the same as in the previous paragraph.
Let Mbe a manifold endowed with a polarized k-symplectic structure (θ, E).Consider
the linear mapping ζ:X(M)−→ A1(M)⊗Rk, defined by: ζ(X) = i(X)θ, for all
X∈X(M).
3

Definition 2. A vector filed X∈X(M) is said to be locally Hamiltonian polarized if
it satisfies the following conditions: (i) Xis foliate; (ii) the Rk-valued 1−form ζ(X) is
closed.
We denoted by H0(M, F) the real vector space of locally Hamiltonians polarized vector
fields
H0(M, F) = {X∈I(M , F)|d(ζ(X)) = 0}
Let Xbe a locally Hamiltonian polarized vector field. Thus, locally, around each point
x∈M, there is an open neighborhood Uof xin M, and an Rk−smooth mapping
H∈C∞(U)⊗Rksuch that ζ(X) = −dH.
With respect to a Darboux’s local coordinate system (xpi, yi)1≤p≤k,1≤i≤n, defined on an
open neighborhood Uof M, the equations of motion of Xare given by







dxpi
dt =−∂H p
∂yi
δp
q
dyi
dt =∂H p
∂xqi
∂H p
∂xqi ∈A0
b(M, F).
(Hamilton’s equation of locally polarized Hamiltonian vector field), and, the mapping
Hand Xhave the following forms respectively
H=
k
X
p=1 

n
X
j=1
aj(y1, ..., yn)xpj +bp(y1, ..., yn)
⊗vp
and
XH=−
n
X
s=1
k
X
p=1 

n
X
j=1
xps ∂aj
∂ys+∂bp
∂ys

∂
∂xps +
n
X
j=1
aj
∂
∂yj
where aj, bp∈A0
b(U, FU).
Definition 3. An element X∈I(M, F) is said to be a polarized Hamiltonian vector
field if the Rk-valued one form ζ(X) is exact. We denote by H(M, F) the real vector
space of polarized Hamiltonian vector fields.
We assume that Mis connected. Then we have an exact sequence of vector spaces
0−→ Rki
−→ C∞M, Rkd
−→ A1(M)⊗Rk−→ 0.
It is clear that ζ(H(M, F)) is vector subspace of A1(M)⊗Rk. Then we take
H(M, F) = d−1(ζ(H(M , F))) .
Therefore, for every mapping H∈C∞(M)⊗Rkwe have the following equivalence :
1. H∈H(M, F) ;
4

2. there exists a unique polarized Hamiltonian vector field XH∈H(M , F) such that
ζ(XH) = −dH.
The elements of H(M, F) are said to be polarized Hamiltonians, and XH, the correspond-
ing polarized Hamiltonian vector fields.
Also, we have the mapping ν:H(M, F)−→ H(M, F) defined by: ν(H) = XH.
Proposition 4. The following diagram:
H(M, F)ζ
−→ A1(M)⊗Rk
տ ր
νH(M, F)−d
is commutative
4 Polarized vectorial Poisson structure subordinate to po-
larized k-symplectic structure
The hypothesis and notations being the same as above.
Recall that ([1], [3]), if H, K∈H(M, F) with associated polarized Hamiltonian vector
fields XH,XKrespectively, then the Lie bracket [XH, XK] is a polarized Hamiltonian
vector field which is associated to polarized Hamiltonian {H, K}defined by
{H, K }={H, K}p⊗vp=−θp(XH, XK)⊗vp
i.e. [XH, XK] = X{H,K}and the correspondence (H, K )7−→ {H, K }from H(M, F)×
H(M, F) into H(M, F), gives to H(M, F) a structure of Lie algebra.
Definition 5. The Lie algebra (H(M, F),{,}) is said to be the polarized Poisson struc-
ture subordinate to the polarized k-symplectic structure (θ, E).
Proposition 6. We have the following properties:
1. H(M, F)is a real Lie algebra.
2. H0(M, F), H 0(M, F)⊂H(M, F).
3. H(M, F)is an ideal of H0(M, F).
4. The sequence of Lie algebras 0−→ Rk−→ H(M, F)−→ H(M, F)֒→H0(M, F)−→
H0(M,F)
H(M,F)−→ 0is exact.
5

Locally, with respect to a Darboux’s local coordinate system (xpi , yi)1≤p≤k,1≤i≤n, defined
on an open neighborhood Uof M, the bracket {H, K}is written in the form {H, K }p⊗vp
where
{H, K }p=
n
X
i=1 ∂H p
∂yi
∂K p
∂xpi −∂H p
∂xpi
∂K p
∂yi
=∂
∂yi⊗vp∧∂
∂xpi ⊗vpdH l⊗vl, dK r⊗vr
where (vp)1≤p≤kis the dual basis of the standard basis (vp)1≤p≤kof Rk.
By using the Poisson polarized bracket, the Hamilton’s equations of polarized Hamilto-
nian vector field XHtake the new form







dxpi
dt =nx1i,...,xki , H op
δp
q
dyi
dt =nyiδp
1,...,yiδp
q,...,yiδp
k, Hop
∂H p
∂xqi ∈A0
b(M, F).
Therefore, for every point xof Uwe have
{H, K }(x) = (XK·H) (x) = hdH(x), XK(x)i
=−(XH·K) (x) = − hdK(x), XH(x)i
And, Let us assume that the function His fixed in H(M, F) and when Kvaries,
{H, K }(x) only depends upon dK(x). Similarly, when we assume that Kis fixed in
H(M, F), we can show that when Hvaries, {H, K}(x) only depends upon dH (x). There-
fore, for every point xof Mthere exists a bilinear, skew symmetric mapping
P(x) : T∗
xM⊗Rk×T∗
xM⊗Rk−→ Rk
such that P(x) (dH(x), dK (x)) = {H, K}(x) for all Hand Kin H(M, F), Pis called
polarized vectorial Poisson tensor.
With respect to the local coordinate system, (xpi , yi)1≤p≤k,1≤i≤n, defined on an open
neighborhood Uof M, the vectorial tensor Pis
P=
k
X
p=1
n
X
i=1  ∂
∂yi⊗vl∧∂
∂xpi ⊗vl⊗vp
Remark 7.The vectorial tensor Pvanishes on the annihilator A1
b(M)⊗Rkof Ein the
space A1(M)⊗Rk.
6

5 Polarized vectorial Poisson manifolds
Let (M, F) be a foliated manifold.
Definition 8. A polarized Poisson structure on (M, F) is a pair (H(M , F),{,}) in which
H(M, F) is a vector subspace of C∞(M)⊗Rk.and {,}is an R−bilinear mapping from
H(M, F)×H(M , F)into H(M, F) satisfying:
1. (H(M, F),{,}) is a Lie algebra.
2. {H, K }= 0,∀H, K ∈A0
b(M)⊗Rk.
3. For each H∈H(M, F), there is a foliate vector field XHon Msuch that
XH(K) = {K, H},∀K∈H(M, F).
A polarized vectorial Poisson structure on (M, F) can be defined on Mby a C∞(M)−bilinear
skew symmetric map
P:A1(M)⊗Rk×A1(M)⊗Rk−→ C∞M, Rk
satisfying the following properties:
1. ∀H, K ∈H(M, F), P(dH, dK)∈H(M, F).
2. The correspondence (H, K )7−→ P(dH, dK) = {H, K }from H(M, F)×H(M, F)
into H(M, F), gives to H(M, F) a structure of Lie algebra.
3. Pvanishes on A1
b(M)⊗Rk: annihilator of Ein the space A1(M)⊗Rk.
4. For every H∈H(M, F) there is a foliate vector field XHsuch that
P(dH, dK) = −XH(K),∀K∈H(M, F).
6 Some properties of the polarized vectorial Poisson struc-
tures
Let (H(M, F), P ) be a polarized vectorial Poisson structure on a foliated manifold
(M, F). Let (x1, ..., xp, y1,...,yq) be distinguished coordinates of a local foliate chart.
Locally the polarized vectorial tensor Ptake the form
P=Aijr
ls  ∂
∂xi⊗vl∧∂
∂yj⊗vs⊗vr
+Bijr
ls  ∂
∂xi⊗vl∧∂
∂xj⊗vs⊗vr
And for all α∈A1(M)⊗Rk,we associate a C∞(M)−linear mapping
P(α, ·) : A1(M)⊗Rk−→ C∞(M)⊗Rk
7

such that P(α, ·)(β) = P(α, β), and a mapping
Ξ : X(M)−→ LC∞(M)A1(M)⊗Rk,C∞(M)⊗Rk
that we define by
Ξ (X) (β) = hβ, X i=
k
X
p=1
βp(X)vp=
k
X
p=1
(βp⊗vp) (X).
Locally, with respect to distinguished coordinates (x1, ..., xp, y1, . . . , yq) we have :
Ξ∂
∂xl(β) =
k
X
p=1
∂βp
∂xlvp
•The mapping Ξ is an isomorphism if and only if k= 1.
•In particular Ξ (XH) (dK ) = −P(dH, dK) = − {H, K }={K, H}for all H, K ∈
H(M, F).
7 Polarized vectorial Poisson structure subordinate to the
natural polarization k−symplectic structure of hom G,Rk+1
Let (G,[,]) be a real n-dimensional Lie algebra equipped with a basis (ei)1≤i≤n. We
denote by ωi1≤i≤nits dual basis and Ck
ij the structure constants of G: [ei, ej] = Ck
ijek.
And let
hom G,Rk+1=G∗⊗Rk+1
the space of linear maps of Ginto Rk+1. The vector space hom G,Rk+1is generated
by the linear maps:
ωi⊗vq, ωi⊗w(1 ≤q≤k; 1 ≤i≤n)
where (vq, w)1≤q≤k;is the canonical basis of Rk+1 .
Each element Xof hom G,Rk+1can be written as
X=xq
iωi⊗vq+yiωi⊗w=






x1
1. . . x1
n
.
.
..
.
..
.
.
xk
1
.
.
.xk
n
y1··· yn







.
The linear mapping X:G−→Rk+1transforms u=ujejinto
X(u) = xq
iωi⊗vqujej+yiωi⊗wujej
=xq
iuivq+yiuiw
8

So, in terms of matrices, we have
X(u) = 






x1
1. . . x1
n
.
.
..
.
..
.
.
xk
1
.
.
.xk
n
y1··· yn










u1
.
.
.
un


.
The space hom G,Rk+1is an n(k+ 1)-dimensional smooth manifold. We equip this
space with the coordinate system (xq
i, yi)1≤q≤k;1≤i≤n.
There is a natural polarized k-symplectic structure (θ, F) on hom G,Rk+1defined by
θ= n
X
i=1
dxq
i∧dyi!⊗vq
and Fis the foliation defined by dy1= 0,··· , dyn= 0.
The associated polarized Hamiltonians are the Rk−valued smooth functions
H∈C∞hom G,Rk+1⊗Rk
defined on hom G,Rk+1by the following expressions
H(X) = ai(y1,··· , yn)xq
i+bq(y1,··· , yn)⊗vq
where a1,...,an, bq(q= 1,··· , k) are real basic functions.
The polarized Poisson bracket of H(X) = ai(y1,··· , yn)xq
i+bq(y1,··· , yn)⊗vqand
K(X) = a′i(y1,··· , yn)xq
i+b′q(y1,··· , yn)⊗vqis given by
{H, K }q(X) = Pn
i=1 ∂H q
∂yi
∂K q
∂xq
i
−∂H q
∂xq
i
∂K q
∂yi
=Pn
i=1 xq
j∂aj
∂yi+∂ bq
∂yia′i−aixq
j∂a′j
∂yi+∂ b′q
∂yi
=Pn
i=1 xq
ja′i∂aj
∂yi−ai∂ a′j
∂yi+a′i∂ bq
∂yi−ai∂ b′q
∂yi.
The bracket, so defined, allows to provide Hhom G,Rk+1,Fwith a polarized vecto-
rial Poisson structure subordinate to the polarization k-symplectic structure (θ, F).This
structure does not depend on the law of G.
8 Linear polarized vectorial Poisson structure of hom G,Rk+1
In addition to the polarized vectorial Poisson structure subordinate to the natural
k−symplectic polarization on hom G,Rk+1,we can define another polarized vecto-
rial Poisson structure Hhom G,Rk+1,F;{,}Lso-called the linear polarized vector
Poisson structure of (G,[,]) .
9

Let H∈Hhom G,Rk+1,F,X∈hom G,Rk+1and jq:G∗−→ hom G,Rk+1
defined by
jq(ωi) = ωi⊗vq.
For all p, q ∈ {1,··· , k},the composite map
G∗jq
−→ hom G,Rk+1dHp
X
−→ R
is the linear form on G∗defined by
(dHp
X◦jq)ωi=dHp
Xωi⊗vq=∂H p
∂xq
i
(X) = δp
qai.
Hence,
dH1
X◦j1=···=dH k
X◦jk=aiei.
We take
{H, K }L(X) = prRkdH 1
X◦j1, dK1
X◦j1, X
=···
=prRkDhdHk
X◦jk, dKk
X◦jki, XE
therefore,
{H, K }L(X) = prRkh[dHp
X◦jp, dKp
X◦jp], Xi
=prRkaiei, a′jej, X
=prRkDaia′jCl
ij el, XE
=aia′jCl
ij xp
lvp
=P1≤i<j≤nCl
ij aia′j−aja′ixp
lvp
Examples
1. Gis an abelian Lie algebra. In this case {,}L= 0. Consequently, Hhom G,Rk+1,F;{,}L
the abelian polarized vectorial Poisson.
2. Gis Heisenberg’s Lie algebra H1of dimension 3. The Lie algebra law of H1
is given by [e1, e2] = e3. And so, for all H, K ∈Hhom G,Rk+1 ,F,X∈
hom G,Rk+1, where H(X) = ai(y1, y2, y3)xi+b(y1, y2, y3)⊗vqand K(X) =
a′i(y1, y2, y3)xi+b′(y1, y2, y3)⊗vq, we have
{H, K }L(X) = a1a′2−a2a′1xp
3vp.
3. Gis the polarized 1-symplectic nilpotent Lie algebra h3⊕aof dimension 4. The
Lie algebra Gdefined by
dω1=ω2∧ω3
dωi= 0 (i= 2,3,4).
10

So, for all H, K ∈Hhom G,Rk+1,F,X∈hom G,Rk+1, where H(X) =
ai(y1, y2, y3, y4)xq
i+bq(y1, y2, y3, y4)⊗vqand
K(X) = a′i(y1, y2, y3, y4)xq
i+b′q(y1, y2, y3, y4)⊗vq, we have
{H, K }L(X) = a3a′2−a2a′3xp
1vp.
4. Gis the polarized 1-symplectic nilpotent Lie algebra n4of dimension 4. The Lie
algebra Gdefined by
dω3=ω1∧ω2
dω4=ω1∧ω3
dωi= 0 (i= 1,2).
So, for all H, K ∈Hhom G,Rk+1,F,X∈hom G,Rk+1, where H(X) =
ai(y1, y2, y3, y4)xq
i+bq(y1, y2, y3, y4)⊗vqand
K(X) = a′i(y1, y2, y3, y4)xq
i+b′q(y1, y2, y3, y4)⊗vq, we have
{H, K }L(X) = a2a′1−a1a′2xp
3vp+a3a′1−a1a′3xp
4vp.
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