Braz. J. Phys. vol.39 número1

Modified Symplectic Structures in Cotangent Bundles of Lie Groups.

F.J. Vanhecke, C. Sigaud, and A.R. da Silva

Instituto de F´ısica, Instituto de Matem´atica, UFRJ, Rio de Janeiro, Brazil (Received on 10 April, 2008)

In earlier work [1], we studied an extension of the canonical symplectic structure in the cotangent bundle of an affine spaceQ=RN, by additional terms implying the Poisson non-commutativity of both configuration and momentum variables. In this article, we claim that such an extension can be done consistently whenQ is a Lie groupG.

Keywords: Symplectic Mechanics; Noncommutative Configuration Space.

1. INTRODUCTION

As applied to physics, noncommutative geometry is under-stood mainly in two ways. The first one is the spectral triple approach of A.Connes [2] with the Dirac operator playing a central role in unifying, through the universal action principle, gravitation with the standard model of fundamental interac-tions. The second one is the quantum field theory on noncom-mutative spaces [3] with the Moyal product as main ingredient. Besides these, a proposition by several authors [4, 5] was made to generalise quantum mechanics in such a way that the oper-ators corresponding to space coordinates no longer commute:

[_bxk_,xbℓ_{]6=0 . This was implemented by an extension of the} Pois-son structure on the cotangent space such that the brackets

sat-isfyxk,xℓ 6=0. Upon quantisation, the corresponding oper-ators should then also be noncommutative. A particle moving in an affine spaceAN_{, has its configuration, in a fixed} refer-ence frame, given by an element{xk}of the translation group:

Q

=RN with cotangent bundleT⋆(

Q

) =RN×RN. In [1], we examined such an extension of the canonical symplectic two-formω0=dxi∧d pi→Ω=ω0+ωF+ωB:

ωF= 1

2Fi j(x)dx

i_∧dxj_,ω B=

1 2B

kℓ_{(p)d p}

k∧d pℓ (1.1)

This extension is form-invariant under a change of the refer-ence frame lifted to the cotangent bundle:

T⋆(

Q

)→T⋆(

Q

): xi,pk→

x′i=Aijxj+ak,p′k=pℓ(A−1)ℓk

(1.2)

Ω → Ω′=dx′i∧d p′_i+1

2F ′

i j(x′)dx′i∧dx′j+ 1 2B

′kℓ_(p′_{)d p}′ k∧d p′ℓ

(1.3) F_{i j}′(x′) =Fkℓ(x) (A−1)ki(A

−1₎ℓ j,B

′kℓ_(p′_{) =A}k

iAℓjBi j(p)

For a general configuration space

Q

, a diffeomorphismφ:xi→ x′i=. φi_{(x), when lifted toT}⋆₍

_Q

_{), becomes}

eφ: xi,pk

→

x′i=φi_(x),p′ k=pℓ

∂(φ−1_(x′₎₎ℓ

∂x′k

F_{i j}′(x′) =Fkℓ(x)

∂(φ−1₎k_(x′₎ ∂x′i

∂(φ−1₎ℓ_(x′₎ ∂x′j

B′kℓ(p′,x′) =∂φ k_(x)

∂xi

∂φℓ(x) ∂xj B

i j_(p)

In general B′kℓ _{is function of both variables {p}′_,x′_{} and no} intrinsic meaning can be given to the particular form of the ex-tensionΩin equation (1.1).

In this work, we show that such an extension is achieved when

Q

=Gis a Lie group. This is possible because the cotangent bundle T⋆(G) has two distinguished trivialisations, the left-and right trivialisations [7] implemented respectively by the bases of the left- and right invariant differential forms. In section2., inspired by the rigid body motion, we use the left trivialisation with left invariant orbody-coordinatesand

con-struct a left invariant two-form. In the case of constantFi jand Bkℓ_{fields theω}

F term arises from a symplectic one-cocycle, as introduced by Souriau [8, 9], andωBwill be automatically left invariant. The constructed two-formΩis obviously closed but the non degeneracy condition leads in general to a con-strained Hamiltonian system. This is examined in more detail forSU(2)in section3.. Final considerations are made in sec-tion4.. Some elements of Lie algebra cohomology [9, 10] are recalled in the appendix.

2. THE PHASE SPACE{M0≡T⋆(G),ω0}

Let {gα,α =1,2,· · ·,N} be coordinates of a group el-ement g∈G. Natural or holonomic coordinates of points

define left- and right invariant vector fields and one forms:

eL_α(g)=. L_g∗|eeα,eRα(g)=. Rg∗|eeα,εαL(g) . =L∗

g−1|gε α_,εα

R(g) . =

R∗

g−1_|gεα. With canonical group coordinates, in terms of

Lα_β(g,h)=. ∂(g h)α_/∂gβ_andRα

β(g,h)=. ∂(h g)α/∂gβ, they are explicitely given by:

eL_α(g) =Lµα(g,e) ∂ ∂gµ , e

R

α(g) =Rµα(g,e) ∂ ∂gµ

(2.1)

εα_L(g) =Lαµ(g−1,g)dgµ , εαR(g) =Rαµ(g−1,g)dgµ

These bases implement canonical trivialisations of the tangent and cotangent bundle. For the cotangent bundle, which is the arena of symplectic or Hamiltonian formalism, we have a left and a right trivialisation:

λ : T⋆(G)→G×

G

⋆:(g,pg=pµdgµ)→

g,πL=L_g|e∗ pg=πLµεµ

πLµ=hpg,eLµi=pνLνµ(g,e)

ρ : T⋆(G)→G×

G

⋆:(g,pg=pµdgµ)→

g,πR_=R∗

g|epg=πRµεµ

πR

µ=hpg,eRµi=pνRνµ(g,e)

They can be viewed as a change of coordinates of a point

(g,pg)inT⋆(G):

(g,pg)↔(gα,pµ)hol↔(gα,πLµ)B↔(gα,πRµ)S (2.2) In rigid body theory, the coordinates of the left trivialisation are the ”body” coordinates, whence the subscript(,)B. The right trivialisation yields ”space” coordinates with subscript (,)S. Both are related through the coadjoint representation ofGin

G

⋆_:

πR

µ=Kµν(g)πLν=Adνµ(g−1)πLν (2.3) Lifting the left multiplication in G to the cotangent bundle yields a group action: eLa:T⋆(G)→T⋆(G):x= (g,pg)→ y= (ag,p_ag′ =L⋆

a−1|agpg). In body coordinates:

e La

B : (gα,πL

µ)B→((ag)α,πLµ)B. The pull-back of the cotangent pro-jectionκ:T⋆(G)→G:x= (. g,pg)→g, acting on the{εα(g)} yieldeLainvariant one forms onT⋆(G):hεLα(x)|=κ⋆xεαL(κ(x)) and the differentials of the left invariant functionsπL

µonT⋆(G) also yield eLa invariant one forms on T⋆(G). Together they provide a left invariant basis of the cotangent space at x= (gα,πL

µ)B∈T⋆(G):

hεα_L|=. Lαµ(g−1,g)hdgµ|,hεLµ| . =hdπL

µ| (2.4) Its dual basis in the tangent spaceTx(T⋆(G))is given by

|eL_αi=. |∂/∂gµiLµα(g,e),|eµLi .

=|∂/∂πLµi (2.5) The canonical Liouville one-formhθ0|=pαhdgα|and its as-sociated symplectic two-formω0=−dθ0=hdgα| ∧ hdpα|, are obtained as:

hθ0|=πLµhε µ

L|,ω0=hεµL| ∧ hε L µ|+

1 2π

L

µfµαβhεαL| ∧ hε β L| (2.6)

The Hamiltonian vector field associated to a functionA(g,πL₎ on phase space

M

0≡T⋆(G), is defined by:ıXω0=hdA|. Its components are:

Xµ =. hεµ_L|Xi=hdA|eµ_Li

Xα =. hεLα|Xi=−hdA|eLαi −πLµfµαβhdA|eβLi (2.7)

With ıYω0=hdB|, the Poisson bracket of dynamical vari-ables:{A,B}0=. ω0(X,Y), is obtained explicitely in(gα,πLµ) variables as:

{A,B}0=hdA|eLαi ∂B

∂πL α

− ∂A

∂πL α

hdB|eL_αi − ∂A

∂πL α

πL µfµαβ

∂B

∂πL β (2.8) In particular, the basic Poisson brackets are:

n gα,gβo

0=0 ,

gα,πL ν 0=L

α ν(g,e) n

πLµ,gβ o

0=−L β

µ(g,e) ,

πLµ,πLν ₀=−πLκfκµν (2.9)

The flow of a particular observable, the HamiltonianH(g,πL_), determines the time evolution of any observableA(g,πL_)by the equation: dA/dt={A,H}0. We assume a Hamiltonian is of the formH(g,πL_{) =K(π}L_{) +V(g).}

Here, as in rigid body mechanics, thekinetic energyis given by

K=. 1

2

I

αβ_πL

απLβ (2.10)

motion read:

hεα_L|dg/dti = Lα_β(g−1,g)d g β dt = ∂K ∂πL α (2.11)

hεLµ|dπL/dti = dπL

µ dt =−

∂V

∂gαL α

µ(g,e) + ∂K

∂πL ν

πLαfανµ

(2.12) The first of these equations(2.11)relates the angular momen-tumπL

αwith the angular velocity in the body frameΩµL: Ωα_L=. Lαβ(g−1,g)

dgβ dt =

I

αµ_πL

µ;πLµ=IµνΩνL (2.13) while the second(2.12)takes the classical form

dπL µ dt +π

L

κfκµνΩνL=− ∂V

∂gαL α

µ(g,e) (2.14) An example ofV(g)is given by agravitational potential en-ergyas follows. LetL=eαLαbe a constant vector in

G

(the position of the centre of mass in the body frame) andγ=γαεα a constant vector in

G

⋆ _{(the gravitational force in the space} fixed frame). The potential energy is defined as:

V(g)=. −(γ|Ad(g)L) =− K(g−1)γ|L (2.15) where(|)denotes the canonical pairing between

G

and its dual

G

⋆_{. To computehdV|e}L

µiwe use the representation of the Maurer-Cartan form:

D(g−1)dD(g) =D′(g−1dg)

whereDis any representationDofG, with derived representa-tionD′of

G

. In particular,dAd(g) =Ad(g)ad(eµ)εµL(g)and dK(g) =K(g)k(eµ)εµL(g). This yields:

hdV|eL_µi(g) =− K(g−1)γ|ad(eµ)L

=−(Γ(g)|ad(eµ)L) (2.16) whereΓ(g)=. K(g−1_{)γ is the variable gravitational force in} the body-fixed frame. Using the above formulae to compute

dK(g−1), we obtain: dΓµ

dt = (Γ|ad(eµ)ΩL) =Γαf α

µβΩβL (2.17) Equation(2.14)reads:

dπL µ dt +π

L

αfαµβΩβL= (Γ|ad(eµ)L) =ΓαfαµβLβ (2.18) Together with(2.13),

ΩαL .

=Lα_β(g−1,g)dg β

dt =

I

αµ_πL

µ

the equations (2.17) and (2.18) form the so-called Euler-Poisson system.

3. MODIFIED SYMPLECTIC STRUCTURE ONT⋆(G)

In appendix A it is shown that, if Θ= 1

2Θαβεα∧εβ ∈

Λ2₍

_G

⋆_{), obeys the cocycle condition (A.1), then Θ} L(g)=.

(1/2)ΘαβεαL(g)∧ε β

L(g)is a closed left-invariant two-form on G. Including this closed two-form in the canonical two-form, one obtains another symplectic two-form on T⋆(G), which, furthermore, iseLainvariant. So we define:

ωI=ω0−ΘL=hεµL| ∧ hdπ L µ|+

1 2 π

L

µfµαβ−Θαβ

hεαL| ∧ hε β L| (3.1) The Poisson brackets are also modified and (2.8), (2.9) be-come:

{A,B}I = ∂A

∂gµL µ

α(g,e) ∂ B

∂πL α

− ∂B

∂gµL µ

α(g,e) ∂ A

∂πL α

− πLµfµαβ−Θαβ

∂A ∂πL α ∂B ∂πL β (3.2) In particular, the fundamental brackets are:

n gα,gβo

I=0 ,

gα,πLν I=L α

ν(g,e) n

πL µ,gβ

o

I=−L β

µ(g,e) , πLµ,πνL _I=− πLκfκµν−Θµν

(3.3) The modified symplectic structure induces an additional inter-action and the Euler equations become:

Ωα_L=. Lαβ(g−1,g) dgβ

dt =

∂K

∂πL α

=

I

αµπL

µ (3.4)

dπL µ

dt = −hdV|e L µi+

∂K

∂πL α

πL

κfκαµ−Θαµ

(3.5) The relation between the velocity in the body frame and the angular momentum(2.13)is maintained: πL

µ=IµνΩνL, while the second(2.14)takes the interaction into account:

dπL µ dt +π

L

κfκµαΩαL =−hdV|eLµi −ΩαLΘαµ (3.6) For a semisimple Lie algebra

G

, we haveΘαβ=−ξµfµαβand we may define a modified Liouville one-form:

hθI|=π′µhε µ L|,π′µ

.

=πLµ+ξµ (3.7) and the symplectic two-form reads

ωI=−dhθI|=hεµL| ∧ hdπ ′ µ|+

1 2π

′

µfµαβhεαL| ∧ hε β

L| (3.8) This means that such that{gα,p′_µ=pµ+ξβLβµ(g−1;g)}are Darboux coordinates:

hθI|=p′µhdgµ|,ωI=. −dhθI|=hdgµ| ∧ hdp′µ| (3.9) In gα,π′µ

coordinates, the Hamiltonian reads H′=K′(π′)+V(g) =1

2

I

µν_(π′

µ−ξµ) (π′ν−ξν)+V(g) (3.10) and the Euler equations read:

Lα_β(g−1,g)dg β

dt =

∂K′

∂π′ α

=

I

αµ(π′_µ−ξµ) (3.11) dπ′µ

dt = −hdV|e L µi+

∂K′

∂π′ α

4. THE CLOSED TWO-FORMωL

Configuration space coordinates which do not Poisson com-mute, are obtained through the addition of a left-invariant and

closed two-form to(3.1):

ϒL₌. 1 2ϒ

µν_hdπL

µ| ∧ hdπLν| (4.1)

ωL=. ω0−ΘL+ϒL = hεµL| ∧ hdπLµ|+ 1 2 π

L

µfµαβ−Θαβ

hεαL| ∧ hε β L|

+1

2ϒ µν_hdπL

µ| ∧ hdπLν| (4.2)

With the notationSαβ≡ πLµfµαβ−Θαβ, we witeωLin matrix form:

ωL≡ 1 2 hε

α

L| hdπLµ|

∧  

S_αβ δαν

−δµ β ϒµν

 

  hε

β L|

hdπL ν|

  (4.3)

The degeneracy of(ωL)is examined comsidering the equation ı_|XiωL=hdA| (4.4) In the bases(2.4),(2.5): Xα=. hεα

L|Xi,Xµ=. hεLµ|Xiand(4.4) reads:

XαΦαν=hdA|eνLi+hdA|eLµiϒµν,

XµΨµβ=−hdA|eLβi+hdA|eαLiSαβ (4.5) where we introduced the matrices, linear in the momenta:

Φαν=. δαν+Sαµϒµν,Ψµβ=. δµβ+ϒµνSνβ (4.6) They are mutually transposed and the products ΦS =

SΨ,ϒ Φ=Ψ ϒare antisymmetric. The fundamental equation

(4.4), defining Hamiltonian vector fields, has a solution ifΦ

andΨhave inverses, i.e. if

∆=. detΦ≡detΨ6=0 (4.7) The matricesϒ Φ−1_=Ψ−1_ϒandΦ−1_S=SΨ−1_{are then also} antisymmetric. The Hamiltonian vector fields are obtained as:

Xα = (Ψ−1₎α

µ hdA|e µ

Li −ϒµνhdA|eLνi

= hdA|eν_Li+hdA|eL_µiϒµν(Φ−1₎ ν

α

Xµ = (Φ−1)µ α

−hdA|eL_αi −S_αβhdA|eβ_Li

= −hdA|eL_βi+hdA|eα_LiSαβ

(Ψ−1₎β

µ (4.8) The Poisson brackets between the basic dynamical variables are:

n gα,gβo

L=−L α

κ(g,e)Lβλ(g,e)ϒκµ(Φ−1)µ λ

gα,πLν L=L α

κ(g,e) (Ψ−1)κν, n

πL µ,gβ

o

L=−L β

κ(g,e) (Ψ−1)κµ

πL

µ,πLν _L=−Sµκ(Ψ−1)κ_ν (4.9)

For a HamiltonianH=K+V, the equations of motion are:

Ωα_L=. Lα_β(g−1,g)dg β

dt =

_∂K

∂πL ν

+hdV|eL_µiϒµν

(Φ−1₎ να

dπL µ dt =

−hdV|eL_βi+ ∂K ∂πL α

S_αβ

(Ψ−1)β_µ

SinceΦ,Ψare linear inπL_{,∆is a polynomial inπ}L_{of degree} at most equal toN, the dimension of the Lie group. It defines an algebraic variety in

G

⋆_:

Π1=. {(g,πL)|∆(πL) =0} (4.10) and its complement

V

∆=.

G

⋆\Π1defines a manifold

M

₀′=. G×

V

∆ (4.11)

with symplectic structure given byωL, restricted to

M

0′. If it happens thatΠ1itself is an algebraic manifold, an imbedded submanifold is obtained:

M

1=. G×Π1 (4.12)

with imbedding in

M

0=. G×

G

⋆: j1:

M

1֒→

M

0. The system is then constrained to

M

1and we may look for solutions of

(4.4)restricted to

M

1. Such solutions may exist if further con-ditions are imposed on the Hamiltonian. To proceed systemati-cally, we follow the algorithm of Gotay, Nester and Hinds [11]. To keep things simple, this will be done in the next section for the semi-simple groupSU(2).

5. A CASE STUDY:SU(2)

The dynamical variables are functions on

M

0=. SU(2)× su(2)⋆. A basis {eα} of the Lie algebrasu(2)may be cho-sen such that its structure constants are the Kronecker sym-bols[eα,eβ] =eµεµαβ. The Killing metricηαβ=. εµανενβµ=

in terms ofτλ, adual magnetic field in momentum space, as Yµν=τλελµν. Definingπ′κ=. πLκ+ξκ,ωLreads:

ωL≡ 1 2 hε

α

L| hdπLµ|

∧  

π′

κεκαβ δαν

−δµ

β τλελµν  



 hε

β L|

hdπL ν|

 

(5.1) The fundamental equation(4.4):ı_|XiωL=hdH|becomes:

Xαπ′_κεκαβ−Xβ=Hβ,Xν+Xµτλελµν=Hν

whereH_β= (∂. H/∂gα)Lα_β(g,e),Hν=. ∂H/∂πL ν

. The ma-trices(4.6)are given explicitely byΦαν=. C1δαν+ταπ′νand Ψµ

β =. C1δµβ+π′µτβ, where C1= (. 1−π′·τ). They obey

Φαν

δνβ−τνπ′β

=C1δαβandΨµβ δβν−π′βτν=C1δµν. It follows that(4.5)implies:

Xα(1−π′·τ) = Hα−π′α(τβHβ)−εαµνHµτν (5.2) Xµ(1−π′·τ) = −Hµ+τµ(π′νHν)−εµαβHαπ′_β (5.3)

5.1. The non degenerate case

The determinant of the matricesΦandΨis given by∆= (C1)2. Obviously the planeΠ1=. {(g,πL)|(1−π′·τ) =0}is an algebraic manifold in

G

⋆_{. Its complement}

_V

∆=.

G

⋆\Π1 defines a manifold

M

′

0

.

=G×

V

∆with symplectic structureωL, retricted to

M

′

0. On

M

0′,ΦandΨhave inverses:

(Ψ−1₎β

ν= (C1)−1

δβν−π′βτν

,

(Φ−1)ν β

= (C1)−1

δνβ−τνπ′β

(5.4) For a HamiltonianH=K(πL_{) +V(g), the Hamiltonian vector} fields are read off from(5.2)and(5.3)with ensuing equations of motion:

Ωα_L =. Lα_β(g−1,g)dg β

dt =

_∂K

∂πL ν

+hdV|eL_µiτλελµν

(Φ−1₎ να

dπL µ dt =

−hdV|eL_βi+ ∂K ∂πL α

π′κεκαβ

(Ψ−1)β_µ (5.5)

For a purely kinetic Hamiltonian, we obtain:

Ωα_L= ∂K ∂πL µ

(Φ−1₎ µ

α , dπ

L µ dt =Ω

α

Lπ′βεβαµ (5.6)

5.2. The degenerate case

The equation C1≡(1−π′·τ) =0 defines a two dimen-sional plane Π1 in su(2)⋆∼=R3. The primary constrained manifold, defined by

M

1=. SU(2)×Π1, is imbedded in

M

0=. SU(2)×su(2)⋆_{. On}

_M

_{1, the closed two-formω}

Lis degenerate and the pairing ofπ′_∈su(2)⋆_{withτ∈su(2)equals 1. So|τi 6=} 0 and, without loss of generality, we take {τα_{} ={0,0,τ}.} In what follows, greek indices {α,β,µ,ν,· · · } shall vary in {1,2,3}, while latin indices {a,b,m,n,· · · } assume only the values{1,2}. The imbedding is given by:

a j1 :

M

1֒→

M

0:

x1≡(gα,πmL)→x0= j1(x1)≡(gα,πLm,πL3=1/τ−ξ3) (5.7) with its differential or push-forward:

j1⋆:T

M

1→T

M

0:(x1;Xα,Xm)→(x0;Xα,Xm,X3=0) (5.8) The pull-back transforms forms on

M

0into forms on

M

1:

j1⋆:^•

(T⋆

M

0)→

^•

(T⋆

M

1) (5.9)

In particular the pull-back ofωLto the five dimensional mani-fold

M

1is

e

ωL|1=. j1⋆(ωL) (5.10)

The restriction ofωL to

M

1, not to be confused with its pull-back, is denoted byωL|1=. ωL◦j1. In matrix representation:

ωL|1 = 1 2 hε

α

L| hdπLµ|

∧       

0 1/τ −π′

2 1 0 0 −1/τ 0 π′

1 0 1 0

π′

2 −π′1 0 0 0 1 −1 0 0 0 τ 0

0 −1 0 −τ 0 0

0 0 −1 0 0 0

      

       

hεβ_L|

hdπL ν|

       

(5.11)

Let(T

M

0)|1

.

={(x,X)∈T

M

0|x∈

M

1}be the subbundle of T

M

0 restricted to

M

1. Following the GNH algorithm [11], we look for a vector field|Xiin(T

M

0)|1, tangent to

M

1and solution ofı|XiωL|1=hdH| ◦j1.

Explicitely :

−(1/τ)X2+π′2X3−X1 = hdV|eL1i

X1−τX2 = ∂K/∂πL1 X2+τX1 = ∂K/∂πL2 X3 = ∂K/∂πL 3

Two independent null vectors ofωL|1, solution ofı|ZiωL|1= 0, are given by:

|Z1i = |eL₁i+ (1/τ)|∂/∂πL

2i −π′2|∂/∂πL3i

|Z2i = |eL₂i −(1/τ)|∂/∂πL1i+π′1|∂/∂πL3i (5.12) Consistency requires{hdH|Zai=0}for(a=1,2)andπ′

3= 1/τ.

C21≡π′2(∂K/∂πL3)−π′3(∂K/∂πL2)− hdV|eL1i = 0 C22≡π′₃(∂K/∂πL

1)−π′1(∂K/∂πL3)− hdV|eL2i = 0(5.13) These two equations define a secondary constrained manifold

M

2⊂

M

1, on which a particular solution of(??)is

|XPi=|eL1i∂K/∂πL1+|eL2i∂K/∂πL2+|eL3i∂K/∂πL3+|∂/∂πL3iC23 (5.14) whereC23≡π′1(∂K/∂πL2)−π2′(∂K/∂πL1)− hdV|eL3i. The gen-eral solution|XGiof(??), on

M

2, still contains two arbitrary functionsζ1andζ2:

(XG) =ζ1       

1 0 0 0 1/τ

−π′ 2

      

+ζ2       

0 1 0 −1/τ

0

+π′ 1

      

+

      

∂K/∂πL 1

∂K/∂πL 2

∂K/∂πL 3 0 0 C23

      

(5.15) This vector must be tangent to

M

1and

M

2. This leads to three equations

hdC1|X_Gi=0 ;hdC21|X_Gi=0 ;hdC22|X_Gi=0 (5.16)

If these three equations determine or not the two arbitrary func-tionsζ1andζ2, will depend on the kinetic energyK(πL)and on the particular form of the potentialV(g). If they do so, the system will have a solution. If not, they will define a tertiary constraint manifold

M

3and the analysis must proceed.

6. CONCLUSIONS

In this work, we analysed the consistency of a modification of the symplectic two-form on the cotangent bundle of a group manifold. This was done in order to obtain classical, i.e. Pois-son, noncommuting configuration (group) coordinates. This was achieved in the non degenerate case, with the closed two-formωLwhich is then symplectic. We do not address here the general quantization problem of such a system and refer e.g. to [12] for a general review on quantization methods. It should be stressed that, whatever the quantisation scheme, any such obtained framework has little to do withnon commutative ge-ometry, either in the sense of A.Connes or as a quantum field theory on non-commutative spaces.

APPENDIX A: THE SYMPLECTIC ONE-COCYCLE

A one-cochain θ on

G

with values in

G

⋆_{, on which}

_G

acts with the coadjoint representationk,θ∈C1(

G

,

G

⋆,k), is a linear map θ:

G

→

G

⋆_{:u →θ(u). Its components are}

θα,µ=. hθ(eµ)|eαi. It is a one-cocycle, θ∈Z1(

G

,

G

⋆,k), if its coboundary,(δ1θ)(u,v)=. k(u)θ(v)−k(v)θ(u)−θ([u,v]), vanishes.

h(δ1θ)(u,v)|wi =. − hθ(v)|[u,w]i+hθ(u)|[v,w]i − hθ([u,v])|wi=0 h(δ1θ)(eµ,eν)|eαi =. −θκ,νfκµα+θκ,µfκνα−θκ,αfκµν=0

The one-cocycleσis called symplectic ifΣ(u,v)=. hσ(u)|viis antisymmetric, Σ(u,v) = −Σ(v,u)orΣ[αµ]=. σα,µ=−σµ,α. Any antisymmetricΘdefined in terms ofθ∈C1(

G

,

G

⋆,k)as

Θ[αβ]=θα,βis actually a 2-cochain on

G

with values inRand trivial representation:Θ∈C2(

G

,R,0). Furthermore, whenθ∈ Z1₍

_G

_,

_G

⋆_{,k),Θis a 2-cocycle ofZ}2₍

_G

_,R,0):

(δ2Θ)(u,v,w)=.−Θ([u,v],w)+Θ([u,w],v)−Θ([v,w],u) =0

(δ2Θ)(eα,eβ,eγ)=. −Θκγfκαβ+Θκβfκαγ−Θκαfκβγ=0 (A.1) In general letΘ=1₂Θαβεα∧εβ∈Λ2(

G

⋆), obey the cocycle condition (A.1). Acting withL⋆_g−1_|gyields the left-invariant

two form:

ΘL(g)=. L⋆g−1|gΘ= 1 2Θαβε

α L(g)∧ε

β

L(g) (A.2)

Using the cocycle relation and the Maurer-Cartan structure equations, it is seen thatΘL(g)is a closed left-invariant two-form onG.

When

G

is semisimple, Θ is exact. Indeed, the Whitehead lemmas state thatH1₍

_G

_{,R,0) =0 andH}2₍

_G

_{,R,0) =0. In} particular, Θ ∈B2(

G

,R,0) is a coboundary and there ex-ists an element ξ of C1(

G

,R,0)≡

G

⋆ _{such that Θ(u,v) =}

(δ1(ξ))(u,v) =−ξ([u,v])or

Θαβ=−ξµfµαβ (A.3)

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