ESI The Erwin Schrodinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, Austria

Remarks on Virasoro and Kac–Moody algebras

J. Grabowski

G. Marmo

A. Perelomov

A. Simoni

Vienna, Preprint ESI 241 (1995) July 27, 1995

Supported by Federal Ministry of Science and Research, AustriaAvailable via anonymous ftp or gopher from FTP.ESI.AC.ATor via WWW, URL: http://www.esi.ac.at

Remarks on Virasoro and Kac-Moodyalgebras

J.Grabowski1 G.Marmo2 A.Perelomov3 A.Simoni2

Erwin Schrodinger International Institute for Mathematical PhysicsPasteurgasse 6/7, A-1090 Wien, Austria

1 Generalities

The algebra of vector fields on a circle plays an important role in connectionwith bosonic string. A central extension of it gives the celebrated Virasoroalgebra. It also provides us with a good example of infinite dimensionalLie algebras which is manageable and for which representation theory iswell understood [GO 1986], [GF 1968], [Wi 1988]. From this point of view,having different realizations of this abstract algebra may be quite useful forapplications, for instance to look for extensions of loop algebras [PS 1986],[Mi 1989] which form another important class of infinite dimensional Liealgebras.

1.1 Virasoro algebra

The Virasoro algebra is a central extension of the Witt algebra W, which isdefined in a basis {Xn : n ∈ Z} by the commutation rules

[Xn, Xm] = (m− n)Xn+m.

The standard realization in vector fields is given by the (complex) vectorfields on the circle defined by

Xn = −ieint ∂∂t.

1Permanent address: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland.

2Permanent address: Dipartimento di Scienze Fisiche e Sezione INFN, Mostrad’Oltremare pad.19, 80125 Napoli, Italy.

3Permanent address: Institute of Theoretical and Experimental Physics, 117259Moscow, Russia.

1

In other words, the Witt algebra W is spanned by the functions Fn(t) =−ieint with the Lie bracket [F,G] = FG′−F ′G. A realization in vector fieldson the real line may be also written in the real form

Xn = enx∂

∂x.

The Virasoro algebra is the central extension of W given by the cocycle

ω(Xn, Xm) =c

12n(n2 − 1)δn,−m,

i.e. the commutation rules are

[Xn, Xm] = (m− n)Xn+m +c

12n(n2 − 1)C,

with C being the central element. The above cocycle is cohomologicallyequivalent to the cocycle

ω0(Xn, Xm) = n3δn,−m

which, in the realization by functions Fn = −ieint on the circle, can be writtenas

ω0(Fn, Fm) =i

2π

∫ 2π

0

F ′n(t)F ′′m(t)dt.

Integrating by parts, we can also write it in a visible skew-symmetric form

ω0(F,G) =i

4π

∫ 2π

0

(F ′G′′ − F ′′G′)dt.

By looking at previous formulae it is immediate to notice that a general Wittalgebra may be written in the form

[Xa, Xb] = (b− a)Xa+b,

with a, b being arbitrary real numbers, so that it is graded by reals. Havingnow any subgroup Γ of reals, we can consider the subalgebra W (Γ) spannedby elements indexed by Γ. In particular, W (Z) is the standard Witt algebra.Also the cocycle ω0(Xa, Xb) = a3δa,−b makes sense for all algebrasW (Γ). Suchalgebras are isomorphic if and only if Γ′ = cΓ for a constant c, and we will

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have an infinite family of non-isomorphic Virasoro algebras. In particular,choosing a1, ..., an ∈ R not rationally related (linearly independent over Q)and generating the group Γ by them, we obtain a Virasoro algebra with theWitt part realized in vector fields on an n-dimensional torus as we will showin Remark 3. We can easily provide a realization of W (R) in terms of vectorfields. Defining namely fa = eiax and

Xa = −ifa∂

∂x,

we find at once [Xa, Xb] = (b − a)Xa+b. As for the cocycle we can take theone geometrically expressed by

(∗) ω(Xa, Xb) = limt→∞

i

2t

∫ t

0

(f ′a(x)f ′′b (x)− f ′′a (x)f ′b(x))dx

which gives us ω(Xa, Xb) = a3δa,−b. If we are not on the line but on an openinterval, say, (0, T ), we can write the cocycle in the form

(∗∗) ω(Xa, Xb) = limk→∞

i

2Tk3

∫ T

0

(f ′ka(x)f ′′kb(x)− f ′kb(x)f ′′ka(x))dx.

In the previous construction we can replace the one dimensional manifoldswith any manifold M. Let us denote by X0 a vector field on a manifold M .It is easy to see that the solutions γ of the eigenvalue problem on F(M,C),

LX0 f = i c λf,

form a subgroup Γ ⊂ R. Here c is a quantity which is independent of λ andmay be a nowhere vanishing function of the constants of motion, so after aredefinition of X0 we may consider c = 1. We denote by f kλ the eigenfunctionscorresponding to the eigenvalue λ. We define Xk

m = −i fkmX0 and get

[Xkn, X

rm] = (m− n) X

s(k,r)m+n .

At the moment we do not bother on how (s) is related with (k) and (r). Theset of vector fields associated with the eigenfunctions of a given eigenvalueconstitute an abelian algebra. The ratio of any two eigenfunctions belongingto the same eigenvalue is a constant of the motion for X0.

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It is clear that we have obtained a generalization of the Witt algebrabecause now eigenvalues can have infinite degeneracy. However, if it makessense to consider powers f a with a ∈ R or belonging to some other sub-group of R, we can extract a subalgebra of the kind considered on the onedimensional manifolds.

Let us take an orbit γ for the vector field X0. Now we can define a cocyclefor W which allows for the construction of a central extension, i.e. a Virasoroalgebra.

First of all, let us assume that for a subgroup Γ ⊂ R we have eigenfunc-tions {fa : a ∈ Γ}, LX0(fa) = iafa, with fafb = fa+b and that the timeparametrizing the orbit runs for the infinite future (the same we can do forthe infinite past):

t : (0,+∞) −→ γ.

Then we can proceed as in the case of the real line, i.e. to define the basis ofthe Witt algebra by Xa = −ifaX0 and the cocycle by

ω(Xa, Xb) = lims→+∞

i

2s

∫ s

0

(f ′a(t)f′′b (t)− f ′′a (t)f ′b(t))dt.

Indeed, since on the orbit with the time parametrization X0 = ∂∂t, we have

fa(t) = caeiat with cacb = ca+b and

ω(Xa, Xb) = lims→+∞

ca+b

2s(a− b)ab

∫ s

0

ei(a+b)tdt = c0a3δa,−b.

Of course, if the orbit γ is periodic with the period T, the above reduces to

ω(Xa, Xb) =i

T

∫ T

0

f ′a(t)f′′b (t)dt = i

∫

γ

LX0(fa)L2X0

(fb)θ,

with θ being the normalized X0-invariant 1-form on γ, which is the classicalform of the Virasoro cocycle.

If the time parametrizing the orbit runs only over an interval (vector fieldX0 is not complete), we can use the formula similar to (∗∗).

Our construction associates a realization of Virasoro algebra for any partof an orbit of X0 on M . These various realizations are parametrized by thethe constants of the motion for X0.

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Remark 1. We notice that we might have started with any vector fieldX on M and construct a Lie algebra structure on F(M) by setting [f, g] =fLXg − gLXf . This is known as a Jacobi structure associated with X. Oneach closed orbit γ of X we consider the Gelfand-Fuks cocycle

ω(f, g) = λ

∫

γ

((LXf) d(LXg)− (LXg) d(LXf))θ,

for θ being the canonical 1-form on γ, providing us with a central extensiondepending on γ .

Note that in many cases the manifold M is a complex one and the vectorfield X0 is a meromorphic vector field, i.e it has the form

X0 = aj(z)∂

∂zj,

where zj are local complex coordinates on M and functions aj(z) are mero-morphic functions.

The simplest example is the case of Riemann sphere, which may be consid-ered as stereographic projection of standard sphere S2 in three-dimensionalspace. In this case we have

X0 = zd

dz

and

Xm = zm+1 d

dz= zmX0 .

The case of n-dimensional torus will be considered in Remark 3 below. Ourconstruction is valid also for Riemann surfaces of higher genus, but it isdifferent from construction of Krichever and Novikov [KN 1987ab].

We can give now a simple example of a family of realizations of Virasoroalgebras on R2 − {0}.On R2 − {0} we consider

X0 = −ih(r)∂

∂ϕ, h(r) 6= 0 , r ∈ R+ .

The eigenvalue problemLX0f = c λ f

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has solutions for λ = m, c = h(r), m ∈ Z, with fm = α(r) eimϕ, α(r) isany function of r, thus the eigenvalue has infinite degeneracy. We have

[fm, fn] = α(r) β(r) ei(m+n)ϕ h(r) (n−m).

Let us consider the Gelfand-Fuks cocycle. We have

ω(fm, fn) = µ

∫

r=r0

((LX0fm) d(LX0fn)− (LX0fn) d(LX0fm))θ

= i µ h2(r0)α(r0) β(r0)m3 2π δm+n,0. (1)

Notice that l(r) is in the kernel of ω for any function l(r).To extract a Virasoro algebra, we redefine X0 so that c = 1 , and selectf1 = α(r)eiϕ, fm = (f1)m to get [fm, fn] = (n − m)fm+n. The use of theGelfand-Fucks cocycle provides us with Virasoro algebra.

Remark 2. As for the Witt algebra, we notice that on R2 − {0} we canconsider a different choice for X0, for instance

X0 = x∂

∂x+ y

∂

∂y.

Solutions forLX0f = λf, λ ∈ Z,

are provided by homogeneous monomials, thus every monomial f k,m−km =xkym−k solves

LX0f = mf

and we can write the commutator as

[fk,m−km , f r,n−rn ] = (n−m)fk+r,m+n−(k+r)m+n .

As eigenfunctions fa, with LX0(fa) = iafa, we may take for instance

fa = ((xy)2)ia4 .

Then, we get the Witt algebra W (R) putting Xa = −ifaX0. This time X0

does not have periodic orbits, therefore taking any trajectory, for instancex = y > 0 , we find that

ω(Xa, Xb) = limt→+∞

i

2 ln t

∫ t

1

[LX0(xia), LX0(xib)]dx

x.

6

gives us a cocycle.

Remark 3. We can apply the construction to the case of the irrational flowon the n-dimensional torus, now

X0 =∑

aj∂

∂xj0 ≤ xi ≤ 2π.

The eigenfunctions of this vector field are

Y (l1,...,ln) = ei l1x1 ...ei lnxn

with eigenvalues i∑ajlj ; they are dense on R. We can use the grading given

by the dense subgroup Γ = {a ∈ R : a = ajlj , lj ∈ Z} and denote Y (l1,...,ln)

by Ya getting a general Witt algebra W (Γ). In this case X0 has no nontrivialconstants of the motion, therefore eigenvalues are not degenerate.

The Lie algebra structure is given by

[F,G] = F LX0G−GLX0F .

In the present case we obtain

[Ya, Yb] = (b− a)Ya+b, a, b ∈ Γ

and the extensionω(Ya, Yb) = a3δa,−b

is obtained via the formula (∗) on a dense orbit. It is easy to see that this isa cocycle which can be also written as

ω(F,G) =

∫

T

[LX0F,LX0G]µ,

where µ = dx1 ∧ ... ∧ dxn is an invariant measure on the torus: LX0µ = 0.Up to now we have realized the Witt algebra as a subalgebra of vector

fields on M. One may try to realize the Virasoro algebra in terms of vectorfields on M. We shall show that the Virasoro algebra cannot be realized invector fields on a finite-dimensional manifold.

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Theorem 1 If L is a central extension of a Lie algebra with respect to acocycle ω such that all cocycles from the class of ω have infinite rank, thenL cannot be realized neither as a Lie algebra of vector fields, nor as a Liealgebra of functions with respect to a Poisson bracket on a finite-dimensionalmanifold.

Proof. Let C be the central element and let L0 be a complementary sub-space to < C > – the linear spann of C, i.e.

L = L0⊕ < C >,

so that L0 ' L/ < C > can be interpreted as the Lie algebra which isactually extended. Realizing L in vector fields, we can find a chart and localcoordinates (x0, ..., xn) such that C = ∂

∂x0and for any X ∈ L, X = f iX

∂∂xi,

and we may choose L0 to be actually

L0 = {X ∈ L : f 0X(0) = 0}.

Then, clearly, ω0(X,Y ) = f 0[X,Y ](0), X, Y ∈ L0, is a cocycle in the class of ω.

Since C = ∂∂x0

is central, f iX do not depend on x0 and

f 0[X,Y ] = X(f 0

Y )− Y (f 0X).

Therefore, for L1 = {X ∈ L0 : X(0) = 0} being the subalgebra of L0 ofvector fields vanishing at 0, we have ω0(X,Y ) = 0 for X,Y ∈ L1. Since L1 isat most n-codimensional in L0, the rank of ω0 is at most n; a contradiction.For the case of a Poisson bracket we can use a similar argument. We can

choose a chart and local coordinates (x1, ..., xn) such that C(0) = 1 andL0 = {F ∈ L : F (0) = 0}. Then ω0(F,G) = {F,G}(0) is the cocycle andL1 = {F ∈ L0 : dF (0) = 0} is a subalgebra of L0 of codimension at most nsuch that

ω(F,G) = 0

for F,G ∈ L1. 2

Corollary 1 The Virasoro algebra has neither a realization in terms of vec-tor fields, nor in terms of a Poisson bracket on a finite-dimensional manifold.

8

Proof. It is sufficient to show that the cocycle in the class of Virasoroare of infinite rank. Assume that ω(Xn, Xm) = n3δn,−m − φ([Xn, Xm]), forsome functional φ ∈ W ∗, is of finite rank, i.e. the kernel L of ω is a finitecodimensional subalgebra in W. Realizing the Witt algebra with vector fieldson the circle and using the general algebraic result of [Gr 1993, Theorem4] describing finite codimensional subalgebras in derivation algebras of as-sociative commutative rings, we get that L contains a finite codimensionalsubmodule, which means in terms of our trigonometric polynomials algebrathat we have an associative ideal I in L of finite codimension. Hence

i

2π

∫ 2π

0

(F (t)eint)′(G(t)e−int)′′dt = φ([F (t)eint, G(t)e−int])

for all n ∈ Z and F,G ∈ I. It is easy to see that on both sides we getpolynomials with respect to n. Since in the left hand side we have

n3

∫ 2π

0

F (t)G(t)dt+ ...

and we have no term with n3 on the right, it must be

∫ 2π

0

F (t)G(t)dt = 0

for all F,G ∈ I, which is impossible. 2

1.2 Kac-Moody algebras

Affine Kac-Moody algebras are extensions of current (loop) algebras. Weassume now that on M , along with the vector field X0, we also have vectorfields X1, X2, . . . , Xn closing a Lie algebra g, i.e.,

[Xi, Xj ] = Cki jXk

andLX0fm = imfm.

We further require [X0, Xj ] = 0 , ∀j ∈ {1, . . . , n} and LXjfm = 0 for anyj ∈ {1, . . . , n} and for any eigenfunction. We use these eigenfunctions to

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define: Y ma = fmXa, Y0 = −iX0. If for each eigenvalue there is only one

eigenfunction satisfying previous requirements, we get

[Y0, Yma ] = mY m

a ,

[Y ma , Y

nb ] = Y m+n

k Cka b .

If X0 has a closed orbit γ, we can define a central extension by setting

ω(Y ma , Y

nb ) =

k

2

∫

γ

(< Y ma |[X0, Y

nb ] > − < Y n

b |[X0, Yma ] >) θ,

where < | > is an invariant symmetric product on the algebra g. Anexample of this situation arises for instance on T ∗SU(2), where we considerX0 to be the canonical lift of a left invariant generator, while X1, X2, X3 aregenerators of canonical lifted left action. Eigenfunctions for X0, invariantunder left action, are to be found among left invariant momenta, and anappropriate choice is to be made. If X0 has no closed orbits, we may useformulae similar to those of (∗) and (∗∗).

A typical situation arises from principal bundles on S1. We consider aG−bundle P with base space S1. The vector field X0 is any projectablevector field which commutes with the G−action on P . Eigenfunctions ofπ∗X0 can be pulled back to P to construct vector fields Y m

l .We give an example by considering again the carrier space R2−{0}. We usecoordinates (r, ϕ). The following vector fields

X0 = −i ∂∂ϕ

, Y−1 =∂

∂r, Y0 = r

∂

∂r, Y1 = r2 ∂

∂r,

with eigenfunctions fm = eimϕ, satisfy all previous conditions. An extension

of sl(2,R) is realized by the vector fields

X0 = −i ∂∂ϕ

, Y m−1 = eimϕ

∂

∂r, Y m

0 = eimϕr∂

∂r, Y m

1 = eimϕr2 ∂

∂r.

We note that from any realization of current algebras in terms of vectorfields on a manifold M we can construct a realization in terms of functionswith the Lie algebra structure given by a Poisson bracket. We lift the vectorfields to T ∗M , equipped with the canonical symplectic structure, and obtainin this way an Hamiltonian realization of the given algebra.

10

Another realization of affine Lie algebras in terms of Poisson brackets,which is not the cotangent lift of an action on the base manifold, is given inthe following example. On F(M×g∗) we define the following Poisson bracket

Λ = Pk Cki j

∂

∂Pi∧ ∂

∂Pj+ ΛM ,

where ΛM is any Poisson bracket on M and Pi are a basis of linear functionson g∗. With H0 ∈ F(M) and f ∈ F(M) such that

{H0, f} = f,

we define Fma = fmPa and we get

{H0, Fma } = mFm

a ,

{Fma , F

nb } = Cc

a bFm+nc .

A central extension can be constructed if there is an invariant symmetricproduct in g ( and therefore in g∗), we can set

ω0(F,G) =k

2

∫

γ

(< F |LX0G > − < G|LX0F >)θ ,

where F,G are assumed to be linear functions on g∗. Note that as in the caseof the Virasoro algebra these central extensions cannot be realized in termsof vector fields

Theorem 2 Affine Kac-Moody algebras cannot be realized neither in termsof vector fields nor in terms of Poisson brackets on finite-dimensional man-ifolds.

Proof. It suffices to show that the cocycles in the class of ω0 are of infiniterank. But the cocycle

ω(Fma , F

nb ) = mδm,−n < Fa, Fb > +φ([Fa, Fb]

m+n),

with a given functional φ, is of infinite rank even when reduced to Fa, Fb froma Cartan subalgebra of g. 2

11

1.3 A Virasoro type algebra from a Kac-Moody alge-bra

Consider now the affine A11 algebra being an extension of the loop sl(2,R)

algebra. Choosing for sl(2,R) a basis X i, i = 0,±1, with

[X i, Xj ] = [j − i]3X [i+j]3 ,

where [n]3 = 0,±1 if n = 0,±1(mod 3), we get the basis of the loop algebrabeing {X i

n : i = 0,±1, n ∈ Z} with the commutation rules

[X in, X

jm] = [j − i]3X [i+j]3

n+m .

Changing the gradation to integers by putting

Y3n+i = X in,

we get the commutation rules similar to those of the Witt algebra:

[Yn, Ym] = [m− n]3Yn+m.

It is in fact the only, except for the Witt, Z-graded algebra [Yn, Ym] =f(n,m)Yn+m with the structure constants being translation invariant, i.e.satisfying f(n,m) = g(m− n). A version of the Virasoro cocycle

ω0(Yn, Ym) = [n3]3δn,−m

is in the case of our Witt3 algebra trivial, since [n3]3 = [n]3.We have, however,the nontrivial cocycle obtained from the loop algebra:

ω(X in, X

jm) = nδn,−mδi,−j,

which gives usω(Yn, Ym) = E(n/3)δn,−m,

where E(x) is the integer closest to x (there is no ambiguity for x of the formn/3).In the following sections we are going to consider a family of realizationsof the infinite dimensional algebras, we have discussed, in terms of someinteresting dynamical systems: the harmonic oscillator and the Euler top.

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2 A realization in terms of the harmonic os-

cillator

1. We give now a family of realizations of the Virasoro algebra associatedwith a two-dimensional isotropic harmonic oscillator.

Here M = C2 = R4, we introduce coordinates z1, z2, z1, z2 to get

X0 = i(zk

∂

∂zk− zk

∂

∂zk

).

Obvious eigenfunctions for X0 are provided by

f r,m−rm = zr1 zm−r2 ,

f r,m−r−m = zr1 zm−r2 , (2)

which along withzr1 z

p2 z

q1 z

s2,

r + p− q − s = m, belongs to the eigenvalue m.By using

Y0 =i

2

(zk

∂

∂zk− zk

∂

∂zk

)

and eigenfunctions f1 = z1z2 , fm = (z1z2)m , m ∈ Z, we get Ym = fm Y0 and[Ym, Yn] = (n−m)Ym+n.

We can compute the Gelfand-Fuks cocycle

ω(Ym, Yn) = c(γ)

∫

γ

((LY0fm) d(LY0fn)− (LY0fn) d(LY0fm))

= c(γ)

∫

γ

mnfm fn(n−m)θ, (3)

where θ = −id (z1z2)z1z2

.We find

ω(Ym, Yn) = ic(γ)

∫

γ

mn(n−m)(z1z2)m(z1z2)nd z1z2

z1z2

which is zero , as γ is a closed curve , unless m+n = 0. Therefore ω(Ym, Yn) =ic(γ)4πm3δm+n,0. By making other appropriate choices of eigenfunctions weget different realizations.

13

2. We can also give a realization of the Witt algebra in terms of Poissonbrackets on Cn. As an example we use the one dimensional harmonic oscilla-tor. From here it will be easier to go to creation and annihilation operatorsand provide a realization in the spirit of the Jordan-Schwinger map.

We consider H0 = −izz , {z, z} = i. We notice that a possible choice ofeigenfunctions is provided by fm = zm , m ∈ Z :

{H0, fm} = mfm.

A realization of Witt algebra obtains by using

tm = fmH0

and we find[tm, tn] = i {tm, tn} = (m− n)tm+n.

An extension is obtained by using

ω(tm, tn) =

∫

γ

({H0, tm}d{H0, t

n} − {H0, tn}d{H0, t

m}) .

In the next section we shall consider X0 to represent the dynamics of theEuler top. We will find a realization of the Witt algebra in terms of ellipticfunctions. We will also compute the Gelfand-Fuks cocycle for a particularorbit.

3 Realization in terms of the Euler top

1. Let us begin with consideration of motion on intersection C of twoquadrics Q1 and Q2 in R3

Q1 ≡ x21 + x2

2 + x23 − 1 = 0,

Q2 ≡ x21

α1+

x22

α2− 1 = 0. (4)

As it is well known, the genus of C is equal one and, hence, this curve maybe parametrized by elliptic functions

x1 =√α1 cn(u, k), x2 =

√α2 sn(u, k), x3 =

√α3 dn(u, k).

14

Here cn(k, u) , sn(u, k) and dn(u, k) are standard elliptic functions (see forexample [BE 1955], [WW 1962] ) and

α1 = 1− α3, α2 = 1− (1− k2)α3.

Let us take the standard metric in R3 :

ds2 = dx21 + dx2

2 + dx23.

Then it is easy to calculate induced metric on C :

ds2 = g(u) du2,

g(u) = A− k2sn2u, A = 1− (1− k2)α3.

So the function g(u) is meromorphic double periodic function second orderwith periods 2K and 2iK ′ and g(u) has two zeros, say u = a and u = −a,and one pole of second order at the point u = iK ′ in the fundamental domain

0 ≤ <u < 2K, 0 ≤ =u < 2K ′.

The position of zero u = a depends on constant A and when A changes fromk2 to 1, a changes from K to K + iK ′,a = K + iα. The quantity α may befound from transcendental equation

dn(α, k′) = k A−12 , k′ =

√1− k2,

which follows from relation

sn(K + ix, k) = [dn(x, k′)]−1.

Now we may factorize the function g(u)

g(u) = Bf(u)f(−u),

where

f(u) =σ(u− a)

σ(u+ iK)e(ζ(a)+ζ(iK))u, ζ(u) =

σ′(u)

σ(u)

is meromorphic one-periodic function

f(u+ 2K) = f(u)

15

with one zero u = a and one pole u = iK ′ in fundamental domain. Hereσ(u) is standard Weierstrass function which entire function with zeros at thepoints of the lattice um,n = 2mω + 2nω′.We may define now a vector field on C :

X0 =1

f(u)

d

du.

Let us find eigenfunctions of this vector field

X0Ψλ = λΨλ.

We havedΨλ

Ψλ

= λf(u) du,

so

Ψλ = CeλF (u), F (u) =

∫f(u)du.

But f(u) is periodic function of the variable u, so

f(u) = f0 + f(u), f0 = f =1

2ω

∫ 2ω

0

f(u)du, ¯f = 0

and Ψλ(u) should be periodic in u. ¿From this we obtain

λf0 2ω = i 2πm , λm =iπ

f0ωm,

and, after a redefinition of the vector field so that f0 = πω, we have

Ψλm = CeimF (u) = C(Ψ1(u))m = CΨm(u) , λm = m

Ψ1(u) = eiF (u).

So now we are ready to define the algebra of vector fields on the circle

Xm = Ψm(u)X0 = Ψm(u)1

f(u)

d

du.

Let us calculate the commutator

[Xm, Xn] = [Ψm(u)X0,Ψn(u)X0] = (n−m)Xm+n,

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i.e. it is a realization of the standard algebra of vector fields on the circle.But the concrete realization of this algebra is related to the function

F (u) =

∫f(v) dv =

∫σ(v − a)

σ(v)ebvdv, a = ω + iα, 0 < α ≤ iω′,

so depends essentially on parameters α. The integral under considerationmay be calculated analytically ,but give sufficiently complicated function, sohere we consider only the symplest case iα = ω′. In this case

f(u) = dn(u, k)

and we have

F (u) =

∫ u

dn(v, k)dv = −i ln(

cn(u) + i sn(u)),

what may be checked directly. Finally we have

Ψm(u) =(

cn(u) + i sn(u))m

and

Xm = −i(

cn(u) + i sn(u))m 1

dn(u)

d

du.

Note that when k → 0 we obtain the standard expression

Xm = −i eimu ddu

.

Let us consider another limiting case

f(u) = cn(u, k) .

Then we have

F (u) =

∫ u

cn(v, k)dv = −i ln(dn(u) + i k sn(u))

and

Xm = −i(dn(u) + i k sn(u))m1

cn(u)

d

du.

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Now:[Xm, Xn] = k(m− n)Xm+n.

It is easy to calculate the cocycle for this case and we have

ωγ(Xm, Xn) = 2n3T (γ)δm+n,0 , T (γ) = 2π.

2. Note that the following three vector fields

X0 = 1dn(u)

ddu,

X1 = 1dn(u)

(cn(u) + i sn(u)) ddu, (5)

X−1 = 1dn(u)

(cn(u)− i sn(u)) ddu,

form the elliptic realization of the SU(1, 1) algebra.Let us give another elliptic realization of this algebra. Let us define

X1 = sn(u)d

du,

X2 = cn(u)d

du, (6)

X3 = dn(u)d

du.

Then we have

[X1, X2] = −X3 ,

[X2, X3] = (1− k2)X1 = k′2X1 , (7)

[X3, X1] = X2 .

Let us introduce

X0 = X3, X+1 = X2 + ik′X1, X−1 = X2 − ik′X1,

then we obtain standard basis in SU(1, 1). The Casimir operator for thiscase has the form

C =1

2(k′

2X2

1 +X22 −X2

3 )

and, if we substitute the explicit expressions (6) for X1, X2, X3, we obtainC ≡ 0.

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Now we can consider a product bundle R3×R3 and double our construc-tion. From the X0 on the first R3 and the given SU(2) algebra realization onthe second R3 we can build up an extension of SU(2)-Kac-Moody as givenin general terms in section 2. More explicitly, we set

Y0 = 1dn(u)

ddu,

Y m1 = (cn(u) + i sn(u))msn(v) d

dv, (8)

Y m2 = (cn(u) + i sn(u))mcn(v) d

dv, (9)

Y m3 = (cn(u) + i sn(u))mdn(v) d

dv,

and we get an extension of the SU(1, 1)-Kac-Moody algebra in terms ofelliptic functions.

It is also possible to use the vector fields X0, X1, X−1 in formula (5) toget a different realization of the same algebra.

4 Conclusions

In this paper we have shown how, by using an appropriate vector field X0 ona generic manifold M, it is possible to select subalgebras of vector fields on Mwhich give rise via central extensions along trajectories of X0 to parametricrealizations of Virasoro or Kac-Moody algebras. We have also discussedtheir realizations in terms of Poisson brackets on T ∗M. We hope that thisapproach may provide a more direct way to the construction of generalizationof Virasoro algebra for Rieman surfaces of higher genus. [KN 1987].

We have also shown that the centrally extended algebras cannot be real-ized as algebras of vector fields on finite-dimensional manifolds.

In addition our realization in terms of oscillators opens the way to theconstruction of q-oscillator deformation of these algebras, different from theone by [RS 1990]. We shall come back to some of these issues in future papers.

AcknowledgmentsWe thank Loriano Bonora and Peter Michor for reading the manuscript andmaking very useful suggestions.We wish to thank A. A. Kirillov for a nice afternoon discussion on infinitedimensional Lie algebras.

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One of the authors (A.P.) would like to thank INFN and University of Napolifor warm hospitality.

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