' 2. PB- and PK-algebras Let EM be a m-dimensional differentiable manifold with local coordinates x=

POISSON LIE ALGEBRA STRUCTURES ASSOCIATED WITH THREE-DIMENSIONAL

LIE ALGEBRAS

By

Fumitake MIMuRA and Akira IKusHIMA

(Received November 25, 1992)

1. Introduction

On a m-dimensional differentiable manifold with a second rank skew-symmetric differentiable tensor field co, F. A. Berezin [1, 2] introduced the Poisson Lie algebras (Poisson-Berezin algebras, briefly PB-algebras) provided with the infinite dimen- sional Lie algebra structure; and illustrated an interesting example of PB-algebras associated with the structure constants of arbitray m-dimensional Lie algebras g.

Moreover, A. A. Kiri11ov [3] investigated the local Lie algebra with one-dimensional fibre, which gave birth to the generalized PB-algebras (Poisson--Kirillov algebras, briefly PK-algebras) defined by the pair of the structure constants of g and a set of differentiable functions F`(i --- 1, •••, m) on the manifold.

In this paper, the centers of PB-algebras are completely determined by setting g as the three-dimensional Lie algebras. By using of the elements of the center, it is effectively determined the set of functions Fi which, together with the structure con- stants of g, give rise to the bracket of PK-algebras (cf. [5] for the case that Fi are the constants). Moreover, there is given a procedure of constructing the bracket of PK-algebras from an arbitrary set of functions Fi. And the procedure is illustrated by some examples.

, For the convenience, differentiability is assumed to be of sufficiently high order and the summation convention is employed throughout.

' 2. PB- and PK-algebras

Let EM be a m-dimensional differentiable manifold with local coordinates x=

(ti,•••, rM) and ER(EIJI) the ring (with respect to the operations of addition and multiplication) of differentiable functions on EIn. Then, in terrns of the components toi"(x) of the tensor field tu on E!n, the generalized Poisson bracket on M(E"l) is defined by [2]

(1) [f, g] = tu 'if,ig,j for f, gE M(EM).

2 Fumitake MiMuRA and Akira lKusHiMA

Here note the convention of the commas:f,i--- Of/0x', g,j= 0g/Oxj. Under this bracket operation, M(EM) makes a Poisson Lie algebra, i.e., (1) defines an infinite dimensional Lie algebra structure (PB-algebra structure) if and only if (here the parenthesis of indices denotes the summation over all cyclic permutations of the indices)

(2) co (ilsl cotk,) = coisco;k+ cojsco ,k,i+coksco gJg = o.

Let C2 be the structure constants of a m-dimensional Lie algebra g and put

(3) coi'-=C2xk.

Then (2) is satisfied identically, since it is equivalent to the Jacobi identity for the structure constants

(4) cliisicgk) =o;

and the bracket (1) has the form

(5) [f, g] = Ck"x kf,ig,j for f, gE Sn (EIJI).

With a set of differentiable functions Fi on E!n, the Poisson bracket (1) can be generalized as [3]

(6) [f, g] =co`"f,ig,j+Fi(fg,,-gf,i) for f,gE SJI(E!n),

which defines an infinite dimensional Lie algebra structure (PK-algebra structure) if and only if

(7) F(ico]'k)+to(iis[to;,'k)=o, (8) F3, ofk-F;, to ik +Fkto ,v,=O.

Similarly as before, let to"- be given by (3). Then, in viewing that (2) is equiva- lent to the Jacobi identity (4), the equations (7) and (8) are reduced respectively to

(9) F(t C," k'xS=O,

(10) (F ;-,C,]'k-Fk Cgk) xs+.F kCLj =o;

and the bracket (6) has the form

(11) [f, g] = CE'tkf,ig,j+F'(fg,i-gf,i) for f, gE [R(EM).

3. Centers of the PB-algebras

Let (S(E!n) be the PB-algebra with the bracket (5) on ER(EM) and tb(EIn) the

center of (S5(EM). An element fE ij(EM) is a differentiable function on EM such that

CLjjvkf,ig,j = O for all g E YI(SM), i.e., it is a solution of the first-order system of

partial differential equations

(12) Ct"x"f,i=O (1'=1, •••, m),

which may be not always effectively solved to determine the center fo (EI)?).

However our discussion is continued by setting g as the three-dimensional Lie algebras. We set also EM= R3 for the covenience. The structure constants Ckij of g : C,23 C,3i C,i2

(13) P= C,23 C,3i C,i2

C,23 C,3i C,i2

take the following forms under a suitable isomorphism of g (Levy--Nahas [4]):

Pi.. ooo

ooo, ooo 100

'

100

P2= OOO

ooo

1 OO

P5= O1O P6= O -1 O

OOO O OI

,P3=

,P7=

OIO -1 OO

ooo

O1O,P,- -1RO 100 OOI

1 OO ,P4= O -1 O o oo

1 1 0 8-

ooo

'

'

where R # -1 in PR8. Let (SScr(R3) be the PB-algebra associated with the structure constants Pa and foa(R3) the center of ecr(R3), respectively for cr=1, •••,8; and denote the coordinates of R3 by (x', x2-, x3) = (x, y, z) as well as (f,i, f,2, f,3) =

(fX, fU, f2) •

THEoREM 1. Elements f of the centers S)cr(R3) of the PB-algebras (S5a(R3) associated with Pa (cr = 1, •••, 8) are of the follozving forms, respectively (O is an arbitraiy dzfferentiable function of the indicated variables) :

, fo'(R3):f=Åë(x, y, z);

fo2(R3) : f= Åqb (x);

fo3(R3) : f = o(x -'y) (x t O) or f== O (xy-') (y iE O);

'

fo4(R3) : f= O(x2-y2);

ij5(R3) : f= ÅqP (x2+y2);

ij6(R3) : f= O(x2.y2+z2);

ij7(R3) : f= O(x2+y2+z2);

foi (R3) : f == Åë(log (x2+zy2) + ft arctan `Vt]ll2L) (x t o) or

4 Fumitake MiMuRA and Akira lKusHiMA

f== Åë(log (x2+Ry 2) - ft arctan rfy) (yl O) for RÅr O,

f= o( (x+fiy) i+i/V=7 (x-V;Iy) "'/V=X) for RÅq O,

f=Åë(xex-iu) (xio) forR=O.

PRooF. Corresponding to the structure constants of Pa, the differential system (13) is written respectively as

pi: Ci'jxkf,i -= O (1' = 1, 2, 3);

P2: CE'xkf,i-= O, C"2xkf,i"--' -xf,=O, C"'3xkf,i=xfu=:O;

P3: Ct-ix kf,i= xf, = O, C"'2x kf,i -- yfz = O, Cti3x kf,i=-xfx-yfg=O;

P`: CE'xkf,i=-yf,=O, Ct'2xkf,i= ntxfz = O, C"3xkf,i-- xfy+yfx=O;

P5: Cg' 'x kf,i = yf, =: O, C"2x kf,i = -xfz = O, Ct' 3x kf,i ---' xfy -yfx =O;

P6 : C"' 'x kf, i = -yf, -zf, = O, Ct2x kf, , = zfx -xfz = O, C"' 3x kf, : =: xfu +yfx == O;

P' : C"' 'x kf,i = yf, -2f, = O, C"2x kf, i --' zfx -xf2 = O, C2'3x kf,i = xfu-yfx = O;

PR8 : Ck' ix kf, i = (x+Zy) f, == O, Cti 2x "f, i -- ' (x- y) f, = O,

Ct'3jckf,i -- - (x+Ry)f.+ (ar'y)fu = O.

Now, the solution can be obtained resectively in each case (cr) for the structure con- stants of Pa as follows.

(i) For the no conditions,fis obviously an arbitrary function on R3.

(ii) The solution f=Åë(x) follows immediately from fx=fy=O if x;O, while xt O can be delated in the consequence by the continuity of f.

(iii) According to x #O or y;O, the subsidiary equation (briefly s-equation) of the last one: x-idx=y-idy have the solution x-iy= const. or xy-i = const, and thereforef==Åë(x-'y) orf==Åë(xym') by the first or the second one, respectively.

(iv) The first (also the second) equation implies thatfis a function of x and y (note the continuity offas used in (ii)). Thereforef=O(x2-y2) follows from the solution x2-y2 = const. of the s-equation of the last one : xdu-ydy =O.

(v) Similarly to (iv), the solutionf =O(x2+y2) is obtained through the s-equa- tion : xdu+ydy =O.

(vi) By the transformation of variables

u=x2-y2+z2, v =y, w=z,

with the Jacobian (functional determinant) D (u, v, w) /D (x, y, 2) == 2x tO, the equa-

tions are rewritten respectively as

h

.

yf.+zfv=O, xf.=O, xf,=O;

which have the solutionf=O(u) ==Åë(x2'y2+z2) (note also the continuity off).

(vii) Similarly to (vi), the solutionf=O (x2+y2+z2) is obtained by the trans- formation of variables

u == x2+y2+z2, v = y, w =z.

(viii) If a7=O, the s-equation of the last one: ('x+y) djc= (x+Ry) dy is, by putting x-iy =u, rewritten as

2.du +(R.2,Zli + R.22+i) du ==O•

This equation has the solutions, for ZÅrO, in viewing of Ru2+1=R (u2+ (1/V71-)2):

2

log x2 +log (Ru2+1) + A arctan V]l'u == const., i.e., log (x2+Ry2) + ft arctan :V) ÅrillL = const. (x to);

for Jl ÅqO, in viewing of Zu2+1 =R (u2- (1/A) 2) :

log x2 + log IRu2+11 - Å} log :iP l//Vv:=l]i- == const , i.e ,

(x +J=]I-y) i+i/V=Jr (x-V=Jl-y) i-i/V=JI = const. ;

and for R =O, since the equation is reduced to jvmiclx+du =O:

log l x1+u = const. i. e., xe"-'y= const. (x ;o) . If y ;O, also by putting u = xy-i, the sequation is rewritten as !2sgcd +(u,2+UR-u,2+R)du ==O;

which has the similar solutions as above for RÅrO, while the term arctan(V]I-x-'y) is replaced with -arctan(axy-i) in the solution. This can be seen also by using the relation arctant + arctant-' = z/2 ( t År O) or - z/2 ( t Åq O) in the above solution of the sÅíquatlon.

Since R ,E -1, either jc+Ry 7EO or x'y 7EO is satisfied if (x, y) 7E (O, O), so that fz ==O follows from the first or the second equation if (x, y) 4 (O, O). Thus the re-

spective solutionsf =O(x, y) are written by the above solutions of the s-equations.

REMARK. In (Nakanishi [5]), the center, i. e., fo6(R3) in Theorem 1, played an

important role for the study of infinitesimal automorphism of the PB-algebra associ-

ated with P6, i.e., with 6t(2, R).

6 Fumitake MiMuRA and Akira lKusHiMA

4. Poisson Lie brackets of PK-algebras

4. 1. We first assume that the differentiable functions F' i'n the bracket (11) belong to the center to (E!Jl) of the PB-algebra (SS (wa), i.e., (see (12))

Ck`'Ffixk =-CfiFfixk=O (1',s=1,•••,m).

Then, in viewing that (10) is reduced to

the bracket (11) defines the PK-algebra structure if and only if (9) and (14) are

satisfied.

The matrix P of (13) defined by the structure constants Ck"j of the three-dimen- sional Lie algebra g is now rearranged as P = (ptj), in which the elements are related by Ckij=pksei"S where Ei"S are the usual alternators. This relation makes (9) and (14) into more simple forms (cf. [5]). In fact (9) is equivalently rewritten as

(15) F('C,"kÅrxS =xSp,, (Ejk`F`+ek'tFj+E`J'`Fk) =O,

which are satisfied identically if at least two indices in i, 7' , k are the same. So it can be assumed that the indices i, 1' , k are all distinguished. Then, since i, 1' , k,ttake the values 1, 2 or 3, the alternators Ejk', Eki` and Eijt do not vanish if and only if t= i,

t=j and t=k respectively. Therefore the equation (15) is equivalent to xS(p,iF'+p,jFj+p,kFk) = O (not summing up for i, 1', k).

Since the indices i, 1' , k are all distinguished, the above equation is equal to

(16) xSpsiF `= O, i. e., XP 'F =O,

where X= (xi, x2, x3) and F== (Fi, F2, F3) and t denotes the transposition.

Moreover the equation (l4) is equivalently rewritten as Fipi,ejkS = O, in which EjkS it!

O if and only if the indices 1' , k, s are all distinguished. Therefore

(17) F'pi, == O, i. e., FP =O,

from which it follows that F' = O for P=P2; F' = F2= O for jP =P3, •••, P5 and PR8;

F'=F2 =F3=O for P=P6, P7. And for such F', F2 and F3, (16) is satisfied iden- tically. Thus the Poisson Lie brackets of PK-algebras are now obtained as follows

(cf. [5]: the case of Fi=const.; note that the series of Pcr in [5] is rearranged here as Pa).

THEoREM 2. The Poisson Lie brackets [f, g] forf, g E ER (E!n) of the PK-algebras associated with the structure constants of Pa and a set of dzfferentiable functions

(ÅqPa, ilJa, Aa) in the center foa(R3) of the PB-algebras Scr(R3) (cr =1, ''', 8) have the

follozvingforms, respectiwely :

P' : [f, g] =Åqb'(fg.-gf.) +gJ'(fg,-gf,) +A'(fg,-gf.), P2 : [f, g]=x(f, gz-fz gy) +02(fgu-gfu) +rp2(fg.-gf.), P3 : [f, g] ==x(f, g.-f. g,) -y(fy g, -f, gy) + 03 (fg.-gf.), P` : [f, g] =x(fy g,-f, gu) -y(f, g.-f. g,) +O` (fg,-gf.), P5 : [f, g] =x(f. gy-fy g.) +y(fz gx'fx g.) +Åë5 (fg.-gf.), ,

P6 : [f, g] =x(fu gz -fa gg) -y(fz gx-fx g,) +z(f. gu -fu g.), P7 : [f, g] =x(fu gz -f2 gu) +y(fz gx -fx g2) +2(fx gu -fy gx),

B8 : [f, g] = (x-y) (f, g. -fz gy) + (x+Ry) (fz gx-fx gz) +08 (fgz-gfz)r in zvhich the brackets forP6 andP7 are those of the PB-algebras.

4.2. In general, by differentiating (9) by k and summing up for k, it follows that

F,k, Cg'jxS+FiCd'k-Fj Cf-k+ (F,i, Cg-k-F;', qik) xS+FkCi'j=O ; and so, assuming (9), the equation (10) is equivalent to

(18) F5k Cg7xs+,FiCMk-Fjct'k= o.

Particularly let the structure constants Ckij satisfy CsijxSt O for some i, 7' and Ckik =O for all i (e.g., see Pa (a 7E3, 4)). Then (18) is reduced to F,kkCs`jxS= O, so that F;'i=O except for (xi) such that CsijxS= O, and therefore F,ii=O on the consider- ing neighbourhood by the continuity of Fi.

In the following examples of the three-dimensional case (EM == 3), the functions Fi(i=1, 2, 3) of (x', x2, x3) == (x, y, z) are denoted by (F', F2, F3) == (F, G, H) as well as (F,i, F,2, F,3) = (Fx, Fu, Fz). Since (18) is skew-symmetric for the indices i, 1' and indentical fori=1', it may be considered for (i, 1')= (1, 2), (2, 3), (3, 1).

ExAMpLE 1: P=P3. In viewing of Ck'k=Ck2"=O and Ck3k=2, (18) is identical

' for (i, 1')= (1, 2), and

(Fx+ Gu+H2) y-2G = O for (i, 1' ) = (2, 3),

' (F.+G,+H,)x-2H :O for (i, 1')== (3, 1).

Since (16) implies yF==xG, by putting F =xP(x, y, z) and G=yP(x, y, z) (xy iL O, which can be delated in the consequence by the continuity of F, G and H), both of the above equations lead to

xP.+yP,+H. = O.

s Fumitake MiMuRA and Akira lKusHiMA

So that H is integrated as

H= 'xfPx d2 - yfP, dz+iV (x, y),

and therefore the final forms of F, G and H are

F=xÅqb. (x, y, 2), G ==yÅqP,(x, y, z),

H =-xrpx (x, y, z) -yÅë, (x, y, z) +rp (x, y),

where O (x, y, z) =fPdz and ilJ =rp (x, y) are arbitrary differentiable functions.

ExAMpLE 2: P=:P`. Similarly as above, since Ck'k=Ck2k =O and Ck3k==2, (18)

is identical for (i, 1') = (1, 2), and

(x-y) (F.+ G,+H,) +2G = O for (i, 1') = (2, 3), (x+Zy) (Fx+ G,+H,) -2F =: O for (i, j) = (3, 1).

And since (16) implies (x-y)F+(x+Ry)G=O, by putting F== (x+Ry)P(x, y, z) and G=-(x-y)P(x, y, z) ((x-y) (x+2y) 7E O, which can also be delated in the consequence), both of the above equations lead to

(x+Ry) P.- (x-y) P,+H. = O.

So that H is integrated as

H =- (x+Ry)fP. dz+ (x-y)fjP,dz+rp (x, y),

and therefore the final form of F, G and H are

F= (x+Ry) (I), (x, y, z), G= (x-y) Åë. (x, y, z), H=- (x+Ry) O. (x, y, 2) + (x-y) Åqb, (x, y, z) +rp (x, y),

where Åë(x, y, z) =fPdz and ilJ=ilJ (x, y) are arbitrary differentiable functions.

ExAMpLE 3: P=P6. This is the case that CgSjjcS i7k O, i.e., C,23xS =x, C,3iarS=

-y, C,'2xS ==z; and Ct'k == O for all i=1, 2, 3. So that F;'i = O, i.e.,

Fx+Gu+H2 = O•

Since (16) implies xF-yG+zH= O, by putting (x2+z2t O)

F= XYP.(,X+' ,Yi Z) , H= YZP.(,X+' ,Yi Z) +ygf (x, y, z),

and so G =P+zilf (y g6 O, which can also be delated in the consequence); the above equation leads to

(19) xyPx+ (x2+ z2) P,+yzP.+ (x2+z2) (zrpu+yrpz) = O,

with the s-equations (note that the last term disappears if zilJy+yill. i O)

dx- dy -dz- -dP

inj ' x2+z2 - Y2 - (x2+22) (zrp,+yrp.) '

If ilJ =ilf (x, y2 --- z2) (this is the case of zil'y+yilJziO), by the independent

solutions x2-y2+z2 =const and x-iz=const. (x#O) which follow respectively from

xdx- -ydy -zd2 !Lt-!Zt x2y -y(x2+z2) yz2' XY Y2'

the solution of (19) is written as P= Åë(x2-y2+z2, x-'z) and therefore . xyO(x2-y2+22, x'iz)

F- x2+22 '

G =ÅqP (x2-y2+z2, x-iz) +zllJ (x, y2-z2), - yzÅë(x2-y2+z2, x-iz)

+yilf (x, y2-z2) . H

- x2+z2

As an example of ilJ such that zilJu+uilJz f O, we first take ilJ =2. Then, since one more solution P+y2/2 =const. of the s-equation follows from

dy - -dP x2+z2 (x2+z2)y'

the solution of (19) is of the from A(x2-y2+z2, x-'z, P+y2/2) == O; accordingly P =o (x2-y2+z2, x-'2) -y2/2 and therefore

- ryO(x2-y2+z2, x-'z) xy3

F- x2+z2 -2(x2+z2)'

2

G =Åqp (x2-y2+z2, x-'z) - -!IS-+22,

xy , f+2,z2, x"Z) -2(x43i 2,) +yz.

The other example of rp is rp =log (x2+z2). Since one more solution is P+2z=

' const. P is of the fromP= O(x2-y2+z2, x-iz) -2z and therefore - xyÅë(x2-y2+z2, x-'z) 2xyz

F- x2+z2 -x2+z2'

G = Åqb (x2-y2+z2, xm'z) - 2z + 2log (x2+z2),

H= YZÅë(X2'xY, i+zi2' X-'Z) - x;Y+2Z, +y log (x2+z2) .

lo Fumitake MiMuRA and Akira lKusHiMA

Here notice that x-'z (x t O) may be replaced with xz-' (z t O) in the above Åë.

REMARK. Particularly for ilJ :O and (I) =Åqb(jc2-u2+z2), the above solutions F, G and H in the case of ilJ :ilf(x) are reduced respectively to

F= XyÅë(x2-y2+z2) x2+z2 '

H-

G = Åë(x2-y2+22) , - yzO(x2-y2+22)

x2+z2 '

which differ nothing, without the change of variables y and z, from the coefficients of the vector field (1.13) in [6] obtained for the study of infinitesimal automor- phism.

Acknowledgement

The authors would like to express their deep thanks to Professor T. N6no for his constant guides and encouragements in the course of the work.

References

[1] E A. Berezin, Some remarks about the associated envelope of a Lie algebra, Funktsional Anal. i Prilozhen 1 (1967), 1-14 = Functional Anal. Appl. 1 (1967), 91-102.

[2] F. A. Berezin, Quantization, Izv. Akad. Nauk Ser. Mat 38 (1974), 1116-1175=Math. USSR- Izv. 38 (1974), 1109-1164.

[3] A. A. Kirillov, Local Lie algebras, Uspeki Mat. Nauk. 31 (1976), 57r76= Russian Math Surveys 31 (1976), 56-75.

[4] M. Levy-Nahas, Deformation and contraction of Lie algebras, J. Math. Phys. 8 (1967), 1211- 1222.

[5] F. Mimura and A. Ikushima, Structure of generalized Poisson algebras, Bull. Kyshu Inst.

Tech. Math. Natur. Sci. 27 (1980), 1-10.

[6] N. Nakanishi, On the structure of infinitesimal automorphism of linear Poisson manifolds I, J. Math. Kyoto Univ. 31 (1991), 71-82.

Department of Mathematics Kyushu Institute of Technology Tobata, Kitadyushu, 804 and

Kyushu Junior College of Science and Engineering

Kokura-Kita, Kitakyushu, 802