Bull. Kyushu Inst. Tech.
(M. & N. S.) No. 27, 1980, pp. 1-10
STRUCTURE OF GENERALIZED POISSON ALGEBRAS
By
Fumitake MiMuRA and Akira IKusHiMA
(Received Oct. 16, 1979)
1. Introduction
The problem of quantization in mechanics originated the problem of making the ring of Cco functions on a manifold into an infinite dimensional Lie algebra, called Poisson algebra, by defining a Poisson bracket operation. The classical Poisson bracket operation was generalized by F. A. Berezin [1, 2] in the study of quantization, and moreover, the operation was generalized by A. A. Kirillov [3, 4] in the study of local Lie algebra with a one-dimensional fibre. This paper deals with the structure of (generalized) Poisson algebras in Lie algebraic viewpoint. For the first place, a brief review is given of the Pois- son algebras defined by F. A. Berezin (P. B. algebras) and its generalization by A. A.
Kirillov (P. K. algebras) with some considerable examples. In which a further discussion is made of Poisson algebras defined by the structure constants of a Lie algebra, while such a Poisson algebra was first considered by F. A. Berezin. Particularly, Poisson algebras are determined completely in the case that the structure constants are those of 3-dimen- sional Lie algebras. Next, isomorphisms of Poisson algebras are investigated and, as a consequence, a method is obtained for determining the Poisson algebras. And finally, some Poisson algebras are derived by means of the method.
2. P.B.algebras
Let EM be a m-dimensional Cco manifold with local coordinates x=(xi,.,.,x,m) and M(E!Jl) the ring with respect to the usual operations of addition and multiplication of Cco functions on E"l. Then the classical Poisson bracket operation (see Example 1) is gener- alized by
[f, g]o=Cijf,ig,j, (1)
where Cij=CiJ' (x) are Cco functions on EM (F.A. Berezin [2, p. 1110]). Here note the conventions of summation (summing up for repeated indices) and comma (f,i==0flOxi, g,j----OglOxJ') which are used throughout. Under this bracket operation, the ring 9{(swl) has an infinite dimensional Lie algebra structure if and only if
c(ilslCl'.k)=O, (2)
elements f, g and h in wt(EM), the bracket operation satisfies [f; gh]o= [f, g]oh+[f, h]og•
So, this algebra is named Poisson-Berezin algebra (briefly P. B. algebra) and denoted by So(C).
3. Examples of P. B. algebras
ExAMpLE 1. Let EM be 2n-dimensional manifold, i.e., m=:2n and Cij the com- ponents of a matrix s=(-g. Eo") where E. is the unit matrix of order n. Then the Poisson bracket (1) has the classical form (i=1,..., n)
[f, g]o=f,ig,n+i-g,if,n+i•
This P. B. algebra is denoted by So(s).
ExAMpLE 2. Let CV be the structure constants of a m-dimensional Lie algebra g and define Cij=Ck'xk. Then the condition (2) is equivalent to
clilsl C,J'k) == O, (3)
which is just the Jacobi identity for the structure constants. Hence So(CYxk) has the P. B. algebra structure under the Poisson bracket (i, j, k== 1,..., m)
[f, g]o == Ck' f,,g,jxk,
which was first considered by F. A. Berezin in [1, p. 100]. This P. B. algebra is denoted by So(g)•
ExAMpLE 3. Let E"l be 2n-dimensional manifold and Ck' the structure constants of an n-dimensional Lie algebra g. In this situation, let CiJ' be the components of a matrix sg =(-lll. Ig:), where D. is a matrix of order n whose components are given by Dii ---- Ci'xk. Then the condition (2) is equivalent to the Jacobi identity for the structure constants [see (3)]. The Poisson bracket (1) has the form (i, j=1,..., n)
[f, g]o =f,ig,n+i-g,if,n+i+Ck'f,n+ig,n+jxk,
which was discussed in a geometric viewpoint by R. Hermann [5, p.46]. This P. B.
algebra is a generalization of Example 1 and denoted by So(sg).
Structure of Generalized Poisson Algebras 3
4. P. K. algebras
The Poisson bracket operation (1) can be generalized by
[f, g]=C`J'f,g,j+Ai(fg,i-gf,i), (4)
where Ai=Ai(x) and Cij=Cij(x) are Cco functions on EM (A.A. Kirillov [4, p.58]).
Under this bracket operation, the ring 9{(EM) has an infinite dimensional Lie algebra structure if and only if Ai and Ci' satisfy (the description in [4] is modified)
A(icjk)+c(ilslCf,k) == O, (5)
AliCj]k+AkCIJ,'=O, (6)
where the parenthesis and brackets of indices denote the symmetrization of i, J', k and the skew-symmetrization of i, j, respectively. The bracket operation (4) satisfies
[f, gh]=:[f, g]h+[f, h]g,
for arbitrary elements f, g and h in g{(swl). So, this algebra is named Poisson-Kirillov algebra (briefiy P. K, algebra). By differentiating (5) by xi and summing up for i, it follows that
AIk CJ' ]k + AkC:'{ + A[iC{,]k + Alr, CiJ' + C[ilsl C!,],k + Clk1sl C{,k ) = : O.
So that, assuming (5), the equation (6) is equivalent to
A[iC{,]k+A5k Cij + C[ilsl C{,],k +CSIsl C{,k)= O. (7)
RBMARK 1. Let Al=tAi(x) be a one parameter family of Cco functions on EM and define (S,(A, C)=(!i(tA, C). Then, from a replacement Ai by tAi in equations (5) and (6) it follows that S,(A, C) has the P. K. algebra structure for arbitrary tif and only if C(ilslCl',k)=O, A(iCjk)=O and A!kCj]k+AkC:'t==O. Therefore it is concluded (see [6]
for the definition of deformation): Let (So(C) have the P. B. algebra structure. Then, S(A, C) has the P. K. algebra structure if and only if S,(A, C)=:S(tA, C) is a defor- mation of P. B. algebra So(C) under the bracket
[f, g],=Cijf,ig,j+tAi(fg,i-gf,i).
In this viewpoint, deformations of the P. B. algebras are contained in the P. K. algebras.
5. ExamplesofP.K.algebras
ExAMpLE 4. Let Ai==Ofor all i. Then (5) is equivalent to (2) and (6) is valid. The Poisson bracket (4) is reduced to (1), and hence, So(C) =S(O, C).
structure constants Ck' of a m-dimensional Lie algebra g. Then, by virtue of the Jacobi identity (3), the equation (5) is equivalent to
A(iC.'k)=O. (8)
,Moreover, let Ai (i=1,..., m) be some constants. Then the equation (6) is reduced to
AkCÅíj •.•-: O, (9)
and, assuming (8), this equation is equivalent to [see (7)]
A[iCJi]k=O. (10)
If Cik'=O for all i, (10) gives no condition ofAi and hence Ai must be determined by (8).
And if Ckk 7!O for some i, (10) gives the solution of Ai: Ai= ACkk (A: constant), so from (8) it follows that Ai =ZCftk (or =O) corresponding to C5il'ICg'k)=O (or IO) respectively.
Therefore, the Poisson bracket of (S(A, g) has the following forms (i, j, k =1,..., m):
If Cftk ==O for all i [although the constants A` must be determined by (8)], [f, g] =Ck'f,,g,jxk+Ai(fg,i-gf,i);
if Cik :O for some i and CSiltlCg'k)=:O,
[f, g] = Ckjf,,g,jxk +ACkk(fg,i --- gf,i) ; and if CEk 7EO for some i and CSiltlCg'k) 7E O, [f, g] = Ck' f, ig ,jxk (P. B. algebra) .
Particularly, let g be the 3-dimensional Lie algebra and determine completely the P. K. algebra S(A, g)=(S(Ai, Ck'xk) (Ai: constants), In this 3-dimensional case, the structure constants Ck' associate to a matrix P = (pij) :
c?3 Cii Cl2 Pii Pi2 Pi3
c223 Cgi C62 = P2i P22 P23
cg3 Cgi Cg2J P3i P32 P33
which is written as Ci'=pk,siJ'S wher.e Åíijk is the usual alternator. Then the equation (8) is equivalent to
A( ip.,6jk)' == Aip,t6jkt +Aj p,,6ki' +Akp,,eijt=O. (1 1)
If at least two indices in i, J', k are the same, the above equation is satisfied identically.
So, let the indices i, j, k are all distinguished. Here note that the indices i, j, k, ttake the values 1, 2 or 3, and hence, the alternators 6jk', eki' and eijt do not vanish if and only if t == i, t =j and t == k respectively. Therefo re, since 6jki = ek iJ' --- eijk : O, the equation (1 1)
Structure of Generalized Poisson Algebras 5
is equivalent to
Aip,i+Ajp,j+Akp,k ==O (no summation for i, j, k).
Since the indices i, j, k are all distinguished, the above equation differs nothing from
Aip,i==O, i.e., A'P==O, (12)
where A=(Ai A2 A3) and t denotes the transposition. Moreover the equation (9) is equivalent to Aipi,eJ'ks=O. Therefore, since EjkS =O if and only if the indices j, k, s are all distinguished, it is obtained
Aip,,=O, i.e., AP=O. (13)
On the other hand, the matrix P takes the following forms under a suitable isomorphism of g (see [7, pp. 1214-1215]):
P,-
(iig), P,-(ig:), P,-(--g•ig), P.-(--6•ig)(AS--i),
.,=
(g --g, i), .,,.(i g, i), .,=(i "-g, s), .,..(i g, 1)
Let gi be the Lie algebra with structure constants CV corresponding to the matrices Pi and denote Si=S(A,gi). Then, determining the constants Ai by (12) and (13), the Poisson brackets of P. K. algebras Si have the following forms:
si : [f, g] == Ai(fg,, -gf, i) + A2(fg,2 -- gf, 2) +A3(fg,3 -gf, 3) , s2 : [f, g] =f,[2g,3]xi + A2(fg,2 -gf, 2) + A3(fg,3 - gf, 3) , S3: [f, g] =f,[3g,i]x'--f,[2g,3]x2+A3(fg,3--gf,3),
S` : [f, g] =f,[,g,,](x' - x2) +f,[3g,i](xi + Zx2) + A3(fg,3 - gf, 3) , S5 : [f, g] ==f,[,g,,]xi -f,[3g,i]x2 + A3(fg,3 -- gf, 3) ,
or6 : [f, g] =f,[,g,,]x i +f,[3g,i]x2 + A3 (fg,3 -- gf, 3) ,
S': [f, g] =f,[2g,3]xi-f,[3g,i]x2+f,[ig,2]x3 (P. B. algebra), .,
/t '
S8 : [f, g] =f,[2g,3]x' +f,[3g,i]x2 +f,[ig,2]x3 (P. B. algebra) .
Moreover, by rewriting Ai by tAi, the above brackets are reduced to those of defor- mations S:==S(tA, gi) of the P. B. algebras S6=S(O, gi) under the brackets (see Re-
i
where ph. are the components of matrices Pi and pin,6J'ks=CSk are the structure constants of Lie algebras gi. It is observed that the P. B. algebras S3 and Sg cannot be deformed into new algebras under the above brackets, that is, S3 and Sg are rigid under the brackets.
6. P.K.algebrasunderisomorphisms
Let us now consider two P. K. algebras S(A, C) and S(B, D). An isomorphism of the algebras is defined by an invertible Cco function Åë on EM (i.e., Åë:O at every point of EM), satisfying
[dif, Åëg]A,c==Åë[f, g].,., (14)
for arbitrary elementsfand g in wt(EM), where the subscriptions of the brackets denote the respective Poisson bracket operations of the form (4). Local version of the definition can be made similarly by an invertible Cco function Åë on a neighbourhood of a given point in EM, and that of the following theorems can also.
THEoREM 1. An invertible Cco function Åë on EM is an isomorphism of P. K. algebras S(A, C) and or(B, D) if and only if
Bi=Åë,jCJi+ÅëAi, (15)
DiJ=ÅëCiJ. (16)
PRooF. Since CiJ'= --CJ'i, from a straightforward computation of (14) it follows that
Dijf,,g,j+B`(fg,i-gf,i)
= diCij f, ,g,j+(Åë,j Cj`+ ÅëA i) (fg,i -- gf, i) . (1 7) So by puttingfis nonzero constant, this equation yields
Big,i= (di,jCj i + ÅëAi)g,i ,
which gives the equation (15) for arbitrary functions g in wt(EM). And so, the equation (17) is reduced to
Dij f,ig,j --- OCij f, ig,j ,
which gives the equation (16) for arbitrary functions f and g in wt(EM).
Conversely, the equations (15) and (16) show the validity ofthe equation (17). There- fore the proof is completed.
Theorem 1 produces the following theorem which was essentially obtained by A. A.
Structure of Generalized Poisson Algebras 7
Kirillov in [4, p. 64].
THEoREM 2. Let Åë be an invertible Cco function on SM. Then a P. K. algebra S(Ai, Cij) is isomorphic to a P. K. algebra S(di,jCji+diAi, ÅëCiJ'), Particularly, the
P. K. algebra S(Ai, C`j) is isomorphic to a P. B. algebra So(diCij)== (S(O, ÅëCij) if and only ifÅë lies in the center ofS(Ai, Cij), in this case Ai takes the form Ai=:(log lÅël),jCij.
Moreover, the P. K. algebra S(Åë,jCJ'`, ÅëC`j) is isomorphic to the P. B. algebra So(Cij)
=s(o, cij).
PRooF. The first statement of the theorem follows from Theorem 1. So let us prove the particular case. The function Åë lies in the center of (S(Ai, Cij) if and only if
[(P, g]A,c=(di,jC" + ÅëAi)g,,- di,,Aig
vanishes for arbitrary functions g in S{(EM), i.e., Åë,jCji+ÅëAi=O and Åë,iAi =O. The former gives Ai=(log 1ÅëD,jCij which satisfies the latter. Hence the if part is proved.
Conversely, let G5(Ai, Cij) be isomorphic to So(ÅëCij). Then by Theorem 1, there exists an invertible Cco function Y' such that O=!U,jCji+Y'Ai and ÅëCij=!UCij. These equations yield di,jCji+diAi= O if Cij#O (for some i, j) or Ai ==O if Cij ----O (for all i, j).
Hence the only if part is proved.
Moreover, by replacing Åë and CiJ' with Åë-i and ÅëCij respectively, the last statement of the theorem follows from the second one. Therefore the proof is completed.
Let us now remark that the equations (5) and (6) for making ER(EM) into a P. K.
algebra S(A, C) are derived from the Jacobi identity of the bracket (4) [3, p. 159]. And the isomorphism di satisfies [see (14)]
[[di f, Åëg]A,c , Åëh]A,c = Åë[[f, g]B,D , h] B,D ,
for arbitrary elements f, g and h in g{(EM). So that Theorem 2 yields the following theo-
rem
THEoREM 3. Let ÅqP be an invertible Cco function on SI]l. Then S(Ai, CiJ') has the P. K. algebra structure (f and only if S(Åë,jCij+ÅëAi, ÅëCij) has the P. K. algebra structure. Particularly, bi((loglÅë1),jCij, Cij) has the P. K. algebra structure if and only if So(ÅëCij)==S(O, ÅëCiJ') has the P. B. algebra structure. Moreover, S(Åë,jCji, diC`j) has the P. K. algebra structure if and only if (So(Cij)= (S(O, Cij) has the P. B.
algebra structure.
REMARK 2. Let gp be a Cco function on E"l and put eq=Åë. Then the particular case of Theorem 3 is restated as follows: S(q,jCij, Cij) has the P. K. algebra structure if and only if So(eqCij)=S(O, eipCij) has the P. B. algebra structure, where op is a Cco function on EM.
where qj are Cco functions on SM. Then the equations (5) and (6) are reduced to
C(ilsl C{.k)= -- C(ilslCik)q,, (1 8) Ci[sCIJ' lk]q,,k= C(ilSICjk)qk. (19)
Since C(ilslCjk) is skew-symmetric for s and k, by substituting (18) into (19) it is obtained q,,kCi[sCljlk] =o.
This equation is satisfied for qi=q,i, where op is a Cco function on El]l.
Conversely, let denote Cijsk=Ci[SCIjlk], q,k==op.,k--opk,, and the pair of indices (il')
=(12),...,(lm); (23),...,(2m);...; (m-1m) as ct=1,...,m(m-1)/2. Then the above equation is rJwritten as Cctfiqp=O. If det(Cctfi)iO, this equation shows qfi= O, i.e., q,,k
=opk,,. Therefore opi =q,i, where q is a Cco function on S!Jl.
REMARK 4. From Remarks 1 and 2 it follows that: Let q be a Cco function on EM. Then S,(qjCij, Cij)= G5(tq,jCij, CiJ') is a deformation of P. B. algebra So(CiJ')
== (!5(O, Cij) if and only if So(eqCij)=S(O, eqCij) has also the P. B. algebra structure, i.e. q,,C(ilslOk) ,,. o.
7. ExamplesofdeterminingP.K.algebras
First of all remark that the if part of the last statement of Theorem 3 is valid even if the function ÅqP is not invertible. So let (So(C) have the P. B, algebra structure under the Poisson bracket
[f, g], :Cijf,,g,j. (1)
Then S(Åë,jCji, diCij) has the P. K. algebra structure under the Poisson bracket
[f, g] == ÅëCiJ' f,ig,j+Åë,jCij(fg,j -- gfl,•), (20)
where Åë is an arbitrary Cco function on EIJI. This P. K. algebra is denoted by SÅë(C).
Since the Poisson bracket (20) is written as
[f, g] =- Åë[f, g]o+ [f, Åë]og+[Åë, g]of, it follows the relation
Åë[f, g]=[Åëf, Åëg]o• (21)
Therefore the P. K. algebra SÅë(C) is isomorphic to the P. B. algebra So(C), if the function di is invertible. This result follows also from the last statement of Theorem 2.
REMARK 5. Let us now keep in mind the relation of Poisson algebra
Structure of Generalized Poisson Algebras 9.
[f, gh]o= [f, g]oh+ [f, h]og,
where f, g and h are arbitrary elements in wt(SM). Then, independently of the bracket (1), the relation (21) is derived as follows
[Åëf, Åëg]o=[f, Åëg]oÅë+[Åë, Åëg]of
-([f, g]oÅë+[f, Åë]og)Åë+([di, g]odi+[Åë, Åë]og)f = ip([f, g]oÅë+[f, Åë]og+[Åë, g]of)
-Åë[f, g]•
The following examples 6, 7 and 8 of P. K. algebras are obtained from Examples 1, 2 and 3 of P. B. algebras, respectively.
ExAMpLE 6. Let EM be 2n-dimensional manifold. Then SÅë(s)=:S(Åë,jsji, Åësij) has the P. K. algebra structure under the Poisson bracket (i, j--- 1,..., n)
[f, g] = di(f, ig ,. + i - g,if,. + i)
+ di,i(fg,n+i-gf,.+i) - di,.+i(fg,i -- gf]i) •
ExAMpLE 7. Let CY be the structure constants of a m-dimensional Lie algebra g.
Then SÅë(g)==S(Åë,jC'k'ixk, ÅëCkjxk) has the P. K. algebra structure under the Poisson bracket (i, j, k== 1,..., m)
[f, g] = ÅëCkjf,,g,jxk + Åë,iCij(fg,j -- gf,j)xk.
ExAMpLE 8. Let SDI be 2n-dimensional manifold and CV the structure constants of an n-dimensional Lie algebra g. Then (!5Åë(s,)==S(Åë,js6i, ÅësE') has the P. K. algebra structure under the Poisson bracket (i, j, k =1,..., n)
[f, g] =Åë(f,ig,n+i-g,if,n+i+CVf,.+ig,.+ixk) + Åë,i(fg,. + i -- gf,.+ i) - Åë,. + i(fg ,i - gf, i) + di,n + iCY(fg,n +j - gf,n+ j)xk•
REMARK 6. As stated above, if the Cco function Åë on S!Jl is invertible, the P. K.
algebras SÅë(s), SÅë(g) and SÅë(s,) are isomorphic to the P. B. algebras So(s), So(g) and So(s,), respectively.
Acknowledgement
One of the authors (F. Mimura) would like to express his deep thanks to Professor
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Department of Mathematics Kyushu Institute of Technology and
Kyushu Junior College of Science and Engineering
