CONSTRUCTION OF GENERALIZED POISSON ALGEBRAS

(Math. Natur. Sci.) No. 31, 1984, pp. 39-43

CONSTRUCTION OF GENERALIZED

POISSON ALGEBRAS

By

Fumitake MiMuRA and Akira IKusHiMA

(Received Nov, 17, 1983)

1. Introduction

Quantization of dynamical system yields an algebraic problem for making a ring of Coo functions on a manifold into a Poisson algebra which has an infinite dimensional Lie algebra structure. The classical Poisson algebra was generalized by F. A. Berezin [1, 2]

and moreover by A. A. Kirillov [3, 4] in the study of local Lie algebra with a one-dimen- sional fibre. A Lie algebraic structure of generalized Poisson algebra (P. K. algebra) defined by A. A. Kirillov was studied by F. Mimura and A. Ikushima [5].

In this paper, an important sort of the P. K. algebra is defined by means of generalized derivatives on the ring of Cco functions and a further discussion is made for a structure of the P. K. algebra. There is given a procedure for making the ring of Cco functions into the algebra.

2. Generalization of classical Poisson algebra

Let 9n be an m-dimensional Cco manifold with local coordinates x=(xi,..., xm) and wt(EM) the ring with respect to the usual operations of addition and multiplication of Cco functions on SM. Then the classical Poisson bracket operation is generalized by

[f, g], == Cijf,g,j, (1)

where Cij--- -Cji are Cco functions on S)Jl (F. A. Berezin [2, p. 1110]). Here note the conventions of summation (summing up for repeated indices) and comma (f,i=Of/Oxi, g,j =Og/exj), which are used throughout. Under this bracket operation, the ring wt(EM) has an infinite dimensional Lie algebra structure if and only if

c(ilslc{,k) :O,

where the parenthesis of indices denotes the symmetrization of i, 1', k. This algebra is . called Poisson-Berezin algebra (P. B. algebra) and denoted by So(C). Moreover, the Poisson bracket operation (1) can be generalized by

[f, g] = Ciij,,g,j+A`(fg,i-- gf,i) ,

40 Fum'itake MiMuRA and Akira lKusHiMA

where Cij-= --Cji and Ai are Cco functions on EM (A. A. Kirillov [4,p. 58]). Under this bracket operation, the ring wt(EIJI) has an infinite dimensional Lie algebra structure if and only if Ai and CiJ' satisfy

A(icjk)+c(ilslCl,k) == O, (2) Afgcj]s + Asclg' -- o, (3)

where the bracket ofindices denotes the skew-symmetrization of i, J'. This algebra is called Pojsson-Kirillov algebra (P. K. algebra) and denoted by S(A, C). Of course So(C)=

S(O, C). For arbitrary elementf f, g and h in wt(swl); So(C) satisfieg.

[f, gh]o == [f, g]oh + [f, h]og,

but (S(A, C) does not satisfy the following if AilO for some i:

[f, gh] == [f, g]h+ [f, h]g,

(which should be Qmitted in [5, p. 3], while this was not used in the discussion). In fact,

the above equation shows [xi, 1]=-Ai=O by putting f==x` and g=h :1. So, the

generalization.carries only the infinite dimensional Lie algebra structure [f, g] + [g, f] -O,

[[f, g], h] + [[g, h], f] + [[h, f], g] - O.

3. Reformulation of P. K. algebra

Let ipi be given Coo function on S"l, and consider the generalized derivative offin

wt(sM) :

D,(f) ==f,, + ipif•

Then, for arbitrary functions f and g in wt(swl), we shall define the bracket operation

[f, g]D=CijDi(f)D,(g), (4)

where Cij --• -Cji are Cco functions on !M, and find an necessary and suMcient condition that wt(EM) has an infinite dimensional Lie algebra structure under the bracket.

REMARK, The condition Cij=-Cii can be derived from [f,g]D+[g,f]D=O.

In fact, this relation yields [1, 1]D=O by puttingf==g=1. So that, since Di(1)==gbi and Di(xk) == 65• + ipixk, from (4) it fo11ows respectively by putting

f=g == 1: [1, 1].= Cij q5iip,• --- O,

f=xk, g =1: [xk, 1].= C`j((5e• +ipixk)(Pj

, = C"j ipj + CijÅë,ipjxk = Ck J' ipj,

f= l, g =xs: [1, xs]D=Cij(Pi((5i+ipjxS) = Cisdii + Cij (P iÅëjxs = Cisipi,

f = xk , g = xs : [xk, xs]. = Cij(6ts•' + ip ixk) (6,S• + ipjxS)

,,,, cks + ckjdijxs + Cisipixk + Cijipidijxkxs = Cks.

Therefore [xk, xs]D+ [xs, xk]D=O implies Cks+ Csk =O.

Now, since Cij --- -Cji, the bracket operation (4) becomes [f, g]D = C`j(f,i +' ip if) (g ,j + ipjg)

-= Ci jf, ,g ,j + Ci J'(ipifg ,i + ipjgf, i) + Cij diiipjfg

=CiJ'f,ig,j+ipjCji(fg,,--gf,,).

So by putting

Ai ,. ipjCJ'i, (5)

the following theorem can be proved.

THEoREM 1. Under the bracket operation [,]D, the ring wt(EM) has an infinite dimensional Lie algebra structure if and only if

q!) ,Cs(i Cik)+C(ilsl C{,k) == O, (6) ip,,kCk[iCj]s+ipkC(ilslC{,k)=O, (7)

PRooF. By substituting (5) into (2), it follows

A(iCJ'k) + C(i1S1C{.k) = di,CS(iCJ'k) + C(iISiC!,k) == 0;

and into (3), it follows also

AlgCj]S + ASCIg -- ip ,,,Ck[i Cj ]S + ip ,(Ce,[iO]s + CksC,i,J') = ipk,,Ck[iCj]s+ ipkC(i1slcl'.k) = o,

here note the relation

Ce,[iCj ]S + CkSCIg = Ceg CjS - Clr.J' CiS + CkSCI I' =:C(ilslCl',k) (since Cij -- -- Cji).

We denote by SD(C), the infinite dimensional Lie algebra under the bracket operation

[ , ]D of wt(EIJI).

42 Fumitake MiMuRA and Akira lKusHiMA

REMARK. SD(C) is a P. K. algebra such as S.(Cij)=:S(ip,jCji, Cii

Since the term C(ilslCf'.k) is contained in both (6) and (7), we shall first find an example of Cij satisfying C(ilslC{,k)=O.

LEMMA 1. Let Ck',...k. be a system ofconstants which is skew-symmetricfor i,j and symmetric in ki,..., k., and satisfying the relation

C#•1•f;.I CI'52's. == O' (8)

Then Cij '--" Ck{.,.k.xki•••xkn satistv C(ilSilCl',k,)= O.

PRooF. By direct computation, it is verified immediately C(ilS'ICII',k)=nCf,i.l.F2.lCg',-E).,.x'`•••xtnxk2•••xkn =O.

REMARK. Let Cij be structure constants of a Lie algebra. Then CY satisfy CSilslCg'k) :O, which is the case n=:1 in (8). The algebra So(CYxk)=bi(O, CVxk) was considered by F.A.Berezin [1,p. 100]. Let n=2, then (8) becomes C[ll;'IC,",k,), =O.

This case: G)o(CE','xkxS) == S(O, CV,'xkx`) was introduced by E. K. Sklyanin [6, p. 265].

Next we shall consider an example of Cij in which the terms Cs(iCjk) in (6) vanish.

LEMMA 2. Let swl be a 3-dimensional manifold, i.e., the indices i, J', k and s take values 1,2and 3. Then CV satisfy Cs(iCjk)=O.

PRooF. Consider the cases (i) i=j and (ii) i7Ej. VV hen i76j, the cases are considered:

(ii---a) s==i4j, (ii-b) ilj=s, (ii-c) six! i,j, i.e., i,j, s are all distinct each other. So that in

s#ifj#s =k. Since Cij----Cji (Cii =O), for each cases by putting the corresponding indices jn Cs(iCjk) == CsiCJ'k+CsjCki+CskCij, it is verified directly: Cs(iCjk)=O.

Finally, we can get an example of ipi which vanjsh ip,,kCk[iCj]s in (7).

LEMMA 3. Let ipi be oftheform ipi=ip,i. Then dii satisfy di,,kCk[iCj]s==O.

PRooF. From Cii--- -- Cji, it follows that

ipk,,Ck[iCJ']s = ql)k,,CkiCjs -•-- dik,,CkjCis = : ipk,,Ck i CJ' S - ip,,kCSJ' Cik =" ip(k,s)CkiCjs,

which vanish because of dik,,== ips,k= ip,k,s•

The lemmas 1, 2, and 3 enable us to find an infinite dimensional Lie algebra under the bracket operation [, ]D of wt(S"l). For example we have:

THEoREM 2. Let swl be a 3-dimensional manilfold and Ci' a system of structure constants ofa 3-dimensional Lie algebra. Then,for arbitraryfand g in E!)l, the bracket operatlon

[f, g]D = CYxkf,ig,,• + gb,jCiixk(fg,, --- gf,,)

makes the ring wt(E"l) into an infinite dimensional Lie algebra (SD(C)==S(di,,•C'k'ixk, CkJxk).

Finally, by the lemmas 2 and 3 we have the following theorem for making wt(S"l) into P. B. and P. K. algebras.

THBoREM 3. Let S"l be a 3-dimensional manifold. Then, the ring sc(E!Jl) becomes a P. B. algebra (S(Cij) if and ont.v if wt(S)Jl) becomes a P. K. algebra biD(Cij)= S(ip,jCji, Cij).

Acknowledgements

The authors would like to express their deep thanks to Professor T. N6no for his constant guides and ehcouradgements. Also, one of the authors (F. Mimura) is indebted the Grant-in-Aid for Scientific Reserch of the Ministry of Education.

References

[1] F. 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), ll16-1175 == Math. USSR- Izv. 38 (1974), 1109-1l64.

[3] A.A.Kirillov, Structures of Lie algebra that have the locality property, Funktsional Anal.i Prilozhen 9 (1975), 75-76 :i: Functional Anal. Appl. 9 (1975), 158-160.

[4] A.A.Kirillov, Local Lie algebras, Uspeki Mat. Nauk. 31 (1976), 57-76- Russian Math.

Surveys 31 (1976), 56-75.

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

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

[6] E.K.Sklyanin, SomealgebraicstructuresconnectedwiththeYang-Baxterequation, Funktsional Anal. i. Prilozhen 9 (1975), 75-76 == Functional Anal. Appl. 9 (1975), 158-160.

Department of Mathematics Kyushu Institute qf Technology and

Yomiuri Kyushu Junior College

of Scienee and Engineering