ALTERNATIVE APPROACH TO THE GENERALIZED POISSON ALGEBRAS
By
Fumitake MiMuRA and Akira IKusHiMA
(Received November 8, 1993)
1. Introduction
On an m-dimensional differentiable manifold EM with a second rank skew-
symmetric differentiable tensor field tu, Berezin [l] introduced the Poisson Lie algebra structure on the ring gR(EM) (with respect to the operations of addition and multiplica- tion) of differentiable functions on EIJI. And Kirillov [2] generalized the Lie algebra structure on the local Lie algebra with oned-dimensional fibre. It was constructed from the pair (tu, Fi) of co and a given system of m differentiable functions Fi (i = 1,•••,m) on E"l. The conditions on (co, Ft) which make the ring gl(EM) into an infinite dimensional local Lie algebra were given in terms of the polyvector fields on gJn.
Alternatively in this paper, we discuss the Poisson Lie algebra structure in connection with generalized vector fields Xf on swl which correspond to arbitrary functions fE 9{(E"l) in a local corrdinate system x = (xi,•••,xM) such that Xf = ÅqP}O/Oxi
+Åëf where d)}-,a)fEEM. The conditions of (co,Fi) can be reformulated with the generalized vector fields Xf. Particularly assuming that the tensor field co is non-degenerate (i.e., it has non-vanishing determinant of its components), a differential 2-form 9 is constructed from the components so that the conditions on (tu,Fi) are to be equivalent to a symmetry X?(9) = Åëf9 under the vector part X? = Åë}a/Oxi in all Xf, where X?(9) denotes the Lie derivative of sl2 by XfO.
For the convenience, differentiability is assumed to be of sufficiently high order and the summation convention is employed throughout.
2. Generalized Poisson algebras
Following Kirillov [2], in a local coordinate system x=(x',•••,xM) of E!)l, the generalized Poisson product {f, g} of f, gEY{(E"l) is given by the components coij(x) of the tensor field co and the set of functions Fi(x) (i = 1,•••,m) such that
(i) {f, g}-coiJ oO .{ oO.g,+Fi(foO.g, -goO.Åq),
which makes the ring 9{(EIJI) into an infinite dimensional local Lie algebra if and only
if (the description with the polyvectors in [2, LEMMA 3] is rewritten as follows)
(2) Fi O o`O .IIk + coki OoF.l - coji /-aF./:- =: o,
OcokS .OcosJ' .O(Djk .
. + cok' -+ coSi . + FJcokS + FkcoSj + FScojk = O. (3) coji
Oxi Oxi Oxt
Alternatively to the consideration of Kirillov in the space of polyvector fields, we take a generalized vector field on EM of the form
,o
X = Åët(X) o.i + Åë'(X)'
which operates to fEY{(EIJI) as
x(f) == Åëi oOl' ' Åëf'
A bracket [X, Y] of such generalized vector fields X and Y is defined by [X, Y] (f) -= X(Y(f)) - Y(X(f)).
Then, in terms of Xf corresponding to fES{(EIJI):
(4) x, =: (cok' t`;.Ii- +Fif) aO.,-F` oa.C.,
the Poisson Lie product (1) can be expressed as
(5) {f, g}-Xf(g).
THEoREM 1. The lacobi identity for arhitrary f, g, hESRCM):
(6) {{f, g}, h} +{{g, h},f}+{{h, f}, g} -=O
is satis.fied if and only if the generalized vector .fields Xf and X, corresponding to arhitrary .L gEY{(EI.Jl) have the relation
(7) [Xf, Xg] =X{f, g}•
PRooF. In viewing of (5), each terms in (6) is written respectively as {{f, g}, h} == X{f, ,}(h),
{{g, h},f} -= - X,({g, h})- - X,(X,(h)), {{h• f}• g} = - Xg({h, f}) = Xg(Xf (h))•
which turn the left hand side of (6) into
{{f, g}, h} + {{g, h},f} + {{h, f}, g} - (X{f,,} - [Xf, X,])(h).
Here, by putting in Xf of (4):
(8) ÅqP} =- coki -aO-.-f, + Fif,
. of
o.i,
denote Xf as follows .o
(lo) Xf=ÅqP} o.i-ÅqPf•
Then the product (5) is written as
Og
(11) {f, g}-di} o.,-Åëfg,
and X{f,g} has the appearance
o
X{f, g} = CPt-f• g} oxi - Åë{J; g}'
And moreover the bracket [Xf, X,] leads to
[X,, X,] =- [ÅqP} oa., - ÅqP,, cZÅrg oO.,. - cl5,]
== (Åë} OoÅë./l - ÅqPSmaÅë.1' )oO.i-(Åë} OoÅë.g' - Åëb Oa21' )
So that the relation (7) is equjvalently expressed as
o` .b f. -Åëe O og .i) f -= cb,,, ,,,
(i3) Åqz}}O oÅq .P f'-cpeO o(P .t;-cpf,,,,, which will be used effectively for the proof of the following theorem.
THEoREM 2. The conditions (2) and (3) of making ER(EM) into an infinite dimensional Lie algebra are satis.fied if and only if the generalized vector .fr'elds Xf and X, have the relation (7) for arbitrary f, geER(EM).
PRooF. Since the relation (7) is equivalent to (12) and (13), it is a problem that
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