Generalized Abelian Deformations: Application to Nambu Mechanics

Letters in Mathematical Physics 39: 107–125, 1997. 107c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Generalized Abelian Deformations:Application to Nambu Mechanics

GIUSEPPE DITO1 ? and MOSHE FLATO21Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto 606-01, Japan2Departement de Mathematiques, Universite de Bourgogne, BP 400, F-21011 Dijon Cedex, France

(Received: 2 September 1996)

Abstract. We study Abelian generalized deformations of the usual product of polynomials introducedin an earlier work. We construct an explicit example for the case of su(2) which provides a tentativequantum-mechanical description of Nambu mechanics on R3 . By introducing the notions of strongand weak triviality of generalized deformations, we show that the Zariski product is never trivialin either sense, while the example constructed here in a quantum-mechanical context is stronglynontrivial.

Mathematics Subject Classifications (1991): 20N15, 81V99, 81S30, 12D05.

Key words: Abelian deformations, deformation quantization, Nambu structures.

1. Introduction

The generalization of Hamiltonian mechanics introduced more than 20 years ago byNambu [6] has been recently formulated within a geometrical framework [7]. Theimportance of the so-called Fundamental Identity (FI) [5, 7] has been recognizedas a dynamical consistency condition for Nambu’s original formulation. Somehowthe FI plays the role of the Jacobi identity for the Poisson bracket in the usualHamiltonian mechanics, but its consequences are by far completely different asit imposes strong constraints on the underlying geometrical structure [7]. Thereader is referred to [7] for further details on Nambu structures and manifolds,we shall limit ourselves to recall the definition. Let M be an m-dimensionalmanifold. Denote by N the algebra of smooth real-valued functions on M . ANambu bracket of ordern onM is ann-linear map onN taking values inN , denotedby (f1; : : : ; fn) 7! ff1; : : : ; fng, fi 2 N , such that the following properties aresatisfied for any functions f0; : : : ; f2n�1 2 N :

(a) Skew-symmetry

ff1; : : : ; fng = �(�)ff�1 ; : : : ; f�ng; 8� 2 Sn;

? Supported by the European Commission and the Japan Society for the Promotion of Science.

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108 GIUSEPPE DITO AND MOSHE FLATO

(b) Leibniz rule

ff0f1; f2; : : : ; fng = f0ff1; f2; : : : ; fng+ ff0; f2; : : : ; fngf1;

(c) Fundamental Identity

ff1; : : : ; fn�1; ffn; : : : ; f2n�1gg

= fff1; : : : ; fn�1; fng; fn+1; : : : ; f2n�1g+

+ffn; ff1; : : : ; fn�1; fn+1g; fn+2; : : : ; f2n�1g+ � � �

+ffn; fn+1; : : : ; f2n�2ff1; : : : ; fn�1; f2n�1gg;

where Sn is the group of permutations of the set f1; : : : ; ng and �(�) is the sign ofthe permutation � 2 Sn. A Nambu bracket defines a Nambu structure on M . ThenM is said to be a Nambu–Poisson manifold.

The quantization of Nambu structures turns out to be a nontrivial problem, even(or especially) in the simplest cases. Usual approaches to quantization have failedto give an appropriate solution, and in a common work [3] with D. Sternheimerand L. Takhtajan, we were led to introduce a new quantization scheme (Zariskiquantization) in order to give a solution to that old problem. The central ideaof Zariski quantization is to look first for an Abelian deformation of the usualproduct of functions instead of a direct deformation of the Nambu bracket. Thenthe quantization of the Nambu bracket is achieved by plugging in the Abeliandeformed product into the classical Nambu bracket.

Abelian algebra deformations are classified according to the Harrison coho-mology, and the second Harrison cohomology group is trivial for the space ofpolynomials on R

n . One way to overcome this cohomological difficulty is to con-sider generalized deformations not of the usual Gerstenhaber type. In the case ofZariski quantization, linearity with respect to the deformation parameter does nothold: The deformed product operation annihilates the deformation parameter, sothe usual Harrison cohomology is not applicable. This product is obtained by fac-torization of real polynomials into irreducible factors and complete symmetrizationby a partial Moyal product. However, the deformed product is not distributive whenconsidered as a product between polynomials (it is only Abelian and associative).The linearization of this deformed product (by extending it to a semigroup alge-bra), called the Zariski product, provides an Abelian algebra deformation of theusual product on the semigroup algebra generated by the set of real irreduciblepolynomials. By an appropriate presentation of this algebra which allows a coher-ent notion of derivatives, we can define, in a natural way, the deformation of theclassical Nambu bracket and, hence, a quantization of the usual Nambu structureon the semigroup algebra.

The Zariski quantization looks like a field quantization where irreducible poly-nomials are considered as one-particle states. We may wonder if, starting fromthe Moyal product, a quantum-mechanical approach would have been possible ifwe had made a decomposition into linear factors instead of irreducible ones. The

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GENERALIZED ABELIAN DEFORMATIONS 109

answer is negative: Complete symmetrization by the Moyal product of linear fac-tors provides no quantization at all. However, as indicated in [3], the situation iscompletely different if we consider an invariant star product on su(2)� � R

3 (theone which is associated with the quantization of angular momentum, the linearfactors being here the generators of the Lie algebra). In that case, it is possible tofind a quantization by staying within the algebra of smooth functions on R

3 .In this Letter we shall work out some remarks stated in [3] regarding a poss-

ible quantum-mechanical approach (with a finite number of degrees of freedom),nontriviality of the generalized deformations (i.e. not of the Gerstenhaber type)and develop an example for su(2)�. It is shown in Section 2 that some kinds ofR[�]-linear deformations are not interesting within our context, in the sense thatthey cannot be associative without coinciding with the usual product. In Section 3,we shall make precise what we call a quantum-mechanical approach to generalizedAbelian deformations. The questions of nontriviality and the equivalence of thesedeformations are also studied. We introduce the notions of A- and B-equivalences(and the corresponding notions of triviality: strong and weak) for generalizeddeformations, and in particular we show that the Zariski product is nontrivial inboth senses (weak and strong). A detailed example is presented in Section 4 forthe su(2)�-case by giving an explicit form of the deformed product in terms ofdifferential operators, providing a strongly nontrivial deformation of the usualproduct on R

3 . We conclude this Letter by some remarks on some of the spectralityproperties of the generalized Abelian deformations and their application to thequantization of the Nambu bracket.

2. R[�]-Linear Products

The generalized deformation introduced in [3] is not an algebra deformation in thesense of Gerstenhaber. Also, when restricted to the space of polynomials on R

n ,it is not additive due to the factorization into irreducible polynomials involved inthe definition of the product. If, instead, we had performed a decomposition (byaddition of monomials and multiplicative factorization) into linear polynomials,it would have been possible to stay within the framework of Gerstenhaber typedeformations by requiring linearity with respect to the deformation parameter. Weshall show here that this attempt is not interesting: First, we would have foundnothing but a trivial deformation of the usual product (the Harrison cohomologyis trivial here) and we cannot expect to conciliate the associativity of this productwith quantization. We shall show that these kinds of products are associative if andonly if they coincide with the usual product of functions.

Consider Rn with its coordinates denoted by (x1; : : : ; xn). Let P be a Poissonbracket on R

n and � a star product on Rn with deformation parameter � such that

the star bracket [f; g]� � (f � g� g � f)=2�, f; g 2 C1(Rn), defines a Lie algebradeformation of the Lie algebra (C1(Rn); P ) (see [1, 2] for the first extensivepapers on star products). Denote by N the algebra of real polynomials on R

n and

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by N [�] the space of the polynomials in � with coefficients in N . The symmetrictensor algebra of N (with scalars) is denoted by S with symmetric tensor product. S[�] is the space of polynomials in � with coefficients in S .

Following the lines of [3], we define a R[�]-linear map �:N [�]! S[�] by

�(xk11 � � � xknn ) =

�xk1

1

� � � �

�xkn

n

�; 8k1; � � � ; kn > 0: (1)

In particular, �(1) = I , the identity of S[�].Let T :S[�]! N [�] be the unique R[�]-linear map defined by T (I) = 1 and

T (F1 � � � Fk) =1k!

X�2Sk

F�1 � � � � � F�k ; 8k > 1; (2)

where Fi 2 N , 1 6 i 6 k.

DEFINITION 1. Let � be a star product onRn endowed with some Poisson bracket.Let us define a new product �� on N [�] by the following formula:

F �� G = T (�(F ) �(G)); 8F;G 2 N [�]: (3)

It is an R[�]-distributive Abelian product. We shall call it the ��-product associatedwith �.

In general, the product defined by (3) is not associative, and the following showsthat it is associative only in the trivial case:

PROPOSITION 1. Let � be a star product onRn such that its associated ��-productis associative, then the product �� is the usual pointwise product on N .

Proof. It is easy to see that the map � is an algebra homomorphism: �(FG) =�(F ) �(G), 8F;G 2 N [�]. Hence, the product F �� G, F;G 2 N [�], definedby (3) can be written in the following form:

F �� G = T (�(FG)) =Xr>0

�r�r(FG); (4)

where �0(FG) = FG and �r, r > 1, are R[�]-linear maps on N [�] (whoserestrictions to N are R-linear map on N ).

Using (4), the associativity of the product �� can be formulated asXr>0

�rX

u+v=r

u;v>0

�u(�v(FG)H) =Xr>0

�rX

u+v=r

u;v>0

�u(F�v(GH)); (5)

8F;G;H 2 N [�]. Consider the last equation for F;G;H 2 N . By identifying thecoefficients of the different powers of � on both side, we find for r = 1:

�1(FG)H = F�1(GH); 8F;G;H 2 N: (6)

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Set G = H = 1 in (6), one finds �1(F ) = F�1(1), 8F 2 N . By definition,T (�(1)) = 1 and (4) implies �r(1) = 0, 8r > 1, thus �1(F ) = 0, 8F 2 N .

Now suppose that for some k > 2 we have �i(F ) = 0 for all 1 6 i 6 k,8F 2 N . By equating the coefficients of �k+1 on both sides of (5), we find that�k+1(FG)H = F�k+1(GH); 8F;G;H 2 N , and by the same argument used forshowing that �1(F ) = 0, 8F 2 N , we find �k+1(F ) = 0, 8F 2 N . In conclusion,�r(F ) = 0, 8r > 1, 8F 2 N , and (4) gives F �� G = FG, 8F;G 2 N and thisshows the Proposition. 2

3. Nontriviality and Equivalence

In Zariski quantization, the map �:N [�] ! S which replaces the usual productappearing in the decomposition of some polynomial into irreducible factors by thesymmetric tensor product, is not R[�]-multiplicative [3]. In order to keep asso-ciativity, the map � has to annihilate nonzero powers of �. Also the space N [�]endowed with the Zariski product is not an Abelian algebra, it is only an Abeliansemigroup. Indeed the distributivity with respect to the addition ofN is lacking dueto the factorization into irreducible factors. However, if we replace ‘factorizationinto irreducible factors’ by ‘factorization of monomials into linear factors (andlinear combinations)’ in the definition of Zariski products, then we get an Abelianalgebra structure on N [�]. In the case of the Moyal product, for reasons explainedin the Introduction, this does not lead to a new product on N [�], and we have to gothrough the construction developed in [3]. However, there are other star productsthan those of Moyal and, in general, it is possible to find a generalized Abelianalgebra deformation in a quantum-mechanical framework [4]. The example ofsu(2) will be treated to some extent in Section 4 (in this case the linear factors aregenerators of the Lie algebra su(2), linear functions on su(2)� � R

3 ; they have aquadratic expression in R

6 ).After making precise what we call a quantum-mechanical approach by defining

a deformed Abelian associative product on N [�], we shall be concerned with thepossible definitions of equivalence and triviality of Abelian generalized deforma-tions. Also, we shall discuss their consequences for both ��-products studied inthe present Letter and the Zariski products introduced in [3].

Let us first define an Abelian product in a way similar to what is done in [3],with the main difference that it is R-distributive. Let �:N [�] ! N be the naturalprojection of N [�] onto N . Let �0 = � � �:N [�] ! S , where � is defined by (1),and let eT :S ! N [�] be the restriction to S of the map defined by (2).

DEFINITION 2. Let � be a star product on Rn . We shall call the ��-product (‘sun

product’) associated with �, the product �� on N [�] defined by

F��G = eT (�0(F ) �0(G)); 8F;G 2 N [�]: (7)

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Contrary to the ��-products of Section 2, the ��-products annihilate all nonzeropowers of the deformation parameter �. Clearly, the product �� is Abelian anddistributive with respect to the addition in N [�], but is not R[�]-multiplicative, i.e.,in general (aF )��G 6= a(F��G) for a 2 R[�]. However, it is always associativeas shown by the following lemma.

LEMMA 1. Let � be a star product on Rn . Its associated��-product is associative.Proof. Simply notice that �0(F��G) = �0(F ) �0(G), 8F;G 2 N [�], and

thus

F��(G��H)

= eT (�0(F ) �0(G��H)) = eT (�0(F ) �0(G) �0(H))

= eT (�0(F��G) �0(H)) = (F��G)��H; 8F;G;H 2 N [�]: 2

Hence,N [�] endowed with a��-product defines an Abelian associative generalizeddeformation of the usual product on N . This makes (N [�];��) into an AbelianR-algebra. Let us mention that, in general, a ��-product associated with some starproduct differs from the usual product on N , and the star products which havethe usual product as an associated ��-product are all of the Moyal type [4]. Moreprecisely, on the dual of a Lie algebra, there exists one and only one covariant starproduct whose associated ��-product is the usual product of polynomials.

We shall now be concerned with the question of the equivalence of ��-products.Several notions of equivalence can be adapted to the kind of generalized deforma-tions considered in the present Letter. Let us present a first definition of equivalencewhich is similar to the usual notion of equivalence for Gerstenhaber type deforma-tions for associative algebras. First notice that since the map�0 is a homomorphism,it is easy to find that any ��-product can be written in the form

F��G = FG+Xr>0

�r�r(FG); F;G 2 N; (8)

where �r:N ! N are linear maps (the ‘cochains’ of the ��-product). As forassociative deformations, one way to understand equivalence of ��-products isgiven by the following definition.

DEFINITION 3. Two products �� and �0� are said to be A-equivalent, if there

exists a R[�]-linear (formally invertible) map S� :N [[�]] 7! N [[�]] of the formS� =

Pr>0 �

rSr, where Sr:N 7! N , r > 1, are differential operators andS0 = Id, such that

S�(F��G) = S�(F )�0

�S�(G)��=� ; F;G 2 N:

It is straightforward to check that Definition 3 indeed defines an equivalence relationbetween ��-products. We have the associated notion of triviality given by

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DEFINITION 4. A ��-product associated with some star product is said to bestrongly trivial if it is A-equivalent to the usual product, i.e., there exists a R[�]-linear (formally invertible) map S�:N [[�]] 7! N [[�]] of the formS� =

Pr>0 �

rSr,where Sr:N 7! N , r > 1, are differential operators and S0 = Id, such that

S�(F��G) = S�(F ) � S�(G); 8F;G 2 N; (9)

where � denotes the usual product.

The following shows that, in general, a ��-product is strongly nontrivial.

PROPOSITION 2. A ��-product is strongly trivial if and only if it coincides withthe usual product.

Proof. Let �� be strongly trivial. From S�(1) = 1, we have S�(F��1) =S�(F ), 8F 2N . Since S� is invertible, we get F��1 = F , 8F 2N . Expression(8) of the ��-product implies that �r(F ) = 0, 8r > 1, 8F 2N . Therefore, the ��-product coincides with the usual product. The converse statement being obvious,the Proposition holds. 2

Actually, the treatment done above applies almost straightforwardly to the caseof Zariski quantization yielding that the Zariski product is never strongly trivial. Letus denote by �� the Zariski product constructed out from some star product whichis defined on the algebra A� (the reader is referred to [3] for the definitions of ��andA�; the deformation parameter � was taken to be ~ in the preceding reference).The product �� has also the form given by (8) where now the maps �r are linearmaps on A0, the classical algebra, and the ‘usual’ product has to be interpreted asthe product on the algebra A0. Recall that there are derivations defined on A� [3]and, hence, we have a natural definition of ‘differential operators’ acting onA� . Byan obvious adaptation of Definitions 3 and 4, we get the corresponding definitionsof A-equivalence and strong triviality for Zariski products. Notice that the proof ofProposition 2 applies literally to the case of Zariski products (the differential aspectof the intertwining operator is not involved in the proof). However, by construction,a Zariski product constructed out from some star product can never coincide withthe ‘usual product’ on A0 (of course, except in the trivial situation where the starproduct is the usual product of polynomials on R

n ). This leads to the followingtheorem:

THEOREM 1. Let �� be a Zariski product associated with some star product ��on R

n , then �� is strongly nontrivial whenever �� is not the usual product.

Another natural definition of equivalence for the generalized deformations that onecan consider is given by the following definition:

DEFINITION 5. Two products �� and �0� are said to be B-equivalent, if there

exists a R[�]-linear (formally invertible) map S� :N [[�]] 7! N [[�]] of the form

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S� =P

r>0 �rSr, where Sr:N 7! N , r > 1, are differential operators and

S0 = Id, such that

S�(F��G) = S�(F )�0

�S�(G); F;G 2 N:

Since the ��-product annihilates the parameter �, the preceding definition is equiv-alent to saying thatS�(F��G) = F�0�G,F;G 2 N , holds. The difference betweenthis definition of equivalence and that given by Definition 3 lies in the fact that thelatter is formulated in such a way that the terms involving the powers of the para-meter � introduced by the intertwining operator S� are taken into account, whilethe former definition allows the product operation to annihilate them. Also noticethat two ��-products which are B-equivalent are not necessarily A-equivalent.

The notion of triviality corresponding to B-equivalence (weaker than strongtriviality, cf. Proposition 3, hence the choice of terminology) is:

DEFINITION 6. A ��-product associated with some star product is said to beweakly trivial if it is B-equivalent to the usual product, i.e., there exists a R[�]-linear (formally invertible) map S�:N [[�]] 7! N [[�]] of the formS� =

Pr>0 �

rSr,where Sr:N 7! N , r > 1, are differential operators and S0 = Id, such that

S�(F �G) = F��G; 8F;G 2 N;

where � denotes the usual product.

By adequate modifications, these definitions also apply to the case of Zariskiproducts.

Let us study the consequences of these definitions for the ��-products andthen for the Zariski products. Consider a ��-product whose cochains are given bydifferential operators (e.g., the ��-product of Section 4). Then the ��-product hasthe form F��G = �(�(F �G)), F;G 2 N [�], where � = Id +

Pr>1 �

r�r and the�r’s are differential operators acting onN , and �:N [�] 7! N is the projection ontothe classical part. Note that � is formally invertible, and we have �(F �G) = F��G,F;G 2 N , therefore � intertwines the ��-product with the usual product in thesense of Definition 6. Hence, we have shown the following proposition:

PROPOSITION 3. A ��-product is weakly trivial whenever its cochains are givenby differential operators.

For a Zariski product �� , it is true that the corresponding � intertwines �� with theusual product on A0, but the cochains �r are never given by differential operatorsacting on A0. Hence

COROLLARY 1. A Zariski product is weakly nontrivial (except when the definingstar product is the usual product).

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Remark 1. We may wonder why we restrict the intertwining operators to bedefined by differentiable cochains as, in general, the deformations considered arenot defined by differentiable cochains. This is so because eventually the nontrivi-ality for the Abelian generalized deformations should be linked with the usual(differentiable) Hochschild or Harrison cohomologies.

Remark 2. Of course, the map T �� which defines the Zariski product satisfies

T � �(F ) �� T � �(G) = F �� G = T � �(F �G); 8F;G 2 A0;

where � denotes the product on the classical algebra A0. But the map T � � isneither invertible as a formal series nor acting by differential operators. Hence,the map T � � is not an intertwiner trivializing the Zariski product in the senseof Definition 6. Even if we define S� by taking the R[[�]]-linear extension of therestriction of T �� to A0, we get an invertible map but not an intertwining map (inthe sense of B-equivalence) as S� is not given by differential operators on A0.

To sum up, the kind of Abelian deformations defined by a Zariski productconsidered in [3] are both strongly and weakly nontrivial. However, the exampledeveloped in Section 4 is strongly nontrivial but weakly trivial. We think thatthe above considerations may give some hints for the definition of an appropriatecohomology for Abelian generalized deformations.

4. Example of su(2)

Here we shall study in some details an example of the��-product associated with aninvariant star product on the dual of the su(2)-Lie algebra (� R

3 ). This ��-productis strongly nontrivial in the sense of Definitions 3 and 4. An explicit formulafor that ��-product will be given, which permits us to extend the ��-product toC1(R3 ) (valued in C1(R3)[[�]], whereto it extends trivially since the ��-productannihilates �). Let �M be the Moyal product on R

6 :

F �M G =Xr>0

�rP(r)M (F;G); 8F;G 2 C1(R6 );

where P (r)M is the rth power of the Poisson bracket

PM (F;G) =X

16i63

�@F

@pi

@G

@qi�@F

@qi

@G

@pi

�:

Consider the functions Li(p; q) =P

16j;k63 "ijkpjqk, 1 6 i 6 3, on R6 , where

"ijk is the totally skew-symmetric tensor with "123 = 1. The Li’s realize the Liealgebra su(2) for the Moyal bracket [�; �]M on R

6 .

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For any polynomial F :R3 ! R, the function F (L1; L2; L3) on R6 satisfies

Li �M F

= LiF + �X

16j;k63

"ijkLk@F

@Lj+ �2

0@2

@F

@Li+X

16j63

Lj@2F

@Li@Lj

1A : (10)

We see that Li �M F (L1; L2; L3) is still a polynomial in the variables (L1; L2; L3).It is easily shown that the productF �M G of two polynomialsF;G in (L1; L2; L3)is a polynomial in (L1; L2; L3). Hence, from the Moyal product �M on R

6 , weget a star product on R

3 satisfying (10) for any polynomial F . This star product,that we shall denote by �, is actually an invariant (and covariant) star product onsu(2)� � R

3 .Let �� be the ��-product associated with �. Remark that Li��Lj = LiLj +

2�2�ij , so this star-product does provide quantum terms. In the following, we shallderive an explicit expression for �� . Before stating the main result of this section,we need some combinatorial preliminaries. We shall denote the cochains of thestar product � on R

3 by Cr, i.e., F �G =P

r>0 �rCr(F;G), where

C1(F;G) = P (F;G) �X

16i;j;k63

"ijkLk@F

@Li

@G

@Lj

is the standard Poisson bracket on su(2)� � R3 . We denote by � the Laplacian

operator � =P

16k63 @2=@L2

k.

LEMMA 2. For any n > 1, m > 0 and any indices i1; : : : ; in 2 f1; 2; 3g, we have

(a)X

(i1;:::;in)

@

@Li1(Li2 � � �Lin) = �(Li1 � � �Lin);

(b)X

(i1;:::;in)

Li1�m(Li2 � � �Lin) = (n� 2m)�m(Li1 � � �Lin);

(c)X

(i1;:::;in)

P (Li1 ;�m(Li2 � � �Lin)) = 0;

(d)X

(i1;:::;in)

C2(Li1 ;�m(Li2 � � �Lin)) = (n� 2m)�m+1(Li1 � � �Lin);

whereP

(i1;:::;in)stands for summation over cyclic permutations of (i1; : : : ; in),

and we set Li2 � � �Lin = 1 when n = 1.Proof. Statement (a) is proved by performing the derivations with respect to

the Li’s on both side.

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The following identity is straightforward by induction on k:

�k(LiF ) = Li�k(F ) + 2k�k�1 @F

@Li; 8F 2 N; k > 1; 1 6 i 6 3: (11)

Statement (b) is true for m = 0. For m > 1, by using identity (11), we find that

�m(Li1 � � �Lin) = Li1�m(Li2 � � �Lin) + 2m�m�1 @

@Li1(Li2 � � �Lin):

Summing over cyclic permutations on both sides of the last equation and usingstatement (a) of the Lemma, gives

n�m(Li1 � � �Lin) =X

(i1;:::;in)

Li1�m(Li2 � � �Lin) + 2m�m(Li1 � � �Lin);

which shows that statement (b) is true.In order to show statement (c), we apply (11) to the Poisson bracket P (Li; F )

and find that

�m(P (Li; F ))

=X

16j;k63

"ijkLk@

@Lj�mF + 2m

X16j;k63

"ijk�m�1 @2F

@Lj@Lk:

The first term on the right-hand side is simply P (Li;�mF ), while the second

term vanishes due to the skew-symmetry of "ijk, hence we have found that�m(P (Li; F )) = P (Li;�

mF ), 8F 2 N;m > 0. Setting F = Li2 � � �Lin inthis relation and summing over cyclic permutations yield

X(i1;:::;in)

P (Li1 ;�m(Li2 � � �Lin)) = �m

0@ X

(i1;:::;in)

P (Li1 ; Li2 � � �Lin)

1A ;

whose right-hand side vanishes due to the Leibniz property and skew-symmetry ofthe Poisson bracket. This shows statement (c).

The differential operator D =P

16k63 Lk@=@Lk acting on a homogeneouspolynomial F of degree deg(F ) gives D(F ) = deg(F )F . The second cochain C2

in (10) can be written asC2(Li; F ) = (2+D)(@F=@Li). LetF = �m(Li2 � � �Lin)in the preceding equation and sum over cyclic permutations; this gives

X(i1;:::;in)

C2(Li1 ;�m(Li2 � � �Lin))

= (2 +D)�m

0@ X

(i1;:::;in)

@

@Li1(Li2 � � �Lin)

1A :

ma96-141.tex; 6/02/1997; 8:23; v.5; p.11


The right-hand side of this equation is equal to (2 + D)�m+1(Li1 � � �Lin) bystatement (a) of the Lemma; since �m+1(Li1 � � �Lin) is a homogeneous poly-nomial of degree n � 2m � 2, we have (2 + D)�m+1(Li1 � � �Lin) = (n �

2m)�m+1(Li1 � � �Lin), and this shows statement (d). 2

We are now in a position to prove the central result of this section.

PROPOSITION 4. Let Hn be the subspace of N consisting of homogeneous poly-nomials of degree n. There exist constants a(n; r), n; r > 0, such that the product�� can be written

F��G =Xr>0

�2r�r(FG); 8F;G 2 N;

where �0 is the identity and �r:N ! N , r > 1, are linear maps whose restrictionsto Hn are given by

�rjHn= a(n; r)�r; 8n; r > 0:

Proof. In Section 2, we saw that the product �� can be written in the form

F��G =Xr>0

�r�r(FG); 8F;G 2 N;

where �0 = Id and for r > 1, �r is a linear map on N . Consider the productLi1�� Lin for any ik 2 f1; 2; 3g, 81 6 k 6 n. By definition, it is given by

Li1�� Lin =1n!

X�2Sn

Li�1� � � � � Li�n ;

which can be expressed as

Li1�� Lin =1n

X(i1;:::;in)

Li1 � (Li2�� Lin); (12)

whereP

(i1;:::;in)denotes sum over cyclic permutations. Using the expression (12)

for the product �� and taking into account that Li � F = LiF + �P (Li; F ) +�2C2(Li; F ), 8F 2 N , 1 6 i 6 3, we find that the coefficient of the rth power of� in Li1�� Lin satisfies the following induction relation for k > 2 and n > 1(when n = 1, we set Li2 � � �Lin = 1):

�k(Li1 � � �Lin)

=1n

X(i1;:::;in)

(Li1�k(Li2 � � �Lin)+

+P (Li1 ; �k�1(Li2 � � �Lin)) + C2(Li1 ; �k�2(Li2 � � �Lin))); (13)

ma96-141.tex; 6/02/1997; 8:23; v.5; p.12


and for k = 1 and n > 1:

�1(Li1 � � �Lin) =1n

X(i1;:::;in)

(Li1�1(Li2 � � �Lin) + P (Li1 ; Li2 � � �Lin)): (14)

The second term on the right-hand side of (14) vanishes by skew-symmetry of thePoisson bracket. From 1��1 = 1, we have �1(1) = 0. By induction on n, we easilyfind that �1(Li1 � � �Lin) = 0, 8n > 0, i.e., �1 = 0.

Let Zk, k > 0, be the following property:

�2kjHn= a(n; k)�k; 8n > 0; and �2k+1 = 0; (Zk)

where a(n; k), n; k > 0, is a constant (notice that for n < 2k, �k vanishes on Hn,and in that case the constant a(n; k) can be arbitrary). The result will be provedby induction on k. First, remark that Z0 is true with a(n; 0) = 1, 8n > 0. Supposethat Zk is true for 0 6 k 6 r � 1, for some r > 1. By hypothesis, �2r�1 = 0 and�2r�2jHn

= a(n; r � 1)�r�1, so the induction relation (13) takes the form

�2r(Li1 � � �Lin)

=1n

X(i1;:::;in)

(Li1�2r(Li2 � � �Lin)+

+a(n� 1; r � 1)C2(Li1 ;�r�1(Li2 � � �Lin))); 8n > 1; (15)

and by application of statement (d) of Lemma 2 to the second term on the right-handside, we find that

�2r(Li1 � � �Lin)

=1n

0@ X

(i1;:::;in)

Li1�2r(Li2 � � �Lin)

1A+

+a(n� 1; r � 1)(n� 2r + 2)

n�r(Li1 � � �Lin); 8n > 1; (16)

the case n = 0 is trivially verified by observing that �2r(1) = 0 for r > 1.Now we show by induction on n that any linear map on N which satisfies

along with (1) = 0, the following relation for some given r > 1 and con-stants �n:

(Li1 � � �Lin)

=1n

0@ X

(i1;:::;in)

Li1 (Li2 � � �Lin)

1A+ �n�

r(Li1 � � �Lin);8n > 1; (17)

ma96-141.tex; 6/02/1997; 8:23; v.5; p.13


must be of the form (Li1 � � �Lin) = �n�r(Li1 � � �Lin), i.e. jHn

= �n�r for

some constants �n. Suppose jHm= �m�

r for 0 6 m 6 n�1, then the inductionrelation (17) can be written as

(Li1 � � �Lin)

=�n�1

n

0@ X

(i1;:::;in)

Li1�r(Li2 � � �Lin)

1A+ �n�

r(Li1 � � �Lin);8n > 1;

which by statement (b) of Lemma 2 yields

(Li1 � � �Lin) =

�(n� 2r)

�n�1

n+ �n

��r(Li1 � � �Lin); 8n > 1;

and this shows that must be proportional to �r on every Hn, n > 0 (the casen = 0 being trivially verified when r > 1, since �r(1) = 0).

Let us apply the preceding result to the induction relation (16); we readily findthat �2r(Li1 � � �Lin) = a(n; r)�r(Li1 � � �Lin) and the constant a(n; r) is given by

a(n; r)

=1n((n� 2r)a(n� 1; r) + (n� 2r + 2)a(n� 1; r � 1)); (18)

for 8n > 1. Hence, �2rjHn= a(n; r)�r and in order to complete the proof we only

need to show that �2r+1 = 0. Under the induction hypothesis, we have �2r�1 = 0and we just have shown that �2rjHn

= a(n; r)�r, hence the induction relation (13)for �2r yields 8n > 1

�2r+1(Li1 � � �Lin)

=1n

X(i1;:::;in)

(Li1�2r+1(Li2 � � �Lin)+

+ a(n� 1; r)P (Li1 ;�r(Li2 � � �Lin))):

By statement (c) of Lemma 2, the second term on the right-hand side vanishes andwe are left with

�2r+1(Li1 � � �Lin) =1n

X(i1;:::;in)

Li1�2r+1(Li2 � � �Lin); 8n > 1:

We have �2r+1(1) = 0 and by induction on n we get �2r+1(Li1 � � �Lin) = 0,8n > 0, i.e, �2r+1 = 0. Hence the property Zr is true. 2

The expression of the constants a(n; r) appearing in the statement of Proposi-tion 4 (also, cf. (18)) can be written explicitly with the help of a generating function.Moreover, for n > 2r, the a(n; r)’s are polynomials of degree r in n. This willallow us to find an expression for the product �� in terms of differential operators.

ma96-141.tex; 6/02/1997; 8:23; v.5; p.14


LEMMA 3. For n > 2r, the constants a(n; r) of Proposition 4 are given by

a(n; r) =1n!

�@

@�

�n �(cos(�))�2(tan(�))n�2r

��=0

; n > 2r:

Moreover, there exist polynomials pr, r > 0, of degree r such that for n > 2r wehave a(n; r) = pr(n).

Proof. Let X be any generator of su(2). The �-powers and the ��-powersof X coincide and the �-exponential [1, 2] and the ��-exponential [3] of X areidentical. Here we fix the deformation parameter � to be i~=2, where ~ is real. The�-exponential of X has been explicitly computed in [2]:

exp�

�tX

i~

��Xn>0

1n!

�t

i~

�nX

n

�

= cos�2(t=2) exp�

2Xi~

tan(t=2)�; (19)

for t in a neighborhood of the origin. By Proposition 4 we have (we denote �i~=2by �)

Xn

� =

[n=2]Xr=0

�i~

2

�2r

a(n; r)�rXn

=

[n=2]Xr=0

�i~

2

�2r

a(n; r)n!

(n� 2r)!Xn�2r; (20)

where [n=2] denotes the integer part of n=2.The function �(�; �) = cos�2(�) exp(� tan(�)) is analytic in a neighborhood

of the origin of C 2 and notice that

exp�

�tX

i~

�= exp

�

�tX

i~

�= �

�t

2;

2Xi~

�:

By (20), we have for t in a neighborhood of 0

exp�

�tX

i~

��Xn>0

1n!

�t

i~

�nX

n

�

=Xn>0

[n=2]Xr=0

�t

i~

�n � i~2

�2r a(n; r)

(n� 2r)!Xn�2r

or

�

�t

2;

2Xi~

�=Xn>0

[n=2]Xr=0

a(n; r)

(n� 2r)!

�t

2

�n �2Xi~

�n�2r

;

ma96-141.tex; 6/02/1997; 8:23; v.5; p.15


so (a(n; r))=(n � 2r)!, for n > 2r, appears to be the coefficient of �n�n�2r in theTaylor series around the origin of �(�; �), hence

a(n; r)

=1n!

�@

@�

�n � @

@�

�n�2r �(cos(�))�2(exp� tan(�))

��=�=0

; n > 2r;

or

a(n; r) =1n!

�@

@�

�n �(cos(�))�2(tan(�))n�2r

��=0

; n > 2r:

From the last equality, we see also that a(n; r) is the coefficient of �n in the Taylorseries of cos�2(�)(tan(�))n�2r . Using the Taylor series expansions for cos�1(�)and tan(�):

cos�1(�) =Xn>0

n�2n; tan(�) =

Xn>0

�n�2n+1;

where n = E2n=(2n)! and �n = 22n+2(22n+2�1)B2n+2=(2n+2)!, the constantsEk (resp. Bk) being the Euler (resp. Bernoulli) numbers, we find that:

a(n; r) =X

j1+��+jn�2r+2=rj1;:::;jn�2r+2>0

j1 j2

�j3� � � �jn�2r+2

; n > 2r: (21)

Let b(s; r) = a(s+ 2r; r), s; r > 0, Ak =P

i+j=k

i;j>0 i j , k > 0, and

c(s; k) =X

j1+��+js=kj1;:::;js>0

�j1� � � �js s > 1; k > 0; (22)

and c(0; k) = �0k, k > 0. Notice that �0 = 1, so in particular we have c(s; 0)= 1; s > 0. From (21), we find that b(s; r) =

Prk=0 Ar�kc(s; k) = Ar +Pr

k=1 Ar�kc(s; k). We will show that the c(s; k)’s are polynomials of degree kin s.

For any integer k > 1, we denote by d(k) the number of partitions of k, i.e., thenumber of ways to write k as a sum of strictly positive integers: k = n1 + � � �+np,with n1 > � � � > np. We call p the length of the partition (n1; : : : ; np). Obviouslywe have 1 6 p 6 k. Let B(k; p) be the set of partitions of k of length p. For apartition (n1; : : : ; np) of B(k; p), let mi be the multiplicity with which a giveninteger ai > 0 appears in (n1; : : : ; np), so that k = m1a1 + � � � + mrar anda1 > � � � > ar. To that partition (n1; : : : ; np) we associate a symmetry factor givenby m(n1; : : : ; np) = (m1! � � �mr!)�1.

Let s > 1 be an integer. Let j1; : : : ; js > 0 be integers such that j1 + � � � +js = k. We can associate to the integers j1; : : : ; js, by ordering them, a partition(n1; : : : ; np) of k for some p. Conversely, for a partition (n1; : : : ; np) of length p

ma96-141.tex; 6/02/1997; 8:23; v.5; p.16


of k there are s!(s�p)!m(n1; : : : ; np) ways to associate a set of nonnegative integers

satisfying j1 + � � � + js = k when s > p. Hence, the sum in (22) for s > 1; k > 1can be written as

c(s; k) =kX

p=1

s(s� 1) � � � (s� p+ 1)�

�X

(n1;:::;np)2B(k;p)

m(n1; : : : ; np)�n1� � � �np;

with an obvious interpretation when s < k. Since c(0; k) = 0 for k > 1, thepreceding formula also covers the case s= 0. Also we saw that we have c(s; 0) = 1,s > 0, hence c(s; k), k > 0, is a polynomial in s. The term p = k in the precedingsum corresponds to a polynomial of degree k and since it is the polynomial withmaximal degree in that sum, we conclude that c(s; k) is a polynomial of degreek in s. By retracing the preceding steps, we conclude that there exist polynomialspr, r > 0, of degree r such that for n > 2r we have a(n; r) = pr(n). With theprevious notations, these polynomials are given by p0(n) = 1 and for r > 1:

pr(n) = Ar +rX

p=1

zp;r(n� 2r) � � � (n� 2r � p+ 1); (23)

where

zp;r =rX

k=p

Ar�k

X(n1;:::;np)2B(k;p)

m(n1; : : : ; np)�n1� � � �np : 2

This lemma allows to express the cochains of the ��-product on su(2)� by differ-ential operators and we conclude this section by a theorem which summarizes theconstruction developed here.

THEOREM 2. The��-product associated with the invariant star product on su(2)�

(defined by Equation (10)) admits the following form:

F��G = FG+Xr>1

�2r�r(FG); F;G 2 N;

where �r:N ! N , r > 1, are differential operators given by

�r(F ) =

0@Ar +

rXp=1

zp;rD(D � 1) � � � (D � p+ 1)

1A�r(F ); (24)

for F 2N where D =P

16k63 Lk@=@Lk and where the constants Ar and zp;r aregiven in Lemma 3. This ��-product is strongly (but not weakly) nontrivial and hasa unique extension to F;G 2 C1(R3).

ma96-141.tex; 6/02/1997; 8:23; v.5; p.17


Proof. From Proposition 4, we have �rjHn= a(n; r)�r for n; r > 0. Notice

that �r vanishes on Hn for n < 2r and by Lemma 3 we have for n > 2r,a(n; r) = pr(n), where the pr’s are the polynomials defined by (23). Hence�rjHn

= pr(n)�r. Let D =

P16k63 Lk@=@Lk . Since DjHn

= n and �r mapsHn to Hn�2r for n > 2r and is 0 otherwise, it follows that the restriction of thedifferential operator pr(D � 2r)�2r to Hn coincides with �rjHn

, n > 0, therefore�r = pr(D � 2r)�2r on N . By using the explicit expression for pr(n) given by(23), relation (24) follows. The last statement of the Theorem is straightforward.

5. Concluding Remarks

Spectrality. As indicated in [3], within the framework of generalized deformations,the spectrum of an observable is obtained through its corresponding �-exponential(cf. proof of Lemma 3). Consider the invariant star product on su(2)� (with � =i~=2), Equation (19) gives explicitly the �-exponential of a linear element X , andfrom that one finds that the �-spectrum of X is discrete [2]. For X linear, we havenoticed that

exp�

�tX

i~

�= exp

�

�tX

i~

�;

thus the �-spectrum and the �-spectrum of X coincide. The �-spectrum of X isdiscrete, hence nontrivial; although the ��-product of Section 4 is weakly trivial.The situation changes drastically if we consider, e.g., the square of the total angularmomentumH = 1

2 (L21 +L2

2+L23). Then it is no longer true that the corresponding

spectra coincide.Consider the Zariski product � associated with the invariant star product su(2)�.

The observableH is an irreducible polynomial (over the reals), hence the �-powersand �-powers of H are identical and the �-spectrum of H is nothing but its �-spectrum. The situation is similar for the case of a linear polynomial X on su(2)�.Hence while the ��-product does not give the usual spectrum for the square ofthe total angular momentum (or the rotational kinetic energy), the Zariski productdoes.

Quantized Nambu Bracket. A quantization of the classical Nambu bracket isachieved by replacing the usual product by the ��-product of Section 4. Dueto the properties of the ��-product, it is easy to see that actually the quantizedNambu bracket is given by

[F;G;H]��= eT � �0(fF;G;Hg); F;G;H 2 C1(R3);

where fF;G;Hg denotes the classical Nambu bracket on R3 , i.e., the Jacobian.

Though the Leibniz rule is not satisfied for the ��-product, this quantized Nambubracket does satisfy the Fundamental Identity. Indeed only the weaker form of the

ma96-141.tex; 6/02/1997; 8:23; v.5; p.18


Leibniz rule

F��

�@

@Li(G��H)�G��

@H

@Li�

@G

@Li��H

�= 0;

is required to ensure that the Fundamental Identity is verified by the quantizedNambu bracket.

Finally let us mention that, as for the ��-product case, the quantized Nambubracket is strongly (but not weakly) nontrivial.

Acknowledgements

The authors are indebted to D. Sternheimer for very useful discussions and carefulreadings of the manuscript. We also thank L. Takhtajan and Ph. Bonneau fortheir useful remarks. G.D. wishes to thank H. Araki and I. Ojima for wonderfulhospitality at RIMS. This work benefited from scientific contacts supported by theEU Network Contract ERBCHRXCT 940701.

References

1. Bayen, F., Flato, M., Frønsdal, C., Lichnerowicz, A. and Sternheimer, D.: Deformation theory andquantization: I. Deformations of symplectic structures, Ann. Phys. 111 (1978), 61–110.

2. Bayen, F., Flato, M., Frønsdal, C., Lichnerowicz, A. and Sternheimer, D.: Deformation theory andquantization: II. Physical applications, Ann. Phys. 111 (1978), 111–151.

3. Dito, G., Flato, M., Sternheimer, D. and Takhtajan, L.: Deformation quantization and Nambumechanics, to appear in Comm. Math. Phys. (1996).

4. Dito, G.: In preparation.5. Flato, M. and Frønsdal, C.: Unpublished (1992).6. Nambu, Y.: Generalized Hamiltonian dynamics, Phys. Rev. D 7 (1973), 2405–2412.7. Takhtajan, L.: On foundation of the generalized Nambu mechanics, Comm. Math. Phys. 160

(1994), 295–315.

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