algebras - istituto nazionale di fisica nuclearetheory.fi.infn.it/casalbuoni/report.pdf · algebras...

65

Upload: hanguyet

Post on 11-Apr-2018

222 views

Category:

Documents


3 download

TRANSCRIPT

Page 1: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Notes on Algebras� Derivations

and Integrals

Roberto Casalbuoni

Dipartimento di Fisica� Universit�a di Firenze

I�N�F�N� Sezione di Firenze

Page 2: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Contents

Index � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� Introduction �

��� Motivations � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���� The algebra of functions � � � � � � � � � � � � � � � � � � � � � � � � � �

� Algebras �

��� Algebras � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���� Non existence of the C matrix � � � � � � � � � � � � � � � � � � � � � � ����� Algebras with involution � � � � � � � � � � � � � � � � � � � � � � � � � ����� Derivations � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

����� Derivations on associative algebras � � � � � � � � � � � � � � � ����� Integration over a subalgebra � � � � � � � � � � � � � � � � � � � � � � ���� Change of variables � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� Examples of Associative Self�Conjugated Algebras ��

��� The Grassmann algebra � � � � � � � � � � � � � � � � � � � � � � � � � ����� The Paragrassmann algebra � � � � � � � � � � � � � � � � � � � � � � � ��

����� Derivations on a Paragrassmann algebra � � � � � � � � � � � � ����� The algebra of quaternions � � � � � � � � � � � � � � � � � � � � � � � � ����� The algebra of matrices � � � � � � � � � � � � � � � � � � � � � � � � � � ��

����� The subalgebra AN�� � � � � � � � � � � � � � � � � � � � � � � � ������ Paragrassmann algebras as subalgebras of Ap�� � � � � � � � � ��

��� Projective group algebras � � � � � � � � � � � � � � � � � � � � � � � � � ������� What is the meaning of the algebraic integration� � � � � � � � ������� The case of abelian groups � � � � � � � � � � � � � � � � � � � � ������ The example of the algebra on the circle � � � � � � � � � � � � ��

� Examples of Associative Non Self�Conjugated Algebras ��

��� The algebra of the bosonic oscillator � � � � � � � � � � � � � � � � � � ����� The q oscillator � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Examples of Nonassociative Self�Conjugated Algebras ��

��� Jordan algebras of symmetric bilinear forms � � � � � � � � � � � � � � ����� The algebra of octonions � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 3: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

A �

A�� Jordan algebras � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Page 4: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Chapter �

Introduction

��� Motivations

The very idea of supersymmetry leads to the possibility of extending ordinary clas sical mechanics to more general cases in which ordinary con�guration variables livetogether with Grassmann variables� More recently the idea of extending classicalmechanics to more general situations has been further emphasized with the intro duction of quantum groups� non commutative geometry� etc� In order to quantizethese general theories� one can try two ways� i� the canonical formalism� ii� thepath integral quantization� In refs� ��� �� classical theories involving Grassmannvariables were quantized by using the canonical formalism� But in this case� alsothe second possibility can be easily realized by using the Berezin�s rule for integrat ing over a Grassmann algebra ���� It would be desirable to have a way to performthe quantization of theories de�ned in a general algebraic setting� In this paper wewill make a �rst step toward this construction� that is we will give general rulesallowing the possibility of integrating over a given algebra� Given these rules� thenext step would be the de�nition of the path integral� In order to de�ne the integra tion rules we will need some guiding principle� So let us start by reviewing how theintegration over Grassmann variables come about� The standard argument for theBerezin�s rule is translational invariance� In fact� this guarantees the validity of thequantum action principle� However� this requirement seems to be too technical andwe would rather prefer to rely on some more physical argument� as the one whichis automatically satis�ed by the path integral representation of an amplitude� thatis the combination law for probability amplitudes� This is a simple consequence ofthe factorization properties of the functional measure and of the additivity of theaction� In turn� these properties follow in a direct way from the very constructionof the path integral starting from the ordinary quantum mechanics� We recall thatthe construction consists in the computation of the matrix element hqf � tf jqi� tii��ti � tf� by inserting the completeness relationZ

dq jq� tihq� tj � � �����

Page 5: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

inside the matrix element at the intermediate times ta �ti � ta � tf � a � �� � � � � N��and taking the limit N � � �for sake of simplicity we consider here the quantummechanical case of a single degree of freedom�� The relevant information leadingto the composition law is nothing but the completeness relation ������ Thereforewe will assume the completeness as the basic principle to use in order to de�ne theintegration rules over a generic algebra� In this paper we will limit our task to theconstruction of the integration rules� and we will not do any attempt to constructthe functional integral in the general case� The extension of the relation ����� to acon�guration space di�erent from the usual one is far from being trivial� However�we can use an approach that has been largely used in the study of non commutativegeometry ��� and of quantum groups ���� The approach starts from the observationthat in the normal case one can reconstruct a space from the algebra of its functions �Giving this fact� one lifts all the necessary properties in the function space and avoidsto work on the space itself� In this way one is able to deal with cases in which noconcrete realization of the space itself exists� We will see in Section � how to extendthe relation ����� to the algebra of functions� In Section � we will generalize theconsiderations of Section � to the case of an arbitrary algebra� In Section � we willdiscuss numerous examples of our procedure� The approach to the integration onthe Grassmann algebra� starting from the requirement of completeness was discussedlong ago by Martin ����

��� The algebra of functions

Let us consider a quantum dynamical system and an operator having a completeset of eigenfunctions� For instance one can consider a one dimensional free particle�The hamiltonian eigenfunctions are

�k�x� ��p��

exp ��ikx� �����

Or we can consider the orbital angular momentum� in which case the eigenfunctionsare the spherical harmonics Y m

� ���� In general the eigenfunctions satisfy orthogo nality relations Z

��n�x��m�x� dx � �nm �����

�we will not distinguish here between discrete and continuum spectrum�� However�n�x� is nothing but the representative in the hxj basis of the eigenkets jni of thehamiltonian

�n�x� � hxjni �����

Therefore the eq� ����� reads

Zhnjxihxjmi dx � �nm �����

Page 6: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

which is equivalent to say that the jxi states form a complete set and that jni andjmi are orthogonal� But this means that we can implement the completeness inthe jxi space by means of the orthogonality relation obeyed by the eigenfunctionsde�ned over this space� On the other side� given this equation� and the completenessrelation for the set fj�nig� we can reconstruct the completeness in the original spaceR�� that is the integration over the line� Now� we can translate the completeness ofthe set fj�nig� in the following two statements

�� The set of functions f�n�x�g span a vector space�

�� The product �n�x��m�x� can be expressed as a linear combination of thefunctions �n�x�� since the set f�n�x�g is complete�

All this amounts to say that the set f�n�x�g is a basis of an algebra� The productrules for the eigenfunctions are

�m�x��n�x� �Xp

cnmp�p�x� �����

withcnmp �

Z�n�x��m�x���p�x� dx ����

For instance� in the case of the free particle

ckk�k�� ��p��

��k � k� � k��� �����

In the case of the angular momentum one has the product formula ��

Y m���

���Y m���

��� ������X

L�j�����j

�LXM��L

����� � ������ � ��

����L � ��

� h������jL�ih����m�m�jLMiY ML ��� ����

where hj�j�m�m�jJMi are the Clebsch Gordan coe�cients� A set of eigenfunctionscan then be considered as a basis of the algebra ������ with structure constants givenby ����� Any function can be expanded in terms of the complete set f�n�x�g� andtherefore it will be convenient� for the future� to introduce the following ket madeup in terms of elements of the set f�n�x�g

j�i �

�BBBBB�

���x����x�� � �

�n�x�� � �

�CCCCCA ������

A function f�x� such thatf�x� �

Xn

an�n�x� ������

Page 7: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

can be represented asf�x� � haj�i ������

wherehaj � �a�� a�� � � � � an� � � �� ������

To write the orthogonality relation in terms of this new formalism it is convenientto realize the complex conjugation as a linear operation on the set f�n�x�g� In fact�due to the completeness� ��n�x� itself can be expanded in terms of �n�x�

��n�x� �Xn

Cnm�m�x� ������

orj��i � Cj�i ������

De�ning a bra as the transposed of the ket j�ih�j � ����x�� ���x�� � � � �x�� �n�x�� � � �� ������

the orthogonality relation becomesZj��ih�j dx �

ZCj�ih�j dx � � �����

Notice that by taking the complex conjugate of eq� ������� we get

C�C � � ������

The relation ����� makes reference only to the elements of the algebra of functionsand it is the key element in order to de�ne the integration rules on the algebra� Infact� we can now use the algebra product to reduce the expression ����� to a linearform

�nm �X�

Z�n�x����x�C�m dx �

X��p

cn�pC�m

Z�p�x� dx �����

If the set of equations

Xp

Anmp

Z�p�x� dx � �nm� Anmp �

X�

cn�pC�m ������

has a solution forR�p�x� dx� then we will be able to de�ne the integration over all

the algebra� by linearity� We will show in the following that indeed a solution existsfor many interesting cases� For instance a solution always exists� if the constantfunction is in the set f�p�x�g� Let us show what we get for the free particle� Thematrix C is easily obtained by noticing that�

�p��

exp��ikx�

���

�p��

exp�ikx�

�Z

dk���k � k���p��

exp��ik�x� ������

Page 8: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

and thereforeCkk� � ��k � k�� ������

It follows

Akk�k�� �Z

dq ��k� � q��p��

��q � k � k��� ��p��

��k � k� � k��� ������

from which

��k � k�� �Z

dk��Z

Akk�k���k���x�dx �Z �

��exp��i�k � k��x�dx ������

This example is almost trivial� but it shows how� given the structure constants ofthe algebra� the property of the exponential of being the Fourier transform of thedelta function follows automatically from the formalism� In fact� what we havereally done it has been to dene the integration rules in the x space by usingonly the algebraic properties of the exponential� As a result� our integration rulesrequire that the integral of an exponential is a delta function� One can performsimilar steps in the case of the spherical harmonics� where the C matrix is given by

C���m������m�� � ����m������m��m� ������

and then using the constant function Y �� � ��

p��� in the completeness relation�

The procedure we have outlined here is the one that we will generalize in the nextSection to arbitrary algebras� Before doing that we will consider the possibility ofa further generalization� In the usual path integral formalism sometimes one makesuse of the coherent states instead of the position operator eigenstates� In this casethe basis in which one considers the wave functions is a basis of eigenfunctions of anon hermitian operator

��z� � h�jzi ������

withajzi � jziz �����

The wave functions of this type close an algebra� as hz�j��i do� But this time thetwo types of eigenfunctions are not connected by any linear operation� In fact� thecompleteness relation is de�ned on the direct product of the two algebras

Z dz�dz

��iexp��z�z�jzihz�j � � ������

Therefore� in similar situations� we will not de�ne the integration over the originalalgebra� but rather on the algebra obtained by the tensor product of the algebratimes a copy� The copy corresponds to the complex conjugated functions of theprevious example�

Page 9: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Chapter �

Algebras

��� Algebras

We recall here some of the concepts introduced in ���� in order to de�ne the inte gration rules over a generic algebra� We start by considering an algebra A givenby n � � basis elements xi� with i � �� �� � � �n �we do not exclude the possibility ofn��� or of a continuous index�� We assume the multiplication rules

xixj � fijkxk �����

with the usual convention of sum over the repeated indices� For the future ma nipulations it is convenient to organize the basis elements xi of the algebra in aket

jxi �

�BBBBBBBB�

x�x�

��xn

�CCCCCCCCA

�����

or in the corresponding bra

hxj � � x�� x�� � � � xn � �����

Important tools for the study of a generic algebra are the right and left multipli�

cation algebras� We de�ne the associated matrices by

Rijxi � jxixi� hxjLi � xihxj �����

For a generic element a �P

i aixi of the algebra we have Ra �P

i aiRi� and a similarequation for the left multiplication� In the following we will use also

LTi jxi � xijxi �����

Page 10: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

The matrix elements of Ri and Li are obtained from their de�nition

�Ri�jk � fjik� �Li�jk � fikj �����

The algebra is completely characterized by the structure constants� The matricesRi and Li are just a convenient way of encoding their properties� For instance� inthe case of associative algebras one has

xi�xjxk� � �xixj�xk ����

implying the following relations �equivalent one with the other�

RiRj � fijkRk� LiLj � fijkLk� �Ri� LTj � � � �����

The �rst two say that Ri and Li are linear representations of the algebra� called theregular representations� The third that the right and left multiplications commutefor associative algebras� In this paper we will be interested in algebras with identity�and such that there exists a matrix C� satisfying

Li � CRiC�� ����

We will call these algebras self conjugated� In the case of associative algebras�the condition ���� says that the regular representations �see eq� ������ spannedby Li and Ri are equivalent� Therefore� the non existence of the matrix C boilsdown two the possibility that the associative algebra admits inequivalent regularrepresentations� This happens� for instance� in the case of the bosonic algebra ����In all the examples we will consider here� the C matrix turns out to be symmetric

CT � C ������

This condition of symmetry can be interpreted in terms of the opposite algebra AD�de�ned by

xDi xDj � fjikx

Dk ������

The left and right multiplication in the dual algebra are related to those in A by

RDi � LT

i � LDi � RT

i ������

Therefore� in the associative case� the matrices LTi are a representation of the dual

algebraLTi L

Tj jxi � xjxijxi � fjikL

Tk jxi ������

We see that the property CT � C implies that the relation ���� holds also for theright and left multiplication in the opposite algebra

LDi � CRD

i C�� ������

Page 11: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

In the case of associative algebras� the requirement of existence of an identity is nota strong one� because we can always extend the given algebra to another associativealgebra with identity� In fact� let us call F the �eld over which the algebra is de�ned�usually F is the �eld of real or complex numbers�� Then� the extension of A �callit A��� de�ned by the pairs

��� a� � A�� � � F� a � A ������

with product rule��� a��� b� � ��� �a � b � ab� ������

is an associative algebra with identity given by

I � ��� �� �����

Of course� this is the same as adding to any element of A a term proportional tothe identity� that is

�I � a ������

and de�ning the multiplication by distributivity� An extension of this type existsalso for many other algebras� but not for all� For instance� in the case of a Liealgebra one cannot add an identity with respect to the Lie product �since I� � ���For self conjugated algebras� Li has an eigenket given by

LijCxi � CRijxi � jCxixi� jCxi � Cjxi �����

as it follows from ���� and ������ Then� as explained in the Introduction� we de�nethe integration for a self conjugated algebra by the formulaZ

�x�jCxihxj � � ������

where � is the identity in the space of the linear mappings on the algebra� Incomponents the previous de�nition meansZ

�x�Cijxjxk � Cijfjkp

Z�x�

xp � �ik ������

This equation is meaningful only if it is possible to invert it in terms ofR�x� xp� This

is indeed the case if A is an algebra with identity �say x� � I� ���� because by takingxk � I in eq� ������� we get Z

�x�xj � �C���j� ������

We see now the reason for requiring the condition ����� In fact it ensures that thevalue ������ of the integral of an element of the basis of the algebra gives the solutionto the equation ������� In fact we haveZ

�x�Cijxjxk � CijfjkpC

��p� � �CRkC

���i� � �Lk�i� � fk�i � �ik ������

��

Page 12: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

as it follows from xkx� � xk� Notice that if C is symmetric we can write theintegration also as Z

�x�jxihCxj � � ������

which is the form we would have obtained if we had started with the same assump tions but with the transposed version of eq� ������ We will de�ne an arbitraryfunction on the algebra by

f�x� �Xi

fixi � hxjfi ������

and its conjugated asf ��x� �

Xij

�fiCijxj � hf jCxi ������

where

jfi �

�BBBBB�

f�f���xn

�CCCCCA � hf j � � �f� �f� � � � �fn � �����

where �fi is the complex conjugated of the coe�cient fi belonging to the �eld jC�Then a scalar product on the algebra is given by

hf jgi �Z�x�hf jCxihxjgi �

Z�x�

f ��x�g�x� �Xi

�figi ������

��� Non existence of the C matrix

Consider now the case in which the C matrix does not exist� For associative alge bras this happens when the left and right multiplications span inequivalent regularrepresentations� In this case� let us take an isomorphic copy of A� say A�

x�ix�j � fijkx

�k �����

andRijx�i � jx�ix�i � hxjLi � xihxij ������

with jx�ii � x�i � De�ne the integration over the direct product A�A�

Z�x�x��

jx�ihxj � � ������

or Z�x�x��

x�ixj � �ij ������

giving rise to the scalar product

hf jgi �Z�x�x��

�f�x��g�x� �Xi

�figi ������

��

Page 13: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

��� Algebras with involution

In some case� as for the toroidal algebras ��� the matrix C turns out to de�ne amapping which is an involution of the algebra� Let us consider the property of theinvolution on a given algebra A� An involution is a linear mapping � � A � A� suchthat

�x��� � x� �xy�� � y�x�� x� y � A ������

Furthermore� if the de�nition �eld of the algebra is jC� the involution acts as thecomplex conjugation on the �eld itself� Given a basis fxig of the algebra� the invo lution can be expressed in terms of a matrix C such that

x�i � xjCji ������

The eqs� ������ imply�x�i �

� � x�jC�ji � xkCkjC

�ji ������

from whichCC� � � �����

From the product property applied to the equality

Rijxi � jxixi ������

we get�Rijxi�� � hx�jRy

i � hxjCRyi � �jxixi�� � x�i hx�j � x�i hxjC �����

and thereforehxjCRy

iC�� � xjCjihxj � hxjLjCji ������

that isCRy

iC�� � LjCji ������

or alsoCRy

xiC�� � Lx�i

������

If Ri and Li are � representations� that is

Ryxi

� Rx�i� RxjCji ������

we obtainCRy

xiC�� � CRx�i

C�� � Lx�i������

Since the involution is non singular� we get

CRiC�� � Li ������

and comparing with the adjoint of eq� ������� we see that C is a unitary matrixwhich� from eq� ������ implies CT � C� Therefore we have the theorem�

��

Page 14: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Given an associative algebra with involution� if the right and left multiplicationsare ��representations� then the algebra is self�conjugated�

In this case our integration is a state in the Connes terminology ����If the C matrix is an involution we can write the integration as

Z�x�jxihx�j �

Z�x�jx�ihxj � � ������

��� Derivations

We will discuss now the derivations on algebras with identity� Recall that a deriva tion is a linear mapping on the algebra satisfying

D�ab� � �Da�b � a�Db�� a� b � A �����

We de�ne the action of D on the basis elements in terms of its representative matrix�d�

Dxi � dijxj ������

If D is a derivation� thenS � exp��D� �����

is an automorphism of the algebra� In fact� it is easily proved that

exp��D��ab� � �exp��D�a��exp��D�b� ������

On the contrary� if S��� is an automorphism depending on the continuous parameter�� then from ������� the following equation de�nes a derivation

D � lim���

S���� �

�������

In our formalism the automorphisms play a particular role� In fact� from eq� ������we get

S����jxixi� � �S���jxi��S���xi� ������

and therefore

Ri�S���jxi� � S����Rijxi� � S����jxixi� � �S���jxi��S���xi� ������

meaning that S���jxi is an eigenvector of Ri with eigenvalue S���xi� This equationshows that the basis x�i � S���xi satis�es an algebra with the same structure con stants as those of the basis xi� Therefore the matrices Ri and Li constructed in thetwo basis� and as a consequence the C matrix� are identical� In other words� ourformulation is invariant under automorphisms of the algebra �of course this is not

��

Page 15: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

true for a generic change of basis�� The previous equation can be rewritten in termsof the matrix s��� of the automorphism S���� as

Ri �s���jxi� � �s���jxi� sijxj � sijs���Rjjxi ������

ors�����Ris��� � RS���x ������

If the algebra has an identity element� I� �say x� � I�� then

Dx� � � ������

and thereforeDx� � d�ixi � � � d�i � � �����

We will prove now some properties of the derivations� First of all� from the basicde�ning equation ����� we get

Ridjxi � RiDjxi � D�Rijxi � D�jxixi�� djxixi � jxiDxi � dRijxi� RDxijxi ������

or�Ri� d � � RDxi �����

which is nothing but the in�nitesimal version of eq� ������� From the integrationrules for a self conjugated algebra with identity we getZ

�x�Dxi � dij

Z�x�

xj � dij�C���j� ������

Showing that in order that the derivation D satis�es the integration by parts rulefor any function� f�x�� on the algebraZ

�x�D�f�x�� � � ������

the necessary and su�cient condition is

dij�C���j� � � ������

implying that the d matrix must be singular and have �C���j� as a null eigenvector�Next we show that� if a derivation satis�es the integration by part formula �������

then the matrix of related automorphism S��� � exp��D� obeys the equation

Cs���C�� � sT����� ������

and it leaves invariant the measure of integration� The converse of this theorem isalso true� Let us start assuming that D satis�es eq� ������� then

� �Z�x�

D�Cjxihxj� �Z�x�

Cdjxihxj�Z�x�

CjxihDxj� CdC�� � dT ������

��

Page 16: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

that isCdC�� � �dT ������

The previous expression can be exponentiated obtaining

C exp��d�C�� � exp���dT � ������

from which the equation ������ follows� for s��� � exp��d�� To show the invarianceof the measure� let us consider the following identity

� �Z�x�

sT��jCxihxjsT �Z�x�

CsjxihxjsT �Z�x�

CjSxihSxj �Z�x�

Cjx�ihx�j �����

where x� � Sx� and we have used eq� ������� For any automorphism of the algebrawe have Z

�x��jCx�ihx�j � � ������

since the numerical values of the matrices Ri and Li� and consequently the C matrix�are left invariant� Comparing eqs� ����� and ������ we get

Z�x��

�Z�x�

�����

On the contrary� if the measure is invariant under an automorphism of the algebra�the chain of equalities

� �Z�x��jCx�ihx�j �

Z�x�jCx�ihx�j �

Z�x�

CsjxihxjsT � CsC��sT �����

implies eq� ������� together with its in�nitesimal version eq� ������� From this �seethe derivation in �������� we get

� �Z�x�

D�Cijxjxk� �����

and by taking xk � I� Z�x�

Dxi � � �����

for any basis element of the algebra� Therefore we have proven the following theorem�

If a derivation D satis�es the integration by part rule� eq� ������ the integration isinvariant under the related automorphism exp ��D�� On the contrary� if the integra�tion is invariant under a continuous automorphism� exp ��D�� the related derivation�D� satis�es ������

This theorem generalizes the classical result about the Lebesgue integral relatingthe invariance under translations of the measure and the integration by parts for mula�

��

Page 17: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

����� Derivations on associative algebras

Next we will show that� always in the case of an associative self conjugated algebra�A� with identity� there exists a set of automorphisms such that the measure of inte gration is invariant� These are the so called inner derivations� that is derivationssuch that

D � L�A� �����

where L�A� is the Lie multiplication algebra associated to A� L�A� is de�ned inthe following way� start with the linear space of left and right multiplications andde�ne

M� � MR �MLT �����

that is the space generated by the vectors

Ra � LTb � a� b � A �����

Then

L�A� ��Xi��

Mi �����

where the spaces Mi are de�ned by induction

Mi�� � �M��Mi� ����

Therefore L�A� is de�ned in terms of all the multiple commutators of the elementsgiven in ������

It is not di�cult to prove that for a Lie algebra� L�A� coincides with the adjointrepresentation ����� We will prove now an analogous result for associative algebraswith identity� That is that L�A� coincides with the adjoint representation of the Liealgebra associated to A �the Lie algebra generated by �a� b� � ab� ba� for a� b � A��The proof can be found� for example� in ref� ����� but for completeness we will repeatit here� From the associativity conditions ������ and ������ one gets

�Ra � LTb � Rc � LT

d � � M�� a� b� c� d � A �����

or�M��M�� M� ����

showing thatL�A� � M� � MR �MLT ������

Therefore the matrix associated to an inner derivation of an associative algebra mustbe of the form

d � Ra � LTb ������

We have now to require that this indeed a derivation� that is that eq� ����� holds�We start evaluating

�Rc� d� � �Rc� Ra � LTb � � R�c�a ������

��

Page 18: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

where we have used the fact that the right multiplications form a representation ofthe algebra and that right and left multiplications commute� Then comparing with

RDc � Rca�cb ������

we see that the two agree for b � �a� Then we get

Dxi � xia� axi � ��a� xi� � ��adj a�ijxj ������

This shows indeed that the inner derivations span the adjoint representation of theLie algebra associated to A�

We can now proof the following theorem�

For an associative self�conjugated algebra with identity� such that CT � C� themeasure of integration is invariant under the automorphisms generated by the innerderivations� or� equivalently� the inner derivations satisfy the rule of integration byparts�

In fact� this follows because the inner derivations satisfy eq� ������

CdC�� � C�Ra � LTa �C�� � La � �CT��

LaCT �T � La �RT

a � �dT ������

��� Integration over a subalgebra

Let us start with a self conjugated algebra A with generators xi� i � �� � � � � n� Letus further suppose that A has a self conjugated sub algebra B with generators y��with � � �� � � � � m� m � n� As a vector space the algebra A can be decomposed as

A � B � C ������

The vector space C is generated by vectors va� with a � �� � � �n �m� Since B is asubalgebra we have multiplication rules

y�y� � f���y�

vay� � fa��y� � fa�bvb

vavb � fabcvc � fab�y� �����

By de�nition the integration is de�ned both inA and in B� Our aim is to reconstructthe integration over B as an integration over A with a convenient measure� To thisend� let us consider the matrix S which realizes the change of basis from xi to�y�� va�� that is

y� � S�ixi� va � Saixi ������

This matrix is invertible by hypothesis� and we can reconstruct the original basis as

xi � �S���i�y� � �S���iava �����

Page 19: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

To reconstruct the integration over B in terms of an integration over A� we willconstruct a function on the algebra

P � pixi �����

such that Z�A�

vaP � ��Z�A�

y�P �Z�B�

y� �����

These are equivalent to require

Z�A�AP �

Z�A�BP �

Z�B�

B �����

These are n� � conditions over the n� � unknown pi� We will see immediately thatthere is one and only one solution to the problem� In fact� by using the matrix Swe can make more explicit the previous equations by writing

� �Z�A�

vaP � Saipj

Z�A�

xixj �����

and recalling that the general rule for integrating over a self conjugated algebra Ais Z

�A��CA�ijxjxk � �ik �����

or Z�A�

xixj � �C��A �ij �����

with CA the matrix realizing the equivalence between right and left multiplicationsof the algebra A

Li � CARiC��A �����

Therefore� we get� � Saipj�C

��A �ij ����

That is�SC��

A �ajpj � � �����

and in analogous way

�SC��A ��jpj �

Z�B�

y� ����

from which we obtain�SC��

A ��jpj � �C��B ��� �������

Since both S and C are invertible� the problem has a unique solution given by

pi � �CAS���i��C��

B ��� �������

��

Page 20: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

��� Change of variables

Consider again a self conjugated algebra and the following linear change of variables

x�i � Sijxj �������

The integration rules with respect to the new variables are

Zx�x�ix

�j � �C ����ij �������

where C � satis�esL�i � C �R�

iC��� �������

and the right and left multiplications in the new basis are related to the ones in theold basis in the following manner� From

R�ijx�i � jx�ix�i �������

we getR�iSjxi � SjxiSijxj � SijSRjjxi �������

orR�i � SijSRjS

�� ������

In analogous way� fromhx�jL�i � x�ihx�j �������

we gethxjSTLi � SijxjhxjST � SijhxjLjS

T ������

that isL�i � SijS

T��LjST �������

In the new basis the R and L representations are still equivalent

L�i � C �R�iC

����������

implyingST��LiS

T � C �SRiS��C ���

�������

orLi �

�STC �S

Ri

�STC �S

���������

Therefore we must have �Li � CRiC���

C � STC �SA �������

with A invertible and�Ri� A� � � �������

Page 21: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

We get

�C ����ij �Zx�x�ix

�j �

Zx�SilSjmxlxm �������

from which Zx�xlxm � �S��C ���ST��

�lm ������

and in particular Zx�xi � �S��C ���ST��

�i� �������

The result can also be expressed in terms of the matrix A de�ned in eq� �������

A � S��C ���ST��C ������

obtaining Zx�xixj � �AC���ij �������

��

Page 22: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Chapter �

Examples of Associative

Self�Conjugated Algebras

��� The Grassmann algebra

We will discuss now the case of the Grassmann algebra G�� with generators �� � suchthat � � �� The multiplication rules are

ij � i�j� i� j� i � j � �� � �����

and zero otherwise �see Table ���

� � �

Table �� Multiplication table for the Grassmann algebra G��

From the multiplication rules we get the structure constants

fijk � �i�j�k� i� j� k � �� � �����

from which the explicit expressions for the matrices Ri and Li follow

�R��ij � fi�j � �i�j �

� �� �

�R��ij � fi�j � �i���j �

� �� �

�L��ij � f�ji � �i�j �

� �� �

�L��ij � f�ji � �i�j�� �

� �� �

������

��

Page 23: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Notice that R� and L� are nothing but the ordinary annihilation and creation Fermioperators with respect to the vacuum state j�i � ��� ��� The C matrix exists and itis given by

�C�ij � �i�j�� �

� �� �

������

The ket and the bra eigenvectors of Ri are

ji �

�� hj � �� �� �����

and the completeness reads

ZG�jihj �

ZG�

��

��

� �� �

������

or ZG�i��j � �i�j ����

which means ZG�

� � ��ZG�

� � �����

The case of a Grassmann algebra Gn� which consists of �n elements obtained byn anticommuting generators �� �� � � � � n� the identity� �� and by all their products�can be treated in a very similar way� In fact� this algebra can be obtained bytaking a convenient tensor product of n Grassmann algebras G�� which means thatthe eigenvectors of the algebra of the left and right multiplications are obtained bytensor product of the eigenvectors of eq� ������ The integration rules extended bythe tensor product give Z

Gnnn�� � � � � � � ����

and zero for all the other cases� which is equivalent to require for each copy of G�the equations ������ It is worth to mention the case of the Grassmann algebra G�because it can be obtained by tensor product of G� times a copy G�� � Then we canapply our second method of getting the integration rules and show that they leadto the same result with a convenient interpretation of the measure� The algebra G�is generated by �� �� An involution of the algebra is given by the mapping

� � � � � ������

with the further rule that by taking the � of a product one has to exchange theorder of the factors� It will be convenient to put � � � � � �� This allows us toconsider G� as G� �G�� � �G����� Then the ket and bra eigenvectors of left and rightmultiplication in G� and G�� respectively are given by

hj � ��� � j�i �

��

�������

��

Page 24: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

withRij�i � j�i�i� hjLi � ihj ������

The completeness relation reads

Z�G����

j�ihj �Z�G����

� � �

��

� �� �

�������

This implies

Z�G����

� �Z�G����

� � �Z�G����

�Z�G����

� � � ������

These relations are equivalent to the integration over G� if we do the followingidenti�cation Z

�G�����ZG�

exp��� ������

The origin of this factor can be traced back to the fact that we have

hj�i � � � � � exp���� ������

��� The Paragrassmann algebra

We will discuss now the case of a paragrassmann algebra of order p� Gp� � with gen erators �� and � such that p�� � �� The multiplication rules are de�ned by

ij � i�j� i� j� i � j � �� � � � � p �����

and zero otherwise �see Table ���

� � p�� p

� � � p�� p

� � p �� � � � � �

p�� p�� p � � �p p � � � �

Table �� Multiplication table for the paragrassmann algebra Gp� �

From the multiplication rules we get the structure constants

fijk � �i�j�k� i� j� k � �� �� � � � � p ������

��

Page 25: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

from which we obtain the following expressions for the matrices Ri and Li�

�Ri�jk � �i�j�k� �Li�jk � �i�k�j� i� j� k � �� � � � � � p �����

The C matrix exists and it is given by

�C�ij � �i�j�p ������

In fact

�CRiC���lq � �l�m�p�i�m�n�n�q�p � �i�p�l�p�q � �i�q�l � �Li�lq ������

The ket and the bra eigenvectors of Li are given by

Cji �

�BBB�

p

p��

��

�CCCA � hj � ��� � � � � p� ������

and the completeness reads ZGp�

p�ij � �ij ������

which means ZGp�

� �ZGp�

�ZGp�

p�� � � ������

ZGp�

p � � ������

in agreement with the results of ref� ��� �see also ������

����� Derivations on a Paragrassmann algebra

In order to de�ne a derivation on an algebra it is enough to de�ne it on the genera tors� The action upon the other elements will be obtained by using the distributivitylaw� We will be rather interested in what it is called a g derivation �this is a particu lar case of a �s�� s�� derivation� see ������ In our case we can de�ne p�� g derivationsas follows

Di � i ������

such thatDi�f��� � �Di�f�� � gi�Dif��� �����

These two equations determine uniquely the action of Di upon the generic elementof the algebra� In fact we have�

Di� � �� � gi� ������

��

Page 26: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

and then� by induction� we get easily

Din � �� � gi � g�i � � � �� gn��

i �n�i�� ��� gni�� gi

n�i�� �����

In the case i � �� we get a condition on g�� In fact� if we take n � p � �� we musthave

D�p�� � � ������

But using ����� we get

� � D�p�� �

�� gp���

�� g�p ������

implying that g� is a �p � �� root of unity

gp��� � � ������

However� for i � � we get no restrictions� Take for instance i � �� then

� � D�p�� � �� � g� � g�� � � � �� gp��p�� ������

is automatically satis�ed� The same is true for the i � �� If we introduce the lineartransformation associated to a derivations� i�e�

Dxi � dijxj ������

in our case we get

Din � �� � gi � g�i � � �� gn��

i �n�i�� � �di�nmm ������

from which�di�nm � �� � gi � g�i � � �� gn��

i ��n�i���m� n � � ������

For n � �� using the fact that Di� � �� we have

�di��m � � �����

We will investigate now the integration by part rule� That is we want to determinewhich conditions imply on the derivations the ruleZ

�Dif�� � � ������

that is Z�Di

n � �di�nm�C���m� � �di�np �����

In order to satisfy eq� ������ for any n we must have

�di�np � ���gi�g�i �� � ��gn��i ��n�i���p � ���gi�g�i �� � ��gp�ii ��n�i���p � � ������

��

Page 27: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

This is automatically satis�ed by D�� due to eq� ������� For i � �� we see that theonly way that the eq� ������ holds� is to require that gi is a �p� �� i� root of unity�that is

gp���ii � � ������

It is interesting to notice that a particular set of Di derivations can be generated byD�� In fact� if we put

Di � iD� ������

we getDi

n � �� � g� � g�� � � � �� gn��� �n�i�� ������

which coincides with eq� ������ with gi � g�� In this case only D� satis�es eq�������� Notice that D� depends on the choice of g�� except in the Grassmann case�p � ���

��� The algebra of quaternions

The quaternionic algebra is de�ned by the multiplication rules

eAeB � ��AB � �ABCeC � A� B� C � �� �� � ������

where �ABC is the Ricci symbol in � dimensions� The quaternions can be realized interms of the Pauli matrices eA � �i A� The automorphism group of the quaternionicalgebra is SO���� but it is more useful to work in the so called split basis

u� ��

��� � ie�� u�� �

���� ie�

u� ��

��e� � ie��� u� �

��e� � ie�� ������

In this basis the multiplication rules are given in Table ��

u� u�� u� u�u� u� � u� �u�� � u�� � u�u� � u� � �u�u� u� � �u�� �

Table �� Multiplication table for the quaternionic algebra�

The automorphism group of the split basis is U���� with u� and u�� invariant and u�and u� with charges �� and �� respectively� We de�ne� as usual� the vector

jui �

�BBB�u�u��u�u�

�CCCA ������

��

Page 28: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

The matrices RA and LA satisfy the quaternionic algebra because this is an associa tive algebra� So R� and R� satisfy the algebra of a Fermi oscillator �apart a sign��It is easy to get explicit expressions for the left and right multiplication matricesand check that the C matrix exists and that it is given by

C �

�BBB�

� � � �� � � �� � � ��� � �� �

�CCCA �����

Thereforehuj � �u�� u

����u���u�� ������

The exterior product is given by

juihuj �

�BBB�u�u��u�u�

�CCCA �u�� u

����u���u��

�BBB�u� � � �u�� u�� �u� �� u� u� �u� � � u��

�CCCA �����

According to our integration rules we getZ�u�

u� �Z�u�

u�� � ��Z�u�

u� �Z�u�

u� � � ������

In terms of the original basis for the quaternions we getZ�u�

� � ��Z�u�

eA � � ������

and we see that the integration coincides with taking the trace in the ��� represen tation of the quaternions �see next Section�� That is� given an arbitrary functionsf�u� on the quaternions we get Z

�u�f�u� � Tr�f�u�� ������

By considering the scalar product

hu�jui � u��u� � u���u� � u��

�u� � u��u�� ������

we see thathujui � � ������

and Z�u�hu�jui � u�� � u��

� � � ������

Therefore hu�jui behaves like a delta function�

Page 29: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

��� The algebra of matrices

Since an associative algebra admits always a matrix representation� it is interestingto consider the de�nition of the integral over the algebra AN of the N�N matrices�These can be expanded in the following general way

A �NX

n�m��

e�nm�anm ������

where e�nm� are N� matrices de�ned by

e�nm�ij � �ni �

mj � i�j � �� � � � � N �����

These special matrices satisfy the algebra

e�nm�e�pq� � �mpe�nq� ������

Therefore the structure constants of the algebra are given by

f�nm��pq��rs� � �mp�nr�qs �����

from which

�R�pq���nm��rs� � �pm�qs�nr� �L�pq���nm��rs� � �qr�pn�ms ������

The matrix C can be found by requiring that jCxi is an eigenstate of Lpq� that is

�L�pq���nm��rs��F �e���rs� � �F �e���nm�e�pq� ������

whereF �e��nm� � C�nm��rs�e

�rs� ������

We get�F �e���qm��pn � �F �e���nm�e�pq� ������

By looking at the eq� ������� we see that this equation is satis�ed by

�F �e���rs� � e�sr� ������

It followsC�mn��rs� � �ms�nr ������

It is seen easily that C satis�es

CT � C� � C� C� � � ������

Therefore the matrix algebra is a self conjugated one� One easily checks that theright multiplications satisfy eq� ������� and therefore C is an involution� Moreprecisely� since

e�mn�� � C�mn��pq�e�pq� � e�nm� �����

��

Page 30: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

the involution is nothing but the hermitian conjugation

A� � Ay� A � AN ������

The integration rules give

�C����rp��qs� � �rs�pq �Z�e�e�rp�e�qs� � �pq

Z�e�e�rs� �����

We see that this is satis�ed by Z�e�e�rs� � �rs �����

This result can be obtained also using directly eq� ������� noticing that the identityof the algebra is given by I �

Pn e

�n�n�� ThereforeZ�e�e�rs� �

Xn

�C����rs��nn� �Xn

�ns�nr � �rs �����

and� for a generic matrix

Z�e�A �

NXm�n��

anm

Z�e�e�nm� � Tr�A� �����

Since the algebra of the matrices is associative� the inner derivations are given by

DBA � �A�B� �����

Therefore Z�e�DBA �

Z�e�

�A�B� � � �����

and we see that the integration by parts formula corresponds to the cyclic propertyof the trace�

����� The subalgebra AN��

Consider the algebra AN of the N �N matrices� and its subalgebra AN��� We havethe decomposition

AN � AN�� � C �����

with

C �N��Xi��

�e�i�N� � e�N�i�

� e�N�N� �����

and

AN�� �N��Xi�j��

�e�i�j� ����

Page 31: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Let us put

P �NX

i�j��

pije�i�j� �����

then we require ZAN

CP � � ����

which implies ZAN

e�i�N�P � pNi � � ������

and analogouslypiN � pNN � � ������

The other condition ZAN

AN��P �ZAN��

AN�� ������

gives ZAN

e�i�j�P � pji � �ij ������

Therefore

P �

�N�� �� �

�������

where �N�� is the identity matrix in N � � dimensions� This result can be checkedimmediately by computing the product ANP with A a generic matrix of AN

ANP �AN�� BC D

��N�� �

� �

��AN�� �C �

�������

implying ZAN

ANP � Tr�ANP � � Tr�AN��� �ZAN��

AN�� ������

����� Paragrassmann algebras as subalgebras of Ap��

A paragrassmann algebra of order p can be seen as a subalgebra of the matrix algebraAp��� In fact� ince a paragrassmann is associative it has a matrix representation�the regular one� in terms the �p� ��� �p� �� right multiplication matrices� Ri �seeeq� ������ These are given by

�Ri�jk � �i�j�k �����

De�ningR� � R� ������

we can write� in terms of the matrices de�ned in eq� �����

R� �pXi��

e�i�i��� �����

��

Page 32: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

and

Rk� �

p���kXi��

e�i�i�k� �����

Therefore� the most general function on the paragrassmann algebra �as a subalgebraof the matrices �p � ��� �p � ��� is given by

f�R�� �p��Xi��

aiRp���i� �

p��Xi��

aiiX

j��

e�j�p���j�i� �����

In order to construct the matrix P de�ned in Section ������ let us consider a genericmatrix B � Ap��� We can always decompose it as �see later�

B � f�R�� � �B �����

In order to construct this decomposition� let us consider the most general �p� ����p � �� matrix� We can write

B �p��Xi�j��

bije�ij� �

p��Xi��

pXj��

bije�ij� �

p��Xi��

bi�p��e�i�p��� �����

By adding and subtracting

p��Xi��

bi�p��

i��Xj��

e�j�p���j�i� �����

we get the decomposition ����� with

f�R�� �p��Xi��

bi�p��Rp���i� �����

and

�B �p��Xi��

pXj��

bije�ij� �

p��Xi��

bi�p��

i��Xj��

e�j�p���j�i� �����

Now� we can check that the matrix P such thatZ�f�� �

ZAp��

BP �ZAp��

f�R��P ����

is given byP � e�p����� �����

In fact� we have�Be�p����� � � ����

implyingBe�p����� � f�R��e

�p����� �������

��

Page 33: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Furthermore

Tr�Rk�e

�p������ �p���kXi��

Tr�e�i�i�k�e�p������ � Tr�e�p���k���� � �p�k �������

and therefore Z�k � Tr�Rk

�e�p������ � �p�k �������

showing that e�p����� is the matrix P we were looking for�We notice that the matrices �B and f�R�� appearing in the decomposition �����

can be written more explicitly as

�B �

�BBBBB�

�b��� �b��� � � � �b��p �� � � � �� � � � �

�bp�� �bp�� � � � �bp�p ��bp����

�bp���� � � � �bp���p �

�CCCCCA �������

and

f�R�� �

�BBBBBBBBBBBBB�

ap�� ap ap�� � � � a� a�� ap�� ap � � � a a�� � ap�� � � � a� a� � � � � �� � � � � �� � � � � � ap ap��

� � � � � � ap�� ap� � � � � � � ap��

�CCCCCCCCCCCCCA

�������

The p� �p � �� parameters appearing in �B and the p � � parameters in f�R�� canbe easily expressed in terms of the �p� ��� �p� �� parameters de�ning the matrixB�

In the particular case of a Grassmann algebra we have

R� � e����� �

� �� �

�� �� P � e����� �

� �� �

�� � �������

The decomposition in eq� ������ for a �� � matrix

B � a � b � c � � d � �������

is given by

�B � b�� � � � d �� f�R�� � f� �� � a� b � c � ������

and the integration is Z���f�� � Tr�f� �� �� �������

from which Z���

� � Tr� �� � ��Z��� � Tr� � �� � � ������

��

Page 34: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

��� Projective group algebras

Let us start de�ning a projective group algebra� We consider an arbitrary projectivelinear representation� a � x�a�� a � G� x�a� � A�G�� of a given group G� Therepresentation A�G� de�nes in a natural way an associative algebra with identity�it is closed under multiplication and it de�nes a generally complex vector space��This algebra will be denoted by A�G�� The elements of the algebra are given by thecombinations X

a�G

f�a�x�a� �������

For a group with an in�nite number of elements� there is no a unique de�nition ofsuch an algebra� The one de�ned in eq� ������� corresponds to consider a formallinear combination of a �nite number of elements of G� This is very convenientbecause we will not be concerned here with topological problems� Other de�nitionscorrespond to take complex functions on G such that

Xa�G

jf�a�j �� �������

Or� in the case of compact groups� the sum is de�ned in terms of the Haar invariantmeasure� In the following we will not need to be more precise about this point� Thebasic product rule of the algebra follows from the group property

x�a�x�b� � ei��a�b�x�ab� �������

where ��a� b� is called a cocycle� This is constrained� by the requirement of associa tivity of the representation� to satisfy

��a� b� � ��ab� c� � ��b� c� � ��a� bc� �������

Changing the element x�a� of the algebra by a phase factor ei��a�� that is� de�ning

x��a� � e�i��a�x�a� �������

we getx��a�x��b� � ei���a�b����ab����a����b��x��ab� �������

This is equivalent to change the cocycle to

���a� b� � ��a� b�� ���ab�� ��a�� ��b�� �������

In particular� if ��a� b� is of the form ��ab�� ��a�� ��b�� it can be transformed tozero� and therefore the corresponding projective representation is isomorphic to avector one� For this reason the combination

��a� b� � ��ab�� ��a�� ��b� ������

��

Page 35: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

is called a trivial cocycle� Let us now discuss some properties of the cocycles� Westart from the relation �e is the identity element of G�

x�e�x�e� � ei��e�e�x�e� �������

By the transformation x��e� � e�i��e�e�x�e�� we get

x��e�x��e� � x��e� ������

Therefore we can assume��e� e� � � �������

Then� fromx�e�x�a� � ei��e�a�x�a� �������

multiplying by x�e� to the left� we get

x�e�x�a� � ei��e�a�x�e�x�a� �������

implying��e� a� � ��a� e� � � �������

where the second relation is obtained in analogous way� Now� taking c � b�� in eq��������� we get

��a� b� � ��ab� b��� � ��b� b��� �������

Again� putting a � b��

��b��� b� � ��b� b��� �������

We can go farther by considering

x�a�x�a��� � ei��a�a���x�e� �������

and de�ningx��a� � e�i��a�a

�����x�a� ������

from whichx��a�x��a��� � e�i��a�a

���x�a�x�a��� � x�e� � x��e� �������

Therefore we can transform ��a� a��� to zero without changing the de�nition of x�e��

��a� a��� � � ������

As a consequence� equation ������� becomes

��a� b� � ��ab� b��� � � �������

We can get another relation using x�a��� � x�a���

x�a���x�b��� � ei��a�� �b���x�a��b��� � x�a���x�b���

� �x�b�x�a���� � e�i��b�a�x�a��b��� �������

��

Page 36: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

from which��a��� b��� � ���b� a� �������

and together with eq� ������� we get

��ab� b��� � ��b��� a��� �������

The last two relations will be useful in the following� From the product rule

x�a�x�b� � ei��a�b�x�ab� �Xc�G

fabcx�c� �������

we get the structure constants of the algebra

fabc � �ab�cei��a�b� �������

The delta function is de�ned according to the nature of the sum over the groupelements�

To de�ne the integration over A�G�� we start as usual by introducing a ket withelements given by x�a�� that is jxia � x�a�� and the corresponding transposed brahxj� From the algebra product� we get immediately

�R�a��bc � fbac � �ba�cei��b�a�� �L�a��bc � facb � �ac�be

i��a�c� �������

We show now that also these algebras are self conjugated� Let us look for eigenketsof L�a�

L�a�jCxi � jCxix�a� ������

giving�ac�b�Cx�ce

i��a�c� � ei��a�a��b��Cx�a��b � �Cx�bx�a� �������

By putting�Cx�a � kax�a��� ������

we obtainka��bx�b��a�ei��a�a

��b� � kbei��b���a�x�b��a� �������

Then� from eqs� ������� and �������

ka��b � kb �������

Therefore ka � ke� and assuming ke � �� it follows

�Cx�a � x�a��� � x�a��� �������

givingCa�b � �ab�e �������

This shows also thatCT � C �������

��

Page 37: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

at least in the cases of discrete and compact groups� The mapping C � A � A isan involution of the algebra� In fact� by de�ning

x�a�� � x�b�Cb�a � x�a��� � x�a��� �������

we have �x�a���� � x�a�� and x�b��x�a�� � �x�a�x�b���� In refs� ��� ��� we haveshown that for a self conjugated algebra� it is possible to de�ne an integration rulein terms of the matrix C Z

�x�x�a� � C��

e�a � �e�a �������

From this de�nition and the eq� ������� it follows ��� ���Z�x�jxihxCj �

Z�x�jxCihxj � � ������

Therefore we are allowed to expand a function on the group �jfia � f�a�� as

f�a� �Z�x�

x�a���hxjfi �������

with hxjfi �P

b�G x�b�f�b�� It is also possible to de�ne a scalar product amongfunctions on the group� De�ning� hf ja � �f�a�� where �f�a� is the complex conjugatedof f�a�� we put

hf jgi �Z�x�hf jxCihxjgi �

Z�x�

�f�x�a����g�x� �Xa�G

�f�a�g�a� ������

It is important to stress that this de�nition depends only on the algebraic propertiesof A�G� and not on the speci�c representation chosen for this construction�

����� What is the meaning of the algebraic integration�

As we have said in the previous Section� the integration formula we have obtainedis independent on the group representation we started with� In fact� it is based onlyon the structure of right and left multiplications� that is on the abstract algebraicproduct� This independence on the representation suggests that in some way we are�summing� over all the representations� To understand this point� we will study inthis Section vector representations� To do that� let us introduce a label � for thevector representation we are actually using to de�ne A�G�� Then a generic functionon A�G�

�f��� �Xa�G

f�a�x�a� �������

can be thought as the Fourier transform of the function f � G � jC� Using thealgebraic integration we can invert this expression �see eq� ��������

f�a� �Z�x��

�f���x�a��� �������

��

Page 38: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

But it is a well known result of the harmonic analysis over the groups� that in manycases it is possible to invert the Fourier transform� by an appropriate sum over therepresentations� This is true in particular for �nite and compact groups� Thereforethe algebraic integration should be the same thing as summing or integrating overthe labels � specifying the representation� In order to show that this is the case�let us recall a few facts about the Fourier transform over the groups ����� Firstof all� given the group G� one de�nes the set �G of the equivalence classes of theirreducible representations of G� Then� at each point � in �G we choose a unitaryrepresentation x belonging to the class �� and de�ne the Fourier transform of thefunction f � G � jC� by the eq� �������� In the case of compact groups� instead ofthe sum over the group element one has to integrate over the group by means of theinvariant Haar measure� For �nite groups� the inversion formula is given by

f�a� ��

nG

X� �G

dtr� �f���x�a���� �������

where nG is the order of the group and d the dimension of the representation ��Therefore� we get the identi�cation

Z�x�f� � �g �

nG

X� �G

dtr�f� � �g� �������

A more interesting way of deriving this relation� is to take in �������� f�a� � �e�a�

obtaining for its Fourier transform� �� � x�e� � �� where the last symbol meansthe identity in the representation �� By inserting this result into ������� we get theidentity

�e�a ��

nG

X� �G

dtr��x�a���� �������

which� compared with eq� �������� gives �������� This shows explicitly that thealgebraic integration for vector representations of G is nothing but the sum over therepresentations of G�

An analogous relation is obtained in the case of compact groups� This can alsobe obtained by a limiting procedure from �nite groups� if we insert ��nG� the volumeof the group� in the de�nition of the Fourier transform� That is one de�nes

�f��� ��

nG

Xa�G

f�a�x�a� �������

from whichf�a� �

X� �G

dtr� �f���x�a���� �������

Then one can go to the limit by substituting the sum over the group elements withthe Haar measure

�f��� �ZGd��a�f�a�x�a� ������

Page 39: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

The inversion formula ������� remains unchanged� We see that in these cases thealgebraic integration sums over the elements of the space �G� and therefore it can bethought as the dual of the sum over the group elements �or the Haar integration forcompact groups�� By using the Fourier transform ������� and its inversion ��������one can easily establish the Plancherel fromula� In fact by multiplying together twoFourier transforms� one gets

�f���� �f���� �Xa�G

��Xb�G

f��b�f��b��a�

�Ax�a� �������

from which Z�x�

�f���� �f����x�a��� �Xb�G

f��b�f��b��a� ������

and taking a � e we obtainZ�x�

�f���� �f���� �Xb�G

f��b�f��b��� �������

This formula can be further specialized� by taking f� � f and for f� the involutedof f � That is

�f ���� �Xa�G

�f�a�x�a��� �������

where use has been made of eq� �������� Then� from eq� ������� we get thePlancherel formula Z

�x�

�f ���� �f��� �Xa�G

�f�a�f�a� �������

Let us also notice that eq� ������� says that the Fourier transform of the convolutionof two functions on the group is the product of the Fourier transforms�

We will consider now projective representations� In this case� the product of twoFourier transforms is given by

�f���� �f���� �Xa�G

h�a�x�a� �������

withh�a� �

Xb�G

f��b�f��b��a�ei��b�b

��a� �������

Therefore� for projective representations� the convolution product is deformed dueto the presence of the phase factor� However� the Plancherel formula still holds� Infact� since in

h�e� �Xb�G

f��b�f��b��� �������

using eq� ������� the phase factor disappears� the previous derivation from eq�������� to eq� ������� is still valid� Notice that eq� ������� tells us that the Fouriertransform of the deformed convolution product of two functions on the group� isequal to the product of the Fourier transforms�

��

Page 40: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

G � RD G � ZD G � TD G � ZDN

�G � RD �G � TD �G � ZD �G � ZDN

�a �� � ai � �� ai ���mi

L� � ai � L ai � ki�

mi � Z � � ki � n� �

�q �� � qi � �� � � qi � L qi ���mi

Lqi �

���iN

mi � Z � � �i � n� �

Table �� Parameterization of the abelian group G and of its dual �G� for G � RD�ZD� TD� ZD

n �

����� The case of abelian groups

In this Section we consider the case of abelian groups� and we compare the Fourieranalysis made in the framework of the ATI with the more conventional one madein terms of the characters� A fundamental property of the abelian groups is thatthe set �G of their vector unitary irreducible representations �VUIR�� is itself anabelian group� the dual of G �in the sense of Pontryagin ������ Since the VUIR�s areone dimensional� they are given by the characters of the group� We will denote thecharacters of G by ��a�� where a � G� and � denotes the representation of G� Forwhat we said before� the parameters � can be thought as the elements of the dualgroup� The parameterization of the group element a and of the representation label� are given in Table �� for the most important abelian groups and for their dualgroups� where we have used the notation a � �a and � � �q�

The characters are given by

��a� � �q ��a� � e�iq�a �������

and satisfy the relation �here we use the additive notation for the group operation�

��a � b� � ��a���b� ������

and the dual�����a� � ���a����a� �������

Page 41: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

That is they de�ne vector representations of the abelian group G and of its dual� �G�Also we can easily check that the operators

Dq�q ��a� � �i�a�q ��a� ������

are derivations on the algebra ������ of the characters for any G in Table ��We can use the characters to de�ne the Fourier transform of the function f�g� �

G� jC�f��� �

Xa�G

f�a���a� ������

If we evaluate the Fourier transform of the deformed convolution of eq� �������� weget

�h��� �Xa�G

h�a���a� �Xa�b�G

f�a���a�ei��a�b�g�b���b� ������

In the case of vector representations the Fourier transform of the convolution is theproduct of the Fourier transforms� In the case of projective representations� theresult� using the derivation introduced before� can be written in terms of the Moyalproduct �we omit here the vector signs�

�h��� � �f��� � �g��� � e�i��D�� �D���� �f�����g�������������

������

Therefore� the Moyal product arises in a very natural way from the projective groupalgebra� On the other hand� we have shown in the previous Section� that the useof the Fourier analysis in terms of the projective representations avoids the Moyalproduct� The projective representations of abelian groups allow a derivation on thealgebra� analogous to the one in eq� ������� with very special features� In fact wecheck easily that

�Dx��a� � �i�ax��a� ������

is a derivation� and furthermore Z�x��

�Dx��a� � � ������

From this it follows� by linearity� that the integral of �D applied to any function onthe algebra is zero Z

�x��

�D

�Xa�G

f��a�x��a�

�� � ������

This relation is very important because� as we have shown in ����� the automorphisms

generated by �D� that is exp��� � �D�� leave invariant the integration measure of theATI �see also later on�� Notice that this derivation generalizes the derivative withrespect to the parameter �q� although this has no meaning in the present case� Inthe case of nonabelian groups� a derivation sharing the previous properties can bede�ned only if there exists a mapping � G� jC� such that

�ab� � �a� � �b�� a� b � G ������

��

Page 42: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

since in this case� de�ningDx�a� � �a�x�a� �����

we get

D�x�a�x�b�� � �ab�x�a�x�b� � � �a� � �b��x�a�x�b�

� �Dx�a��x�b� � x�a��Dx�b�� ������

Having de�ned derivations and integrals one has all the elements for the harmonicanalysis on the projective representations of an abelian group�

Let us start considering G � RD� In the case of vector representations we have

xq ��a� � e�iq�a �����

with �a � G� and �q � �G � RD labels the representation� The Fourier transform is

�f��q� �ZdD�af��a�e�iq�a �������

Here the Haar measure for G coincides with the ordinary Lebesgue measure� Also�since �G � RD� we can invert the Fourier transform by using the Haar measure on thedual group� that is� again the Lebesgue measure� In the projective case� eq� �����still holds true� if we assume �q as a vector operator satisfying the commutationrelations

�qi� qj� � i�ij �������

with �ij numbers which can be related to the cocycle� by using the Baker Campbell Hausdor� formula

e�iq�ae�iq�b � e�i�ijaibj��e�iq��a�

b� �������

giving

���a��b� � ��

��ijaibj �������

The inversion of the Fourier transform can now be obtained by the ATI in the form

f��a� �Z�q�

�f��q�xq ���a� �������

where the dependence on the representation is expressed in terms of �q� thought nowthey are not coordinates on �G� We recall that in this case� eq� ������� gives

Z�q�xq ��a� � �D��a� �������

Therefore� the relation between the integral in ATI and the Lebesgue integral in �G�in the vector case is Z

�q��Z dD�q

����D�������

��

Page 43: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

In the projective case the right hand side of this relation has no meaning� whereasthe left hand side is still well de�ned� Also� we cannot maintain the interpretationof the qi�s as coordinates on the dual space �G� However� we can de�ne elements ofA�G� having the properties of the qi�s �in particular satisfying eq� ��������� by usingthe Fourier analysis� That is we de�ne

qi �ZdD�a

��i �

�ai�D��a�

�xq ��a� ������

which is an element of A�G� obtained by Fourier transforming a distribution overG� which is a honestly de�ned space� From this de�nition we can easily evaluatethe product

qixq ��a� �ZdD�b

��i �

�bi�D��b�

�xq ��b�xq ��a� �������

Using the algebra and integrating by parts� one gets the result

qixq ��a� � irixq ��a� ������

where

ri ��

�ai� i�ijaj ������

where �ij � ���e�i�� �e�j��� with �e�i� an orthonormal basis in RD� In a completelyanalogous way one �nds

xq ��a�qi � irixq ��a� ������

where

ri ��

�ai� i�ijaj ������

Then� we evaluate the commutator

�qi� �f��q�� �ZdD�a

h�i�ri �ri

f��a�

ixq��a� ������

where we have done an integration by parts� We get

�qi� �f��q�� � ��i�ijDqj�f��q� ������

where Dqj is the derivation ������� with qj a reminder for the direction along wichthe derivation acts upon� In particular� from

Dqjqi �ZdD�a

��i �

�ai�D��a�

���iaj�xq��a� � �ij ������

we get�qi� qj� � ��i�ij ������

in agreement with eq� �������� after the identi�cation �ij � ��ij���

��

Page 44: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

The automorphisms induced by the derivations ������ are easily evaluated

S����xq ��a� � e��D�qxq ��a� � e�i��axq ��a� � xq����a� �����

where the last equality follows from

ZdD�a

��i �

�ai�D��a�

�e��D�qxq ��a� � qi � �i ������

Meaning that in the vector case� S���� induces translations in �G� Since Dq satis�esthe eq� ������� it follows from ���� that the automorphism S���� leaves invariantthe algebraic integration measure

Z�q�

�Z�q���

�����

This shows that it is possible to construct a calculus completely analogous to theone that we have on �G in the vector case� just using the Fourier analysis followingby the algebraic de�nition of the integral� We can push this analysis a little bitfurther by looking at the following expression

Z�q�

�f��q� qi xq ���a� � �i�

�ai� i�ijaj

�f��a� �������

where we have used eq� ������� In the case D � � this equation has a physicalinterpretation in terms of a particle of charge e� in a constant magnetic �eld B� Infact� the commutators among canonical momenta are

��i� �j� � ieB�ij �������

where �ij is the � dimensional Ricci tensor� Therefore� identifying �i with qi� we get�ij � �eB�ij��� The corresponding vector potential is given by

Ai��a� � ��

��ijBaj �

e�ijaj �������

Then� eq� ������� tells us that the operation �f��q� � �f��q�qi� corresponds to take thecovariant derivative

�i ��ai

� eAi��a� �������

of the inverse Fourier transform of �f��q�� An interesting remark is that a translation

in �q generated by exp��� � �D�� gives rise to a phase transformation on f��a�� First ofall� by using the invariance of the integration measure we can check that

�f��q � ��� � e��D �f��q� �������

��

Page 45: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

In fact Z�q�

�f��q � ���xq ���a� �Z�q���

�f��q�xq�� ���a� � e�i��af��a� �������

Then� we have

Z�q�

�e��

D �f��q�xq ���a� �

Z�q�

�f��q��e���

Dxq���a�

� e�i��af��a� �������

where we have made use of eq� ������ This shows eq� �������� and at the same timeour assertion� From eq� �������� this is equivalent to a gauge transformation on thegauge potential Ai � �ijaj� Ai � Ai � �i�� with � � �� � �a� Therefore� we see hereexplicitly the content of a projective representation in the basis of the functionson the group� One starts assigning the two form �ij� Given that� one makes achoice for the vector potential� For instance in the previous analysis we have chosen�ijaj� Any possible projective representation corresponds to a di�erent choice of thegauge� In the dual Fourier basis this corresponds to assign a �xed set of operators qi�with commutation relations determined by the two form� All the possible projectiverepresentations are obtained by translating the operators qi�s� Of course� this isequivalent to say that the projective representations are the central extension ofthe vector ones� and that they are determined by the cocycles� But the previousanalysis shows that the projective representations generate noncommutative spaces�and that the algebraic integration� allowing us to de�ne a Fourier analysis� gives thepossibility of establishing the calculus rules�

Consider now the case G � ZD� Let us introduce an orthonormal basis on thesquare lattice de�ned by ZD� �e�i�� i � �� � � � � D� Then� any element of the algebracan be reconstructed in terms of a product of the elements

Ui � x��e�i�� ������

corresponding to a translation along the direction i by one lattice site� In generalwe will have

x��m� � ei��m�Um�� � � �UmD

D � �m �Xi

mi�e�i� �������

with a calculable phase factor� The quantities Ui play the same role of �q of theprevious example� The Fourier transform is de�ned by

�f��U� �Xm�ZD

f��m�xU ��m� ������

where the dependence on the representation is expressed in terms of �U � denotingthe collections of the Ui�s� The inverse Fourier transform is de�ned by

f��m� �ZU

�f��U�xU ���m� �������

��

Page 46: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

where the integration rule is Z�U�

xU ��m� � �m�� �������

Therefore� the Fourier transform of Ui is simply �m�e�i�� The algebraic integrationfor the vector case is Z

�U��Z L

dD�q

LD�������

Since the set �U is within the generators of the algebra� to establish the rules of thecalculus is a very simple matter� Eq� ������ is the de�nition of the set �U � analogousto eq� ������� In place of eq� ������ we get

Ui�f��U�U��

i � e���ijDj �f��U� �������

Here Dj is the j th component of the derivation �D which acts upon Ui as

DiUj � �i�ijUj �������

By choosing �f��U� � Uk we have

UiUkU��i U��

k � e�i�ik �������

which is the analogue of the commutator among the qi�s� The automorphisms gen erated by �D are

S����xU��m� � e�� DxU��m� � e�i

��mxU ��m� �������

From which we see thatUi � S����Ui � e�i�iUi ������

This transformation corresponds to a trivial cocycle� As in the case G � RD it givesrise to a phase transformation on the group functionsZ

�U�

�e��

D �f��U�xU ���m� �

ZU�

�f��U��e���

DxU���m�

� ei��mf��m� �������

Of course� all these relations could be obtained by putting Ui � exp��iqi�� with qide�ned as in the case G � RD�

Finally� in the case G � ZDn � the situation is very much alike ZD� that is the

algebra can be reconstructed in terms of a product of elements

Ui � x��e �i�� ������

satisfyingUni � � �������

Therefore we will not repeat the previous analysis but we will consider only the caseD � �� where U� and U� can be expressed as ���

�U��a�b � �a�b�� � �a�n�b��� �U��a�b � e��in

�a����a�b� a� b � �� � � � � n �������

��

Page 47: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

The elements of the algebra are reconstructed as

xU��m� � ei�nm�m�Um�

� Um�� �������

The cocycle is now

���m�� �m�� � ���

n�ijm�im�j �������

In this case we can compare the algebraic integration ruleZUxU��m� � �m�� �������

withTr�xU��m�� � n�m�� �������

A generic element of the algebra is a n� n matrix

A �n��X

m��m���

cm�m�xU��m� �������

and therefore ZUA �

nTr�A� ������

In ref� ���� we have shown that the algebraic integration over the algebra of then� n matrices An is given by Z

An

A � Tr�A� �������

implying ZUA �

n

ZAn

A ������

����� The example of the algebra on the circle

A particular example of a group algebra is the algebra on the circle de�ned by

znzm � zn�m� �� � n�m � �� �������

with z restricted to the unit circle�

z� � z�� �������

This is a group algebra over ZZ� De�ning the ket

jzi �

�BBBBBBBBBBBBB�

�z�i

��z�zi

�CCCCCCCCCCCCCA

�������

��

Page 48: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

the Ri and Li matrices are given by

�Ri�jk � �i�j�k� �Li�jk � �i�k�j �������

and from our previous construction� the matrix C is given by

�C�ij � �C���ij � �i�j�� �������

or� more explicitly by

C �

�BBBBB�

� � � � �� � � � �� � � � �� � � � �� � � � �

�CCCCCA �������

showing thatC � zi � z�i �������

In fact

�C��LiC�lp � �l��m�i�n�m�n��p � �i�p��l � �i�l�p � �Ri�lp ������

In this case the C matrix is nothing but the complex conjugation �z � z� � z����The completeness relation reads now

Z�z�

ziz�j � �ij �������

from which Z�z�zk � �k� ������

Our algebraic de�nition of integral can be interpreted as an integral along a circleC around the origin� In fact we have

Z�z�

��

��i

ZC

dz

z�������

Page 49: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Chapter �

Examples of Associative Non

Self�Conjugated Algebras

��� The algebra of the bosonic oscillator

We will start trying to reproduce the integration rules in the bosonic case� It isconvenient to work in the coherent state basis� The coherent states are de�ned bythe relation

ajzi � jziz �����

where a is the annihilation operator� �a� ay� � �� The representative of a state at�xed occupation number in the coherent state basis is

hnjzi �znpn

�����

So we will consider as elements of the algebra the quantities

xi �zipi � i � �� �� � � � �� �����

We introduce the ket �BBBBB�

�z

z��p

� ��

�CCCCCA �����

The algebra is de�ned by the multiplication rules

xixj �zi�jpi j

� xi�j

s�i � j�

i j �����

from which

fijk � �i�j�k

sk

i j �����

��

Page 50: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

It follows

�Ri�jk � �i�j�k

sk

i j ����

and

�Li�jk � �i�k�j

sj

i k �����

In particular we get

�R��jk �pk �j���k� �L��jk �

pk � � �j���k ����

Therefore R� and L� are nothing but the representative� in the occupation numberbasis� of the annihilation and creation operators respectively� It follows that theC matrix cannot exist� because �R�� L�� � �� and a unitary transformation cannotchange this commutation relation into �L�� R�� � ��� For an explicit proof consider�for instance� R�

R� �

�BBBBB�

�p

� � � � � �� �

p� � � � �

� � �p

� � � �� � � � � � �� � � � � � �

�CCCCCA ������

If the matrix C would exist it would be possible to �nd states hzj such that

hzjR� � zhzj ������

withhzj � �f��z�� f��z�� � � �� ������

This would mean

hzjR� � ��� f��z��p

�f��z�� � � �� � �zf��z�� zf��z�� zf��z�� � � �� ������

which impliesf��z� � f��z� � f��z� � � � � � � ������

Now� having shown that no C matrix exists� we will consider the complex conjugatedalgebra with generators constructed in terms of z�� where z� is the complex conjugateof z� Correspondingly we introduce the vectors

Rijz�i � jz�iz�i � hzjLi � zihzj ������

where� for instance

hzj � ���zp� �z�p� � � � �� ������

and the integration rules give

Z�z�z��

ziz�jpi j

� �i�j �����

Page 51: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

We see that they are equivalent to the gaussian integration

Z�z�z��

�Z dz�dz

��iexp��jzj�� ������

��� The q�oscillator

A generalization of the bosonic oscillator is the q bosonic oscillator ����� We will usethe de�nition given in ����

b�b� q�bb � � �����

with q real and positive� We assume as elements of the algebra A� the quantities

xi �ziqiq

������

where z is a complex number�

iq �qi � �

q � �������

andiq � iq�i� ��q � � � � ������

The structure constants are

fijk � �i�j�k

vuut kq

iq jq ������

and therefore

�Ri�jk � �i�j�k

vuut kq

iq jq � �Li�jk � �i�k�j

vuut jq

iq kq ������

In particular

�R��jk � �j���k

qkq� �L��jk � �j���k

q�k � ��q ������

We see that R� and L� satisfy the q bosonic algebra

R�L� � qL�R� � � ������

For q real and positive� no C matrix exists� so� according to our rules

Z�z�z��q

ziz�j

iq jq � �ij �����

This integration can be expressed in terms of the so called q integral �see ref� ������by using the representation of nq as a q integral

nq �Z �����q�

�dqt e

�qt��q t

n ������

��

Page 52: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

where the q exponential is de�ned by

etq ��Xn��

zn

nq �����

and the q integral through

Z a

�dqtf�t� � a��� q�

�Xn��

f�aqn�qn ������

Then the two integrations are related by �z � jzj exp�i���

Z�z�z��q

�Z d�

��

Zdq�jzj�� e�qjzj

��q ������

The Jackson integral is the inverse of the q derivative

�Dqf��x� �f�x�� f�qx�

��� q�x������

In fact� for

F �a� �Z a

�dqtf�t� � a��� q�

�Xn��

f�aqn�qn ������

one has�DqF ��a� � f�a� ������

as can be checked using

F �qa� � qa��� q��Xn��

f�aqn���qn � a��� q��Xn��

f�aqn�qn � F �a�� a��� q�f�a�

������

��

Page 53: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Chapter �

Examples of Nonassociative

Self�Conjugated Algebras

��� Jordan algebras of symmetric bilinear forms

The Cli�ord algebra CN is de�ned in terms of N � � generators

ei � �e�� ea�� a � �� � � � � N� i � �� �� � � � � N � � �����

such that�ea� eb�� � ��ab �����

and e� is the identity� We will introduce the associated Jordan algebra as the algebracomposed by the elements ei� with product rules �see ref� ����

ei � ej ��

��ei� ej�� �����

In the following we will omit the � in the product of the elements of the algebra�Therefore� the product rules are

eaeb � �abe� �����

e�ea � eae� � ea �����

e�e� � e� �����

This algebra is a non associative one� The structure constants are

fabc � � ����

fab� � fa�b � f�ab � �ab �����

fa�� � f�a� � f��a � � ����

f��� � � ������

��

Page 54: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Correspondingly we get the following expressions for the matrices of the right mul tiplications ��Ri�jk � fjik�

�Ra�bc � �� �R��bc � �bc ������

�Ra��� � �� �R���� � � ������

�Ra��b � �ab� �R���a � � ������

�Ra�b� � �ab� �R��a� � � ������

Notice thatRTi � Ri ������

and that the algebra is commutative

eiej � ejei ������

By using the vectors

jei �

�BBBBB�

e�e���eN

�CCCCCA �����

we have from the very de�nition

Rijei � jeiei ������

By taking the transposed of this equation we get

hejRTi � hejei �����

and using the symmetry of Ri and the commutativity of the algebra we get

hejRi � eihej ������

showing thatLi � Ri ������

ThereforeC � � ������

Then the integration rules follow from

ZCN

eiej � �ij ������

that is ZCN

ei � �i� ������

��

Page 55: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Therefore ZCN

ea � ��ZCN

e� � � ������

If the algebra CN is realized by matrices od dimension d� then

ZCN

�� � �� ��

dTr�� � �� ������

Consider now the most general derivation on CN � Let us put

Dei � dijej �����

The conditions for D being a derivation are

D�e��� � �De� � De� � De� � � ������

implyingd�i � � �����

Then

D�eaeb� � �abDe� � � � eadbiei � daieieb � �ebda� � eadb�� � �dab � dba� ������

giving rise todab � �dba� da� � � ������

That isDe� � �� Dea � dabeb ������

with d an antisymmetric matrix� The corresponding automorphism

S � exp�D� ������

corresponds to an orthogonal transformation of the elements ea leaving invariant e��From this derivation we have found the the most general continuous automorphismof CN is an element of the group SO�N�� The complete group of automorphisms isgiven by O�N� since the algebra is invariant under the parity operation

ea � �ea� e� � e� ������

From our integration rules follow also that for the most general derivation on CN

one has ZCN

Df�e� � � ������

where f�e� is an arbitrary �linear� function on CN � Therefore the integration isinvariant under SO�N� transformations�

The previous algebra has the following interpretation� Given a vector space witha basis ea equipped with a metrics

hv�jv�i � gabva�v

b� ������

��

Page 56: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

we de�ne an algebra by enlarging the elements ea to ei � �e�� ea� where e� is theidentity and product rule

eaeb � gabe� �����

or� for generic elementsv�v� � hv�jv�ie� ������

The structure constants and the matrix elements of the right multiplications aregiven by

fabc � � �����

fa�b � gab ������

fab� � f�ab � �ab ������

fa�� � f�a� � f��a � � ������

f��� � � ������

and

�Ra�bc � �� �R��bc � �bc ������

�Ra��� � �� �R���� � � ������

�Ra��b � �ab� �R���a � � ������

�Ra�b� � gab� �R��a� � � �����

We see that the Ri matrices are given by

R� � �� Ra �

� vawa �

�������

where the vectors va and wa are de�ned by

�va�b � �ab� �wa�b � gab �����

These vectors have the following property

wa � gva ������

where g is the matrix with elements gab� The matrices Ra are not any more sym metric� but since the algebra is abelian we have

La � RTa ������

Therefore the C matrix is de�ned by the relation

CRa � RTaC ������

Writing

C �c zz A

�������

��

Page 57: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

where a is a number� z a vector and A a matrix� One gets the conditions

z � va � va � z� Awa � ava ������

implyingz � �� A � ag�� ������

By choosing a � �� we get

C �

� �� g��

�������

Then we obtain the integration rules

Z�e�ea � ��

Z�e�e� � � �����

From this it is easy to extend the treatment to a complex vector space equippedwith a K!ahler metrics� A basis will be given by e� � �ea� e

�a�� with an algebra

eae�b � �a b � e�bea � � ba� eaeb � e�ae

�b � � ������

We require also that the conjugation� ea � e�a is an involution of the algebra�implying

��a b � �b a �����

Notice that in the case the metrics reduce to a Kronecker delta� the algebra isnothing but the one obtained by taking as a product the anticommutators of Fermicreation and annihilation operators� Since we have �� � �a� �a��

e�e� �

� �a b� ab �

��

� �a b�� ba �

�� g�� ������

where we have made use of eq� ����� and of the commutativity of the algebra� iffollows

g � gy ������

Then we check easily that

C �

� �� g��

�������

with

g�� �

� ���

��y��� �

�������

a hermitian matrix� In particular

Cjei �

�B� � � �

� � ���

� ��y��� �

�CA�B� e�eae�a

�CA �

�B� e�

���e�a��y���ea

�CA ������

��

Page 58: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

In particular for � � �� C is the matrix representation of the involution� In general�one has to introduce the matrix

I �

�B� � � �

� � �� � �y

�CA ������

Then� the involution is given by the product

IC �

�B� � � �

� � �� � �

�CA ������

The integration rules are

Z�e�ea � ��

Z�e�e�a � ��

Z�e�e� � � �����

��� The algebra of octonions

We will discuss now how to integrate over the octonionic algebra �see ������ Thisalgebra �said also a Cayley algebra� is de�ned in terms of the multiplication tableof its seven imaginary units eA

eAeB � ��AB � aABCeC � A� B� C � �� � � � � ������

where aABC is completely antisymmetric and equal to �� for �ABC� � ��� �� ����������� �������� ������� �������� ������ and ������� The automorphism group of thealgebra is G�� We de�ne also in this case the split basis as

u� ��

��� � ie��� u�� �

���� ie��

ui ��

��ei � iei��� u�i �

��ei � iei�� �����

where i � �� �� �� In this basis the multiplication rules are given in Table ��

u� u�� uj u�ju� u� � uj �u�� � u�� � u�jui � ui �ijku

�k ��iju�

u�i u�i � ��iju�� �ijkuk

Table �� Multiplication table for the octonionic algebra�

Page 59: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

This algebra is non associative and in the split basis it has an automorphism groupSU���� The non associativity can be checked by taking� for instance�

ui�uju�k� � ui���jku�� � � �����

and comparing with

�uiuj�u�k � �ijmu

�mu

�k � ��ijk�kmnun �����

We introduce the ket

jui �

�BBB�u�u��uiu�i

�CCCA �����

and one can easily evaluate the matrices R and L corresponding to right and leftmultiplication� We will not give here the explicit expressions� but one can easily seesome properties� For instance� one can evaluate the anticommutator �Ri� R

�j ��� by

using the following relation

�Ri� R�j ��jui � Rijuiu�j � R�

j juiui � �juiui�u�j � �juiu�j�ui �����

The algebra of the anticommutators of Ri� R�i turns out to be the algebra of three

Fermi oscillators �apart from the sign�

�Ri� R�j �� � ��ij� �Ri� Rj�� � �� �R�

i � R�j �� � � �����

The matrices R� and R�� de�ne orthogonal projectors

R�� � R�� �R�

��� � R�

�� R�R�� � R�

�R� � � �����

Further properties areR� � R�

� � � �����

andR�i � �RT

i ����

Similar properties hold for the left multiplication matrices� One can also show thatthere is a matrix C connecting left and right multiplication matrices� This is givenby

C �

�BBB�

� � � �� � � �� � � ��� � �� �

�CCCA �����

where � is the ��� identity matrix� We have CT � C� It follows that the eigenbrasof the matrices of type R� are

huj � �u�� u����u�i ��ui� ����

��

Page 60: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

For getting the integration rules we need the external product

juihuj �

�BBB�u�u��uiu�i

�CCCA�u�� u

����u�j ��uj

�BBB�u� � � �uj� u�� �u�j �� ui �iju� ��ijku�ku�i � ��ijkuk �iju

��

�CCCA ������

According to our rules we get

Z�u�

u� �Z�u�

u�� � ��Z�u�

ui �Z�u�

u�i � � ������

Other interesting properties are

hujui � u� � u�� � �u�� � �u� � � ������

and usinghu�jui � u��u� � u��

�u� � u�i�ui � u�iu

�i ������

we get Z�u�hu�jui � u�� � u��

� � � ������

Showing that hu�jui behaves like a delta function�

Page 61: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Appendix A

A�� Jordan algebras

A Jordan algebra� A� is a nonassociative algebra which enjoys the following twoproperties�� Commutativity� xy � yx�� Jordan identity� �xy�x� � x�yx��where x� y � A� For example� one can obtain a Jordan algebra from an associativeone� by de�ning the Jordan product �which will be denoted in this particular exampleby �� as

x � y ��

��xy � yx� �A���

Noticing thatx � y � y � x� x � x � x� �A���

one can verify immediately that this is a Jordan algebra� In dealing with nonasso ciative algebra it turns out convenient to de�ne the associator

�a� b� c� � �ab�c� a�bc� �A���

which represents a measure of the nonassociativity of the algebra� By using theproduct rules we get in a basis �xi�

�xixj�xk � fijlflkmxm � �LTi Rk�jmxm � �LT

xixj�kmxm � �RjRk�imxm �A���

xi�xjxk� � fjklfilmxm � �RkLTi �jmxm � �Rxjxk�imxm � �LT

j LTi �imxm �A���

More symbolically

�ab�c � �LTaRc�b � LT

abc � �RbRc�a

a�bc� � �RcLTa �b � Rbca � �LT

b LTa �c �A���

In particular for associative algebras we get the equivalent identities for the rightand left multiplications

�LTa � Rb� � �� LaLb � Lab� RaRb � Rab �A��

��

Page 62: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Notice that the commutativity condition gives

ab � ba �� Rba � LTb a �A���

that isRa � LT

a �A��

In terms of the associator the Jordan identity can be written as

x�yx�� � �xy�x� �� �x� y� x�� � � �A����

By sending x� x��z �� � F � where F is the �eld over wich the algebra is de�ned�we get

� � �x � �z� y� �x � �z��� � �x� y� x�� � ���z� y� x�� � ��x� y� xz��

� ����x� y� z�� � ��z� y� xz�� �A����

This is satis�ed if��x� y� xz� � �z� y� x�� � � �A����

Again we send x� x � �w in this identity obtaining from the coe�cients of � and�� �the constant term is identically zero� the following relations

� � �w� y� xz� � �x� y� wz� � �z� y� xw� � �

�� � ��w� y� wz� � �z� y� w�� � � �A����

The coe�cient of �� is zero for eq� �A����� therefore we get the further condition

�x� y� wz� � �w� y� xz� � �z� y� xw� � � �A����

Using the commutativity condition �A�� and eq� �A���� we can rewrite the associ ator in the equivalent forms

�a� b� c� � �Ra� Rc�b � �Rab �RbRa�c � �RbRc �Rbc�a �A����

Therefore we can write the condition �A���� in two equivalent ways

�Rx� Rwz� � �Rw� Rxz� � �Rz� Rxw� � � �A����

and

�Rxy � RyRx�wz � �RyRxz � Ry�xz��w � �Rzy �RyRz�xw � � �A���

from which

Rz�Rxy � RyRx� � RyRxz � Ry�xz� � Rx�Rzy � RyRz� � � �A����

��

Page 63: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

exchanging x� y

Rz�Ryx �RxRy� � RxRyz � Rx�yz� � Ry�Rzx � RxRz� � � �A���

and subtracting these two expressions

Rz�R�x�y � �Rx� Ry�� �RyR�x�z �RxR�z�y� �Rx� Ry�Rz�Ry�xz� �Rx�yz� � � �A����

and using again commutativity

�Rz� �Rx� Ry�� � R�xz�y � Rx�zy� � R�x�z�y� � R�Rx�Ryz �A����

or�Rz� �Rx� Ry�� � R�Rx�Ryz �A����

recalling that the condition for the linear mepping D to be a derivation is

�Ri� d� � RDxi � Rdxi �A����

�where as for the right and left multiplications we use the convention dxi � dijxj��we see that the expression

D�x�y� � �Rx� Ry� �A����

is a derivation for any pair x� y � A� Therefore a basis for the derivations is

Dij � �Ri� Rj� �A����

��

Page 64: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

Bibliography

��� F�A�Berezin and M�S�Marinov� JETP Lett� �� ���� ���� ibidem Ann� of Phys��� ��� ����

��� R�Casalbuoni� Il Nuovo Cimento� ��A ���� ��� and ibidem ���

��� F�A�Berezin� The method of second quantization� Academic Press ������

��� A�Connes� Noncommutative geometry� Academic Press �����

��� V�G�Drinfeld� Quantum Groups� in Proceedings of the International Congressof Mathematicians� Berkeley ���� pp� � ���� AMS� Providence� RI�

��� J�L�Martin� Proc� Roy� Soc� ���A ���� ����

�� A�Messiah� Quantum Mechanics� North Holland� Amsterdam ������

��� R�Casalbuoni� Int� J� Mod� Phys� A�� ��� ����� physics���������

�� R� Casalbuoni� Int� Journ� of Mod� Phys�� �� ��� ��� math�ph��������

���� R� Casalbuoni� Int� Journ� of Mod� Phys� A�� ���� ���� physics������

���� R�D�Schafer� An introduction to nonassociative algebras� Academic Press������

���� A�P�Isaev� in Proceedings of "II International Workshop on Classical and Quan tum Integrable Systems� �Dubna� � �� July� ���� q alg#������

���� N�Jacobson� Lie Algebras� Interscience Publish�� N� Y� London ������

���� L�Biedenharn� J� Phys� A�� ���� ���$ A�Macfarlane� J�Phys� A�� ���������

���� L�Baulieu and E�G�Floratos� Phys� Lett� B������� ���

���� A�A� Kirillov� Elements de la Theorie des Representations� %Editions MIR�Moscou ����$ ibidem Representation Theory and Noncommutative HarmonicAnalysis I� Springer Verlag �����

��

Page 65: Algebras - Istituto Nazionale di Fisica Nuclearetheory.fi.infn.it/casalbuoni/report.pdf · Algebras Algebras Non existence of the C matrix Algebras with in v olution Deriv ... sical

��� B� de Wit� J� Hoppe and H� Nicolai� Nucl� Phys� B�� ����� ���$ D� Fairlie�P� Fletcher and C� Zachos� J� Math� Phys� �� ���� ����$ J� Hoppe� Int� J�Mod� Phys� A� ���� �����

���� T�H�Koornwinder� Representations of Lie groups and quantum groups� eds�V�Baldoni and M�A�Picardello� Pitman Research Notes in Mathematical Series���� Longman Scienti�c & Technical ����� pp� �� ���$ see also M�Chaichian�A�P�Demichev and P�P�Kulish� HIP � ��#Th� q alg#������

��� N�Jacobson� Exceptional Lie Algebras� M� Dekker� Inc�� N� Y� �����

���� R�Casalbuoni� G�Domokos and S� K!ovesi Domokos� Il Nuovo Cimento� ��A���� ����

��