Notes on Algebras� Derivations
and Integrals
Roberto Casalbuoni
Dipartimento di Fisica� Universit�a di Firenze
I�N�F�N� Sezione di Firenze
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 � � � � � � � � � � � � � � � � � � � � � � � � � �
�
A �
A�� Jordan algebras � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��
�
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 � � �����
�
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 �����
�
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� ������
�
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� ������
�
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�
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 �����
�
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�� ������
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 ������
��
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 ������
��
��� 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�
��
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
��
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 ������
��
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�
��
����� 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 ������
��
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 �����
�
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 ��� �������
��
��� 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� � � �������
�
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 �������
��
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�� �
� �� �
������
��
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 �
��
�������
��
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 ������
��
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� ������
��
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 � � ������
��
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 ������
��
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�
�
��� 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� �����
��
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� ����
�
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��� �����
��
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����� �������
��
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� � �� � � ������
��
��� 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� ������
��
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��� �������
��
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 �������
��
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��� �������
��
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� ������
�
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�
��
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� �������
�
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 ������
��
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�������
��
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���
��
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� �������
��
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� �������
��
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 �������
��
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
�������
��
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�������
�
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 �����
��
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 �����
�
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 ������
��
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�
������
��
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��� � � ������
��
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� ������
��
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� ������
��
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
�������
��
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 ������
��
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�
�
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� ����
��
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�
�
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��
��
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����
��
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����
��
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 �����
��
��� 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���� ����
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