- Email: [email protected]
On commutation relations for quons
Vol. 41 (1998)
REPORTS ON MATHEMATICAL PHYSICS
No. 2
ON COMMUTATION RELATIONS FOR QUONS* WLADYSLAW MARCINEK Institute of Theoretical Physics, University of Wroclaw, Pl. Maxa Boma 9, 50-204 Wroclaw, Poland (e-mail: [email protected]) (Received August 19, 1996 -
Revised February 13, 1997)
A model of particles equipped with generalized statistics is developed. The commutation relations and the corresponding consistency conditions are given in terms of quantum Weyl algebras. The Fock-like space representation and the corresponding scalar product are also described.
PACS numbers: 03.65.Ca, 03.65.Fd. 1. Introduction In the last years a few different approaches to particle systems equipped with generalized statistics have been developed. For example, particles called quons with the statistics interpolating between fermions and bosons have been considered by Greenberg [l, 21, Mohapatra [3] and Five1 [4]. Their approach is based on the following commutation relations for the creation and anihilation operators (CAO) afaj - Qaiaf = hii&
(I)
where q is a statistics parameter. Such particles have been also studied by D. Zagier [5], Boiejko and Speicher [6, 71, Finkelstein [8] and others. Observe that the above relations for q = 1 describe the commutation relation for bosons, and for q = -1 the anticommutation relation for fermions. In these two cases the following additional relations have appeared aiaj f ajai = 0, where the sign “+” is for fermions and “-” for bosons. For q # f 1 we have only the mixed relations between anihilation and creation operators, there are no relations for operators of the same kind, i.e. for the creation or for the anihilation operators themselves [4, 91. Hence there is a problem with such relations for quons. The paper partially supported by KBN, Grant No 2P 302 087 06, Poland and by the Katholischer Akademischer Ausltinder Dienst, Bonn, Germany.
W. MARCINEK
156
On the other hand, we have a different approach to particles with fractional spin and statistics based on the Wilczek notion of anyons [lo]. According to this concept anyon is a composite system which consists of a particle with charge e and a singular magnetic field (a flux) concentrated completely on a vertical line. The charge is moving around the singular line in the region in which there is no influence of the magnetic field. Every rotation yields a certain phase factor q. The factor q can be in general an arbitrary complex number on the unit circle, i.e. IqI = 1. Hence some authors use the name “quons” for such particles. The name “anyons” is sometimes reserved to the case when q is the m-th root of unity, see for example the papers by Majid [12, 131 which describe anyons by certain &-graded structures. In general, anyons have been studied from different points of view by several authors. The path integral method has been used by Wilczek, Zee and Wu [lo, 11, 161. The connection with the Chern-Simon theory has been studied in [17-191. Anyons as q-deformation of bosons have been considered in [20]. In [21] the anyonic excitations are described by the Zamolodchikov-Faddeev algebra. A simple algebraic model for generalized quons as generalization of Wilczek’s model of anyons has been presented in [22]. We must indicate here that there is an approach to particle systems with some nonstandard statistics in low-dimensional spaces based on the notion of the braid group B,, see [23, 241 for example. In this approach, the configuration space for the system of n identical particles moving on a manifold M is given by the formula &n(M)
= (Mxn - D)/Sn,
where D is the subcomplex of the Cartesian product M x n on which two or more particles occupy the same position and S, is the symmetric group. The group ~1 (Qn(M)) E B,(M) is known as the n-string braid group on M. Note that there exists a subgroup I&(M) of B,(M) which is an extension of the symmetric group S, describing the interchange process of two arbitrary indistinguishable particles. It is obvious that the statistics of the given system of particles is determined by this group, see [23, 241. The mathematical formalism based on the braid group has been developed by Majid, see [25-311 for example. The noncommutative geometry based on the notion of braid group and corresponding commutation relation has been also studied by the author in [32, 331. It is known that recently the deformed commutation relations for the creation and anihilation operators (CAO) have been studied from different points of views by several authors. For example, the deformation of bosons and fermions corresponding to quantum groups SU,(2) has been given by Pusz and Woronowicz [34, 351. The deformation corresponding to superparticles has been considered by Chaichian, Kulisch and Lukierski [36]. Quantum deformations have been also studied by Vokos [37], Arik [38] an d many others. The commutation relations for Hecke braiding have been studied by Kempf [39]. It is interesting that from the algebraic point of view all these deformations of commutation relations for CA0 can be described in terms of the so-called Wick algebras [40]. The construction of Wick algebras is based on a single operator which describes the deformation. The problem of additional rela-
ON COMMUTATION RELATIONS FOR QUONS
157
tions is presented in [40] but this presentation is not quite satisfactionary. There is a nice solution of the problem with additional relations for quons. If we replace the deformation parameter q by an operator C, then we obtain more general commutation relations [9]. If C 2 = id, then it is easy to see that the additional relations exist. This case has been studied in [43-46], see also [47, 481. If C2 # id, then we need a second operator B for the additional relations. In this way the commutation relations of CA0 for particles with generalized statistics can be described in terms of two operators, B and C [15, 421. Such particles are said to be generalized quons. Obviously, the operators B and C are not quite arbitrary. They must satisfy some consistency conditions, see [15, 421 for example. It is interesting that these conditions look like the Wess-Zumino consistency conditions for the noncommutative calculi on the quantum plane [49]. Hence we can use solutions used in such calculi in order to construct our commutation relations [42, 50, 511. From a purely algebraic point of view, the consistency conditions have been studied by Ralowski in [52]. The commutation relations for particles equipped with arbitrary statistics can be described as certain representation of the so-called quantum Weyl algebra W&N). The creation and anihilation operators act on the B-symmetric algebra A. The creation operators act as multiplication in A and the anihilation ones act as a noncommutative contraction. The algebra A plays the role of a noncommutative Fock space. In this way we obtain the Fock-like representation for our particles with generalized statistics [42, 50, 511. A proposal for the general algebraic formalism for the description of particle systems equipped with an arbitrary generalized statistics based on the concept of monoidal categories with duality has been given by the author in [53]. The physical interpretation of this formalism was shortly indicated. A few examples of applications for this formalism have been considered in [54, 551. In this paper we are going to study particle systems equipped with generalized statistics as composite systems which contain particles coupled with a few quanta of certain external field as a result of interactions. The commutation relations for such systems and the Fock-like representation are considered from the point of view of the general theory presented in [53]. We would like to show that there is a well formulated general theory for systems with generalized statistics. The paper is organized as follows. In Section 2 the fundamental assumptions for our algebraic model of composite systems with generalized statistics are presented. The commutation relations for such system are studied in terms of quantum Weyl algebras, and the consistency conditions are described. The solution corresponding to a system of generalized quons of [22] is given as an example. In Section 3 we describe the model for generalized quons as a composite system which contains a charge and a few magnetic fluxes. It is interesting that the composite fermions of Jain [56] can also be described as a particular example of our composite system of particles. The category with duality corresponding to the quantum Weyl algebra is described in Section 4. The Fock-like space representation is considered in Section 5. The positive-definite and nondegenerate scalar product is also given.
158
W. MARCINEK
2. Fundamental assumptions
We are going to study particles with generalized statistics as a result of interaction of usual particles with certain external field. The proper physical nature of this field is not essential at the moment. Our fundamental assumption is that the usual particles, fermions or bosons, are transformed under interactions into a system with generalized statistics which contains particles and a few quanta of the external field. This system is called a composite system of particles. It should be characterized by two operators B and C. For the description of such systems we use the concept of quantum Weyl algebras [59-61] specified for our goal. Let E be a finite-dimensional Hilbert space equipped with a basis zi, i = 1, .. . , N = dim H. The complex conjugate space is denoted by E*. Note that our notation is not covariant. We assume that the pairing (-I.) : E’ @E @ and the corresponding scalar product is given by (U*Io) E (+I) := C&i,
(2)
i=l
where U* = CL,
$x*~ and w = CL,
uizi. We have here the following
DEFTNITION.A linear and invertible operator
C : E* @ E -
E 8 E”,
for which we have CC,;)* = i$i,
i.e.
C* = CT,
(4)
where (CT):*{ = CT:;, is said to be a skew or a cross operator [57]. Let C : E @ E* E* @ E be an arbitrary skew operator. We define a partial dual to C as a linear operator 6’ : E 8 E E @ E such that we have the relation ((*I*)@ idE) 0 (idE= @ 6’) = (idE B (.I.))
o (C 8 idE)
(5)
on E* @E @ E. We have (z*~J&) = 6*ij, hence we obtain
(c)g = c;$.
(6)
We must indicate that the operator C is a primary object for our formalism. The second operator B is also linear and invertible. It is given by the relation B(xi @ xj) = c B;zk k,l
@x1,
(B*)$ = B:;;;.
(7)
DEFINITION.An algebra W E W(B, C),
W = W(B, C) := T(E crsE*)/IB,c,
(8)
ON COMMUTATION
RELATIONS FOR QUONS
159
where IB,c = IC + IB + I,. is an ideal in T(E @ E*), is said to be a quantum Weyl algebra or a Wick algebra with additional relations for the given composite system if it is generated by the following relations
x*ixj = i?j 1 +
c
c;;xkx*‘,
k,d
xixj _
c
B;zkz’
(9)
= 0,
kJ
x=x*3
-
c
--+i*j
*k
B ekeI
x
d
= 0,
x
*k,*l
and fulfills the consistency conditions B(r)B(a)B(r)
= B(2@(1@(2),
B(r)C(a)C(l) = ,92)C(1)B(2), _ (idE@E + C)(idE@E - B) = 0.
(10)
The quantum Weyl algebra W(B, C) is said to be Zeft covuriunt if and only if there exists a certain quantum matrix (semi-)group denoted by M(B, C) together with a coaction 6 : W M 8 W such that &ri :=
&k,
c
&r*i := C(Z-‘):;x*“,
k
(11)
k
where 2 = (2;) E M(B,C) [61, 621. The right covariance can be defined in a similar way. We need a solution to the consistency conditions (10) for the construction of the algebra W(B, C) for our composite system of particles. As was indicated in introduction, we can use here the solutions used in noncommutative geometry in order to construct examples of our algebra. The general solution is not known. Hence we can restrict our attention to some particular cases only. In this paper we use the solution to consistency conditions which has been found by Borowiec and Kharchenko for differential calculi, see [62]. In this solution both operators B and C are diagonal. They are given by the following formulae C(X*~ @ xj) := cjixj @ Zig, B(xi @Ixj) := bijxj 8 xi for i,j = 1, .. . . N, where cij and bij are complex parameters relation I b,j for i # j Cij
= Qi
qi are certain parameters,
for i = j ’
and in addition we have
(no summation)
(12)
such that we have the
(13)
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W. MARCINEK
In our physical interpretation the set {qi : i = 1, . . .. N} of parameters describes the statistics of initial particles. The statistics for the corresponding composite system is described by the coefficients bij. 3. The generalized quon particles As an example of particles with generalized statistics we would like to study a system of charged particles moving on the plane IR2 under the influence of a transverse magnetic field. Our studies are based on Wilczek’s model of anyons. It is known that according to this concept an anyon is a composite system consisting of a charged particle and a singular magnetic field concentrated completely on a vertical line. The particle is moving around the line in the region in which there is no influence of the magnetic field. Every rotation yields a certain phase factor q, where q is an arbitrary complex number such that qij = 1. The statistics of anyons is determined by two factors: by the statistics of the charged particle and by the factor q. Observe that the space on which the particle moves is in fact the space M E IR2\ {m}, where m E R2 is the point of intersection of the magnetic line with the plane R2. This point is said to be singular. We see that only rotations of the particle by 27r around the singular point are essential for the concept of anyons. The above concept can be generalized in the following way. We assume that the magnetic field is concentrated completely in several vertical lines. Every such line is said to be a $ux. We also assume that there is on average N fluxes per particle. Let us denote by rni the i-th point of intersection of the magnetic flux with the plane. It is obvious that only rotations by 27r around an arbitrary singular point rni are important for us. Every such rotation yields a certain phase. Any movement between two different singular points is completely irrelevant because it does not produce phase changes. This means that every charged particle moving in the singular magnetic field is transformed into a composite system consisting of the charge and a certain number of magnetic fluxes. This composite system is said to be a system of generalized quons. We can see that the “effective” space for the particle is M E IR2 \ {ml, . .., mN}. We assume that the i-th singular point is characterized by real numbers qi, -1 5 qi I 1, i = l,... , N, and the initial statistics of charged particles moving around the i-th singular point. We also assume that ‘TVis an operator for 29r left rotation around this point. Let us denote by G N the group generated by all such rotations. It is in general a free group generated by (pi. It is obvious that two rotations aiaj and ajai yield the same phase. Hence we can assume that the group GN is abelian and it can be identified with N copies of the group of integers 2. In this way we have GNSZNSZ$ Now let us consider
. . . $2
(N summands).
(15)
the quantum Weyl algebra W(B, C) corresponding to the system of generalized quons. If we substitute the solution (12) into the definition of W(B, C), then we obtain the quantum Weyl algebra W&N) for quons. In this way W&N) is an algebra generated by xi and x*i (i, j = 1,. . . , N), and subject to
ON COMMUTATION
161
RELATIONS FOR QUONS
relations x*ixi
=
x*ixj = bjixjx*i
1 + qi SAIJ*~ for i = j,
xixj = bijxjxi
for all
X*jX*i
i, j,
for i # j, for all
= bjiX*iX*j
i, j,
(16)
(no summation) [22]. In our physical interpretation the generator xi describes a quantum state with the phase arising as a result of one left rotation around the i-th singular point. The state with the phase corresponding to two rotations around two different points mi and mj can be described by the monomial xixj. Similarly x*i corresponds to the phase arising as the result of one right rotation around the i-th singularity. This means that there is the so-called standard gradation for the algebra W&N) by the group GN, see [47] for details. For this gradation we have grade(x*i) = -oi. grade(xj) = gi, (17) Observe that there is a mapping b : F x r ---+ @ - (0) (called by Scheunert commutation factor) such that we have the following relations
b(a + P,r) = b(a,-/)b(P,71, b(a,PMP, a) = 1, b(a,a) = 1,
[63] a
(18)
for every (Y,p, y E G N. The mapping b is defined by b(cri, gj) := bij for the generators of the grading group. It is easy to see that there is a unique extension of this definition to all pairs of elements of the grading group. One can see that there is a quantum matrix algebra corresponding to our generalized quons. This algebra is denoted by M&N) and is generated by the formula i k zj ‘1 =
bikbjlCjkCilZ/Z;
for 2 = ($). The comultiplication the followmg formulae AZ; = c
(no
Summation)
(19)
A, the counit E and the antipode
z; 8 zj”,
E(Z$ = CY;,
S(zj) = (z-1);.
S are given by
(20)
k
Note that the quantum Weyl algebra W,+(N) is covariant with respect to a coaction of the above quantum matrix algebra. For the left and right coaction we have xi @ zh =
bijCikZb
@Xi,
x*i
@ 2;
=
bikCijZi
@ X*i.
(21)
It is known that, in general, we have the following formula for the commutation factor b on the group GN, bij = (-I)% q% , (22) where Sij are elements of an integer-valued symmetric matrix, Rij are elements of a skew-symmetric integer-valued matrix, and q is a complex parameter [64]. It follows
W. MARCINEK
162
from the condition bii E 1 that Sii = 0. One can see that if we have q = exp(y) for p > 3, then Sij = 0 and the grading group GN = ZN can be reduced to 2: = Z, $ * * . $ 2,. It is interesting that if q = fl, then the grading group GN can be reduced to the group GN c Zz” := Zz@...@z2
(N summands).
(23)
As an example let us consider a quantum Weyl algebra Wq,b(N) corresponding to an electron in a singular magnetic field. In this case we assume that qi = (-l)N+l, and the factor b is given by the relation (24) where q := exp(niN) G (-l)N, L’ij = -L’j, = 1 for i # j, L’ii = 0, i,j = 1,2,. . . , IV. This means that bij = fl and the grading group is G z ZZ $ +1. $ Zs. In this case every electron is transformed into the composite system containing elementary charge e and N magnetic fluxes. If the number N of fluxes is even, then the system is said to be a composite fermion. If N is odd, then we obtain a composite boson [55, 561. The commutation rules can be given in the following form,
x*ixi _
1,
for i = j,
2*W + q-l xjx*i = 0,
for i < j,
xixj + qxjxi = 0,
for i < j.
(__l)N+lxix*i
=
(25)
If q = exp(ri(N + 1)) and N is even, then q = 1 and we obtain the following relations for composite fermions x*izi
+ xi2*i
=
1
,
for i = j,
x*izj _ xjx*i = 0,
for i < j,
xixj - xjxi = 0,
for i < j.
(26)
If N is odd, then we obtain relations for composite bosons x*ixi
x*i5j XiXj
_
xix*i
+ xjz*i +
XjXi
=
1
,
for i = j,
= 0,
for i < j,
0,
for i < j.
=
(27)
4. The category of states In order to define the Fock-like representation for the quantum Weyl algebra W(B, C) we need a B-symmetric algebra A and a noncommutative contraction mapping c. We are going to use the concept of monoidal categories in order to describe these
ON COMMUTATION
RELATIONS FOR QUONS
163
objects. A monoidal carego~~ C s C(@, Ic) is, briefly speaking, a category C equipped with a monoidal associative operation (a bifunctor) @ : C x C --+ C which has a two-sided identity object k, see [26, 28, 30, 66-691 for details. A (right) *-operation in the monoidal category C is a contravariant functor (-)* in C such that (-)** = idc, (U @IV)* = V* @U*, where U and V are arbitrary objects of the category category C is a family of maps
(28)
C. A (right) pairing in the
satisfying some compatibility axioms, see [53]. Let C be a monoidal category equipped with a (right) *-operation (-)* and a (right) pairing g, then such category is said to be a category with (right) duality. One can introduce a (left) duality structure in a similar way. Note that the right and left duality in the category C can be understood in general as two independent structures. But it is possible to assume that these two structures are related. Such a relation can be established by a set of natural isomorphisms called a generalized cross (or twist) symmetry. Note that the generalized cross symmetry is in fact an equivalence for the right and left duality in C. Hence, in the category equipped with such generalized cross symmetry we can identify the right and left duality. The generalized cross symmetry can be given by a set of natural isomorphisms 9 = {!&*,v : u* @3v v @au*} (30) such that %AYW,W
= (RP,w
@idv)o(idu
@%,w), (31)
%A*,VBW
= Wv
@%P,w)o(%P,v
@idw),
for all objects U, V, W in C. Note that the generalized cross symmetry is in general not a braid symmetry. For the braid symmetry in the category with duality we need some additional transformations like !P~{!P~,~:u@~-vvu}
(32)
for arbitrary objects U, V in C and !P E (!Pu*,v* : u* @v*
-
v* 63 u*}
(33)
for arbitrary objects U*, V* in C*. We need also some new commutative diagrams for all these transformations and pairings. In fact, a family of natural isomorphisms
~={%cw:u~w--+w~u},
(34)
W. MARCINEK
164
such that we have the following relations RW,W
= (~~,w~idv)o(idv~~~,w),
Q%,VQW
= Wv
(35) @J%,w)o(%,v
@idw),
is said to be a braidingor a braid symmew on C. The corresponding category is said to be a braided monoidal category [26, 66-681. It is denoted by C _=C(@, Ic,S). If in addition !P$,v = id, (36) for all objects U, V E C, then !P is said to be a symmetry or tensor symmetry and the corresponding category C is called a symmetric monoidal or tensor category, see [45, 46, 691. The monoidal category with unique duality and braid symmetry is said to be rigid [26, 28, 301. For the quantum Weyl algebra W(B, C) there is a rigid braided monoidal category with duality C(B, C, g). Let us briefly recall the construction of such category, see [53] for more details. Note that the category C contains: the basic field k = @ of complex numbers, the space E and its complex dual E*, all tensor products of E and E*, all direct sums, and some quotients. The *-structure is given as the usual complex conjugation operation. The pairing (29) can be introduced as an extension of the initial pairing (2). We have here the following four linear mappings
(37)
where (B*)fl = B$. All above mappings can be extended to tensor products of spaces E and E* by formulae (35). In this way we obtain ~~*C3k,E@’
..-
CC’)
o . . . o c(1)
o . . . o C(“+‘-l)
o . . . o C(“-l)
o C(“+‘)
o
. . . 0 Cc”),
(38)
for the generalized twist symmetry, and
(39)
for the additional symmetry. In this way we obtain the braid symmetry in our category C. An algebra d G An(E) := TE/IB is said to be a B-symmetric algebra over E [59, 701 if it is generated by the relation m o (idE@E - B)(& 8 zj) = 0,
(40)
16.5
ON COMMUTATIONRELATIONSFOR QUONS where m is the multiplication
in the algebra, or equivalently Zisj _
c B$V k,l
= 0.
(41)
On the other hand we have a B-symmetric algebra A’ over the space E*. It is an algebra defined as the quotient A* 5 d&E*) := TE*/I*, such that we have the following relations ~;;a;Z*kg*l = 0. 2*+1 _ c *k,*l
Obviously, A and A* are quadratic, associative and unital algebras over E and E*, respectively, see Baez [69]. They are graded A = @&,v A”,
do G k,
A1 G E,
A* = @nENd*n,
No = k,
d*l E E*.
We denote by P : TE -
(42)
A and P* : TE* --+ A* the quotient mappings. We have
+ (B CZJ idEBk--a) o (idE @ B @IidEmk-3) + . . . , (43)
P; = idE*ak + B* @ idE*Bk-2
+ (B* ~3idE.Bk--2) o (idEe @ B* 8 idElk+) + . . . , for the k-th homogeneous
components
of P and P*, respectively. The multiplication
m in A can be given by the relation
_ m(f @ 9) = for f E E@“, tj E E@,
f = P&),
%+df
8
(44)
6)
and g = Pl(ij). For example we have
zizj=m(zi~~zj):=~i~ij+~~ Let us define a right contraction
skasl.
mapping c : E* @JTE --+ TE.
DEFINITION. We define the k-th component ck : E* QDEmk -+ contraction mapping c by the following formulae cl = (*I’),
Emkvl of the tight
ck := cl @ idEm”-, + (idE. @ c&l) o (C @ idEak-l).
(45)
The left contraction mapping can also be defined, see [57]. Note that the above contraction is the same as the evaluation mapping from the previous papers, see [51, 50, 581. It is easy to see that we have ck+l
on E* @ E@‘” @ E@.
:=
Ck Cl+ idEB’
+ (id E’ ‘%cl) 0 (*E*,E@” @ i&B!)
(46)
166
W. MARCINEK
LEMMA. The contraction mapping c ti braid commutative, i.e. we have (47)
Proof:We prove the lemma only for k =
1 = 1. We have
C2 [ib '8(i&c@ - B)] = [(cl@ id.!r) + (idE 63ci) o (G @ idE)] [idE* 8 (idEBE - B)]
= (ci @ idE) [idi+ @ ((idE@E + 6) o (id - B))] = 0, 0
where the relation (5) has been used.
Let us consider in particular the braided monoidal category corresponding to our quon systems. In this case we obtain a category of GN-graded spaces denoted by C(@, q, b, GN). If we substitute (12) into (37) we obtain the braid symmetry for quons. For example we have
The algebra A for the quon system is an algebra generated by {xi (i = 1, . . ., N} and by the relation zizj = bij& (no summation). (49) This algebra can be regarded as the algebra of all braid equivalent quantum states for left rotations. LEMMA. For the contraction mapping c corresponding to generalized quons we have the relations Ck_i(idE* ‘8 ck) [(idE*@E* - !&*,E*) ‘8 idAk] = 0 (50) on E*@E*@Ak. Proof: We prove the lemma only for k = 2. We calculate ci 0 (idE* ‘8 c~)(z*~ @ x*j @ xm @ 2”) = = =
(51)
On the other hand, we have cl 0 (idEe @ cl) o (B* 8 idE,s,,E)(%*i@ x*j @Jzrn @Ixn) = bij
c~(z*~ @(Pz*~
= bij@m@n
+ bmiSinZm))
+ bijbmajin@.
(52) 0
ON COMMUTATION
167
RELATIONS FOR QUONS
It follows from the above braid commutativity (50) that the contraction mapping c can be projected to the algebra A. Hence, the projected contraction is a mapping Q : E* 631A -+ A given by the relations 01 :=
Cl
E
ak(idE* ‘8 pk) := 4-1 We can prove that for the contraction rule [57]
(.I.),
(53)
o ck.
Q we have the following deformed
Qk+l 0 (idE* 8 m) = m 0 [(ok @ idAl) + (idA& 8 (wl)o (@E-,Ak
@
idAl)] ,
Leibniz
(54)
defined on E* @ dk 18Al. The braiding @E*,Ak is given by @E.,Ak (idE* @ pk) = (% @ idE-)!PE* ,A”.
(55)
It follows from (50) that for quons we also have the relation ol+r 0 (idEB @Jal) 0 [idE*mE=@A- (@P,E* @JidA)] = 0
(56)
on E* ~3E* 63A. We have for example os(z*i 8 &r’)
= @“xl + ck&“Zk.
(57)
5. The Fock-like representation We are going to construct now a representation a : W(B, C) ---+ End(d) of the quantum Weyl algebra W(B,C) on the algebra A. For the generators xi and x*j (i,j = l,..., N) we define azif - m(2 8 f),
!Y&*ifZ cX(x*i8 f)
(58)
for every f E A. We can extend this representation to the whole algebra W(B, C). Obviously, our commutation relations for the creation and anihilation operators should be consistent with the defining relations for the algebra W(B, C). One can also consider a representation of the quantum Weyl algebra W on the algebra A* in a similar way. Let us define a P-bracket by the relation [&, hj]p G [e, .],P(& 8 ;Lj) := c o (id - P)(&, &&), where ki stands for a,*., or a,., c is the composition
(59)
mapping and
P(a*i @ aj.j) := a*(?<@&),
(60)
where 2 stands for Zig or zi, and the braiding 9 on the right-hand side of the above formula is given by (37). We have here the following important theorem.
168
w. MARCINEK
THEOREM. On the algebra A we have the following commutation rdatiom for the representation of the quantum Weyl algebra W(B, C), [U,*i, U2J]P = 6ij, Proof:
[t&l, CS,ci]#= 0,
[&*i 7a,*Li]@ = 0,
(61)
Using (58) we obtain (a,*;
0
U,j)f
0 U,j
(U,i
E
QZ+I 0
(ib
)f - mo(idE
(U,*i OU,*j )f - a~_1 o
@rn)(~*~8 xj
@Im)(xi
@xj
CZJ f),
@ff),
(62)
(idEe @ CX~)(Z*~ 8 x*j 8 f),
for f E Al. We also have 0
C
!P(U,*i @ U,j)f
CO!P(U,i
@O&)f
c 0 !P(U& @ +j)f
E
m o (idE @ Q~+I) 0 (.C ~23 idA)(x*i @ xj C3f),
E m
o
(idE am> 0 (B @ idA)(xi @ xj ~3f),
(63)
= cyl+l o (idEe @ cul)(B* @IidA)(x*i @ x*j @ f).
It follows immediately from the definition (59) of the @-bracket that we have [;li,;lj3*f=
[&O&j
-CO!F(&,@iij)]f.
(64)
NOWwe can calculate the first relation (61) as follows
l!Pf = [qfl 0 (idE*
[Us-i, azi
= V-Lo (aI
@
f) @ m)(x*$ @ xj @If) - m o (idE @JCII)o (c @IidA)] (x*~ @ xj ~23
idA)(x*i 8 xj @ f) = bij f,
where f E A and the relation
(65)
(54) has been used. For the second relation (61) we
obtain [%i, %j]tP.f = m o (m ~3idA) [((idE@E - B) @ idA)(xi @ xj ~3f)] = 0,
(66)
where the associativity and B-commutativity of the algebra A have been used. Finally, we calculate the third commutation relation (61) [U,*i , a,.&f
= alp1 o (idEe @ a~) [idE*wE*BA - (B* @id/t)] (CC** C3x*j Q3f) = 0,
where the relation (56) has been used.
(67) 0
Let us consider the representation of the algebra Wq,b(N) corresponding to quons. In this case we have the following action for the creation and anihilation operators
(68)
ON COMMUTATION
[r&
= gg
RELATIONS FOR QUONS
(i = 1, .*., IV). For the commutation
relations we obtain
aiaj- bijajai = 0,
aaaj -Cijajaf = &jl,
169
ara; - bija;aa = 0,
(69)
where a: := a,.* and ai := a,$. We introduce here the following state vector nlv) = (xi)nl . . . (dy
I%,...,
(70)
up to some normalization factor. For the ground state vector (0) E Id we have as usual a ,*ilO) = 0. The scalar product can be given by the following formula (BJQc := s,,
(,_lP;i)
for 9 E E@‘” and f E EBn, where PA is an operator
(71) defined by induction
PL+l := (id KJPL) o R,,
where P: G id. The operator
R,
(72)
is given by the formula
R n := id + C(1) + @1)~(2)
+ . . . + c(l) . . . c(n-11,
(73)
where @ti) := idE @ . +. @ 0 8 . . . @ idE, c on the i-th place. If the operator bounded operator acting on some Hilbert space such that (i)
c is a
C = C*,
(ii) IICl] 5 1,
(74)
(iii) C? is Yang-Baxter, then-according to Boiejko and Speicher [7]-the scalar product given by (71) is positive definite. Note that the existence of a nontrivial kernel of the operator RI e id + 0 is essential for the nondegeneracy of the scalar product. If the kernel ker RI is nontrivial, then the scalar product (71) is degenerate. Hence we must remove this degeneracy by factoring the mentioned above scalar product by the kernel of RI. In this way we obtain the following product (#B,C
:= (@jc
(75)
for s = P,(g), t = P,(i). One can prove that this scalar product defined. For the quon system we obtain (xcl%)q,b
=
Xn(r)
c
(%I lzr(j,)) -** (xi, Ixr(j,)),
[52, 531 is well
(76)
?rES, where xa,xT Edn,
xa = xil-..xin,
X, = xjl...xjn,
and
(77) (i,j)EJ(r)
(i,j)EJ(a)
170
W. MARCINEK
J(T) := ((4 j) : 1 I 6 j I n}, r(i) > x(j), ni is the number of elements of the set K(r) := {(i,j) E J(T): i = j}. It follows from the mentioned above theorem of Boiejko and Speicher that the corresponding scalar product is positive definite if all qi are real, -1 5 qi < 1, and bij = bij. Acknowledgments The author would like to thank A. Borowiec and J. Lukierski for discussion and critical remarks, and C. Juszczak and R. Ralowski for any other help. REFERENCES 0. W. Greenberg: Phys. Rev. Lett. 64 (1990), 705. 0. W. Greenberg: Phys. Rev. D43 (1991), 4111. R. N. Mohapatra: Phys. Lett. B242 (1990), 407. D. I. Fivel: Phys. Rev. Lett. 65 (1990), 3361. [5] D. Zagier: Commun. Math. Phys. 147 (1992), 199. [6] M. Boiejko and R. Speicher: Interpolation between bosonic and fermionic relations given by generalized Rrownian motions, Preprint FSB 132-691, Heildelberg (1992). [7] M. Boiejko and R. Speicher: Math. Ann. 300 (1994), 97. [8] R. J. Finkenlstein: q-Generating functions and the q-oscilators, Proceedings of the XXI International Conference on Differential Geomem’c Methods in Theoretical Physics, China, 5-9 June 1992, ed. N. Yang, M. L. Ge and X.W. Zhon, pp. 201-205, World Scientific, Singapore 1993. [9] R. Scipioni: Phys. Lett. B327 (1994) 56. [lo] F. Wilczek: Phys. Rev Lett. 48 (1982), 114. [Ill F. Wilczek and A. Zee: Phys. Rev. Lett. 51 (1983), 2250. [12] S. Majid: Anyonic quantum groups, in Spinors, Twistors and Clifford algebras, ed. Z. Oziewicz et al, Kluwer, Dordrecht 1993, p. 327. [13] S. Majid: Czech. J. Phys. 44 (1994) 1073. [14] Yu Ting and Wu Zhao-Yan: Science in China A37 (1994), 1472. [15] S. Meljanac and A. Perica: Mod. Phys. Lett. A!J (1994), 3293. [16] Y. S. Wu: Phys. Rev. Lett. 52 (1984) 2103. [17] Y. S. Wu: Phys. Rev. Lett. 53 (1984), 111. [18] D. P. Arovas, R. Schrieffer, F. Wilczek and A. Zee: Nucl. Phys. B251 (1985), 117. [19] J. K. Kim, W. T. Kim and H. Ihin: J. Phys. A27 (1994) 6067. [20] V. Bardek, M. Doresic and S. Meljanac: Phys. Rev. D49 (1994), 3059. [21] Yue-lin Shen and Mo-lin Ge: Zamoiodchihov-Pad Algebra in 2-component Anyons, Nankai University Preprint, China, August 1995. 1221 W. Marcinek: On quantum Weyl algebras and generalized quons, in Proceedings of the symposium: Quantum Groups and Quantum Spaces, Warsaw, November 20-29, 1995, Poland, ed. R. Budzyiiski, W. Pusz and S. Zakrzewski, Banach Center Publications, Warsaw 1997. [23] Y. S. Wu: J. Math. Phys. 52 (1984), 2103. [24] T. D. Imbo and J. March-Russel: Phys. Lett. B252 (1990), 84. [25] S. Majid: Int. J. Mod. Phys. A5 (1990), 1. [26] S. Majid: J. Math. Phys. 34 (1993), 1176. [27] S. Majid: J. Math. Phys. 34 (1993), 4843. [28] S. Majid: J. Math. Phys. 34 (1993), 2045. [29] S. Majid: Algebras and Hopf Algebras in Braided Categories, in Advances in Hopf Algebras, Plenum 1993. [30] S. Majid: J. Geom. Phys. 13 (1994), 169. [31] S. Majid: AMS Cont. Math. 134 (1992), 219. [32] W. Marcinek: Rep. Math. Phys. 33 (1993), 117. [l] [2] [3] [4]
ON COMMUTATION RELATIONS FOR QUONS [33] [34] [35] [36] [37] [38] [39]
[40] [41] (421
[43] [44] [45] [46]
[47] [48]
[49] [50] [51]
[52] [53] [54]
171
W. Marcinek: J. Math. Phys. 35 (1994), 2633. W. Pusz: Rep. Math. Phys. 27 (1989), 394. W. Pusz and S. L. Woronowicz: Rep. Math. Phys 27 (1989), 231. M. Chaichian, P. Ku&h, J. Lukierski: Phys. Lett. B262 (1991), 43. S. P. Vokos: J. Math. Phys. 32 (1991), 2979. M. Arik: The q-difference operator, the quantum hyperplane, Hilbert spaces of analytic functions and q-oscillators, Istambul Technical University Preprint (1991). A. Kempf: Lett. Math. Phys. 26 (1992), 11. P. E. T. Jorgensen, L. M. Schmith, and R. F. Werner: J. Funct. Anal. 134 (1995) 33. D. B. Fairle and C. K. Zachos: Phys. Lett. B256 (1991), 43. W. Marcinek: On the deformation of commutation relations, in Proceedings of the XIII Workshop in Geometric Methods in Physics, July 1-7, 1994 Bialowieia, Poland, ed. J. Antoine, Plenum, 1995. D. Gurevich: Soviet. J. Contempory Math. Anal. 18 (1983). D. Gurevich: Quantum Yang-Baxter equation and a generalization of the formal Lie theory, in Seminar on Supermanifokis, Report No 19 (1987), 33. V.V. Lyubashenko: Vectorsymmetries, in Seminar on Supermanifolds, No 19 (1987), 1. D. Gurevich, A. Radul and V. Rubtssov: Noncommutative differential geometry and Yang-Baxter equation, Preprint I.H.E.S./M/91/88 (1988). W. Marcinek: Rep. Math. Phys. 34 (1994), 325. W. Marcinek: Int. J. Mod. Phys. Al0 (1995), 1465-1481. J. Wess and B. Zumino: Covariant differential calculus on the quantum hyperplane, CERN Preprint 5697/90 (1990) and Proceedings Supplement. Volume in honor of R. Stora, Nucl. Phys. 18 (1990), 303. W. Marcinek and Robert Ralowski: J. Math. Phys. 36 (1995) 2803. W. Marcinek and R. Ralowski: Particle operators from braided geometry, in Quantum Groups, Formalism and Applications, XXX Karpacz Winter School in Theoretical Physics, 1994, Eds. J. Lukierski et al., 149-154 (1995). R. Ralowski: On deformations of commutation relation algebras, Preprint IFI’ UWr 890, (1995), q-alg/9506004. W. Marcinek: Rep. Math. Phys. 38 (1996), 149-179. W. Marcinek: Topology and quantization, in Proceedings of the IVth International School on Theoretical Physics, Symmetry and Structural Properties, Zajqczkowo k. Poznania, August 29 - September 4, 1996, Poland, ed. B. Lulek, T. Lulek and R. Florek, pp. 415-424, World Scientific, Singapore 1997 and hep-th/97
05 098, May 1997. [55] W. Marcinek: On algebraic model of composite fermions and bosons, in Proceedings of the IXth Max Born Symposium, Kmpacz, September 25 - 28, 1996, Poland, ed J. Lukierski et al., PWN, Warszawa 1997. [56] J. K. Jain: Phys. Rev. Lett. 63 (1989) 199, Phys. Rev. B40 (1989), 8079; 41 (1990), 7653. [57] W. Marcinek: On *-Algebras and Noncommutative Geometry, Preprint KL-TH 95/19 (1995). [58] J. C. Baez: Adv. Math. 95 (1992), 61. [59] A. Borowiec, V. K. Kharchenko and Z. Oziewicz: Calculi on Clifford-Weyl and exterior algebras for Hecke braiding, Conferencia dictata por Zbigniew Oziewicz at Centro de Investigation en Matematicas, Guanajuato, Mexico, Seminario de Geometria, jueves 22 de abril de 1993, and at the Conference on Differential Geometric Methods, Ixtapa, Mexico, September 1993. [60] E. E. Demidov: Uspekhi Mat. Nauk 48 (1993) 39 (in Russian). [61] E. E. Mukhin: Commun. Alg. 22 (1994), 451. [62] A. Borowiec and V. K. Kharchenko: Coordinate calculi on associative algebras, in Quantum Groups, Formalism and Applications, Ed. J. Lukierski et al, PWN, 1994, pp. 231-272, q-alg/9501018. [63] M. Scheunert: J. Math. Phys. 20 (1979), 712. [64] Z. Oziewicz: Lie algebras for arbitrary grading group, in Differential Gebmehy and Its Applications, ed. J. Janyska and D. Krupka, World Scientific, Singapore 1990. [65] S. MacLane: Categories for Working Mathematician, Graduate Text in Mathematics 5, Springer, Berlin 1971.
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(661 A. Joyal and R. Street: Braided tensor categories, Preprint. [67] D. N. Yetter: Math. Proc. Camb. Phi. Sot. 1108 (1990), 263. [68] Y. I. Manin: Quantum Groups and Noncommutative Geometry, University of Montreal Preprint (1989). [69] J. C. Baez: Lett. Math. Phys. 23 (1991), 1333.
REPORTS ON MATHEMATICAL PHYSICS
No. 2
ON COMMUTATION RELATIONS FOR QUONS* WLADYSLAW MARCINEK Institute of Theoretical Physics, University of Wroclaw, Pl. Maxa Boma 9, 50-204 Wroclaw, Poland (e-mail: [email protected]) (Received August 19, 1996 -
Revised February 13, 1997)
A model of particles equipped with generalized statistics is developed. The commutation relations and the corresponding consistency conditions are given in terms of quantum Weyl algebras. The Fock-like space representation and the corresponding scalar product are also described.
PACS numbers: 03.65.Ca, 03.65.Fd. 1. Introduction In the last years a few different approaches to particle systems equipped with generalized statistics have been developed. For example, particles called quons with the statistics interpolating between fermions and bosons have been considered by Greenberg [l, 21, Mohapatra [3] and Five1 [4]. Their approach is based on the following commutation relations for the creation and anihilation operators (CAO) afaj - Qaiaf = hii&
(I)
where q is a statistics parameter. Such particles have been also studied by D. Zagier [5], Boiejko and Speicher [6, 71, Finkelstein [8] and others. Observe that the above relations for q = 1 describe the commutation relation for bosons, and for q = -1 the anticommutation relation for fermions. In these two cases the following additional relations have appeared aiaj f ajai = 0, where the sign “+” is for fermions and “-” for bosons. For q # f 1 we have only the mixed relations between anihilation and creation operators, there are no relations for operators of the same kind, i.e. for the creation or for the anihilation operators themselves [4, 91. Hence there is a problem with such relations for quons. The paper partially supported by KBN, Grant No 2P 302 087 06, Poland and by the Katholischer Akademischer Ausltinder Dienst, Bonn, Germany.
W. MARCINEK
156
On the other hand, we have a different approach to particles with fractional spin and statistics based on the Wilczek notion of anyons [lo]. According to this concept anyon is a composite system which consists of a particle with charge e and a singular magnetic field (a flux) concentrated completely on a vertical line. The charge is moving around the singular line in the region in which there is no influence of the magnetic field. Every rotation yields a certain phase factor q. The factor q can be in general an arbitrary complex number on the unit circle, i.e. IqI = 1. Hence some authors use the name “quons” for such particles. The name “anyons” is sometimes reserved to the case when q is the m-th root of unity, see for example the papers by Majid [12, 131 which describe anyons by certain &-graded structures. In general, anyons have been studied from different points of view by several authors. The path integral method has been used by Wilczek, Zee and Wu [lo, 11, 161. The connection with the Chern-Simon theory has been studied in [17-191. Anyons as q-deformation of bosons have been considered in [20]. In [21] the anyonic excitations are described by the Zamolodchikov-Faddeev algebra. A simple algebraic model for generalized quons as generalization of Wilczek’s model of anyons has been presented in [22]. We must indicate here that there is an approach to particle systems with some nonstandard statistics in low-dimensional spaces based on the notion of the braid group B,, see [23, 241 for example. In this approach, the configuration space for the system of n identical particles moving on a manifold M is given by the formula &n(M)
= (Mxn - D)/Sn,
where D is the subcomplex of the Cartesian product M x n on which two or more particles occupy the same position and S, is the symmetric group. The group ~1 (Qn(M)) E B,(M) is known as the n-string braid group on M. Note that there exists a subgroup I&(M) of B,(M) which is an extension of the symmetric group S, describing the interchange process of two arbitrary indistinguishable particles. It is obvious that the statistics of the given system of particles is determined by this group, see [23, 241. The mathematical formalism based on the braid group has been developed by Majid, see [25-311 for example. The noncommutative geometry based on the notion of braid group and corresponding commutation relation has been also studied by the author in [32, 331. It is known that recently the deformed commutation relations for the creation and anihilation operators (CAO) have been studied from different points of views by several authors. For example, the deformation of bosons and fermions corresponding to quantum groups SU,(2) has been given by Pusz and Woronowicz [34, 351. The deformation corresponding to superparticles has been considered by Chaichian, Kulisch and Lukierski [36]. Quantum deformations have been also studied by Vokos [37], Arik [38] an d many others. The commutation relations for Hecke braiding have been studied by Kempf [39]. It is interesting that from the algebraic point of view all these deformations of commutation relations for CA0 can be described in terms of the so-called Wick algebras [40]. The construction of Wick algebras is based on a single operator which describes the deformation. The problem of additional rela-
ON COMMUTATION RELATIONS FOR QUONS
157
tions is presented in [40] but this presentation is not quite satisfactionary. There is a nice solution of the problem with additional relations for quons. If we replace the deformation parameter q by an operator C, then we obtain more general commutation relations [9]. If C 2 = id, then it is easy to see that the additional relations exist. This case has been studied in [43-46], see also [47, 481. If C2 # id, then we need a second operator B for the additional relations. In this way the commutation relations of CA0 for particles with generalized statistics can be described in terms of two operators, B and C [15, 421. Such particles are said to be generalized quons. Obviously, the operators B and C are not quite arbitrary. They must satisfy some consistency conditions, see [15, 421 for example. It is interesting that these conditions look like the Wess-Zumino consistency conditions for the noncommutative calculi on the quantum plane [49]. Hence we can use solutions used in such calculi in order to construct our commutation relations [42, 50, 511. From a purely algebraic point of view, the consistency conditions have been studied by Ralowski in [52]. The commutation relations for particles equipped with arbitrary statistics can be described as certain representation of the so-called quantum Weyl algebra W&N). The creation and anihilation operators act on the B-symmetric algebra A. The creation operators act as multiplication in A and the anihilation ones act as a noncommutative contraction. The algebra A plays the role of a noncommutative Fock space. In this way we obtain the Fock-like representation for our particles with generalized statistics [42, 50, 511. A proposal for the general algebraic formalism for the description of particle systems equipped with an arbitrary generalized statistics based on the concept of monoidal categories with duality has been given by the author in [53]. The physical interpretation of this formalism was shortly indicated. A few examples of applications for this formalism have been considered in [54, 551. In this paper we are going to study particle systems equipped with generalized statistics as composite systems which contain particles coupled with a few quanta of certain external field as a result of interactions. The commutation relations for such systems and the Fock-like representation are considered from the point of view of the general theory presented in [53]. We would like to show that there is a well formulated general theory for systems with generalized statistics. The paper is organized as follows. In Section 2 the fundamental assumptions for our algebraic model of composite systems with generalized statistics are presented. The commutation relations for such system are studied in terms of quantum Weyl algebras, and the consistency conditions are described. The solution corresponding to a system of generalized quons of [22] is given as an example. In Section 3 we describe the model for generalized quons as a composite system which contains a charge and a few magnetic fluxes. It is interesting that the composite fermions of Jain [56] can also be described as a particular example of our composite system of particles. The category with duality corresponding to the quantum Weyl algebra is described in Section 4. The Fock-like space representation is considered in Section 5. The positive-definite and nondegenerate scalar product is also given.
158
W. MARCINEK
2. Fundamental assumptions
We are going to study particles with generalized statistics as a result of interaction of usual particles with certain external field. The proper physical nature of this field is not essential at the moment. Our fundamental assumption is that the usual particles, fermions or bosons, are transformed under interactions into a system with generalized statistics which contains particles and a few quanta of the external field. This system is called a composite system of particles. It should be characterized by two operators B and C. For the description of such systems we use the concept of quantum Weyl algebras [59-61] specified for our goal. Let E be a finite-dimensional Hilbert space equipped with a basis zi, i = 1, .. . , N = dim H. The complex conjugate space is denoted by E*. Note that our notation is not covariant. We assume that the pairing (-I.) : E’ @E @ and the corresponding scalar product is given by (U*Io) E (+I) := C&i,
(2)
i=l
where U* = CL,
$x*~ and w = CL,
uizi. We have here the following
DEFTNITION.A linear and invertible operator
C : E* @ E -
E 8 E”,
for which we have CC,;)* = i$i,
i.e.
C* = CT,
(4)
where (CT):*{ = CT:;, is said to be a skew or a cross operator [57]. Let C : E @ E* E* @ E be an arbitrary skew operator. We define a partial dual to C as a linear operator 6’ : E 8 E E @ E such that we have the relation ((*I*)@ idE) 0 (idE= @ 6’) = (idE B (.I.))
o (C 8 idE)
(5)
on E* @E @ E. We have (z*~J&) = 6*ij, hence we obtain
(c)g = c;$.
(6)
We must indicate that the operator C is a primary object for our formalism. The second operator B is also linear and invertible. It is given by the relation B(xi @ xj) = c B;zk k,l
@x1,
(B*)$ = B:;;;.
(7)
DEFINITION.An algebra W E W(B, C),
W = W(B, C) := T(E crsE*)/IB,c,
(8)
ON COMMUTATION
RELATIONS FOR QUONS
159
where IB,c = IC + IB + I,. is an ideal in T(E @ E*), is said to be a quantum Weyl algebra or a Wick algebra with additional relations for the given composite system if it is generated by the following relations
x*ixj = i?j 1 +
c
c;;xkx*‘,
k,d
xixj _
c
B;zkz’
(9)
= 0,
kJ
x=x*3
-
c
--+i*j
*k
B ekeI
x
d
= 0,
x
*k,*l
and fulfills the consistency conditions B(r)B(a)B(r)
= B(2@(1@(2),
B(r)C(a)C(l) = ,92)C(1)B(2), _ (idE@E + C)(idE@E - B) = 0.
(10)
The quantum Weyl algebra W(B, C) is said to be Zeft covuriunt if and only if there exists a certain quantum matrix (semi-)group denoted by M(B, C) together with a coaction 6 : W M 8 W such that &ri :=
&k,
c
&r*i := C(Z-‘):;x*“,
k
(11)
k
where 2 = (2;) E M(B,C) [61, 621. The right covariance can be defined in a similar way. We need a solution to the consistency conditions (10) for the construction of the algebra W(B, C) for our composite system of particles. As was indicated in introduction, we can use here the solutions used in noncommutative geometry in order to construct examples of our algebra. The general solution is not known. Hence we can restrict our attention to some particular cases only. In this paper we use the solution to consistency conditions which has been found by Borowiec and Kharchenko for differential calculi, see [62]. In this solution both operators B and C are diagonal. They are given by the following formulae C(X*~ @ xj) := cjixj @ Zig, B(xi @Ixj) := bijxj 8 xi for i,j = 1, .. . . N, where cij and bij are complex parameters relation I b,j for i # j Cij
= Qi
qi are certain parameters,
for i = j ’
and in addition we have
(no summation)
(12)
such that we have the
(13)
160
W. MARCINEK
In our physical interpretation the set {qi : i = 1, . . .. N} of parameters describes the statistics of initial particles. The statistics for the corresponding composite system is described by the coefficients bij. 3. The generalized quon particles As an example of particles with generalized statistics we would like to study a system of charged particles moving on the plane IR2 under the influence of a transverse magnetic field. Our studies are based on Wilczek’s model of anyons. It is known that according to this concept an anyon is a composite system consisting of a charged particle and a singular magnetic field concentrated completely on a vertical line. The particle is moving around the line in the region in which there is no influence of the magnetic field. Every rotation yields a certain phase factor q, where q is an arbitrary complex number such that qij = 1. The statistics of anyons is determined by two factors: by the statistics of the charged particle and by the factor q. Observe that the space on which the particle moves is in fact the space M E IR2\ {m}, where m E R2 is the point of intersection of the magnetic line with the plane R2. This point is said to be singular. We see that only rotations of the particle by 27r around the singular point are essential for the concept of anyons. The above concept can be generalized in the following way. We assume that the magnetic field is concentrated completely in several vertical lines. Every such line is said to be a $ux. We also assume that there is on average N fluxes per particle. Let us denote by rni the i-th point of intersection of the magnetic flux with the plane. It is obvious that only rotations by 27r around an arbitrary singular point rni are important for us. Every such rotation yields a certain phase. Any movement between two different singular points is completely irrelevant because it does not produce phase changes. This means that every charged particle moving in the singular magnetic field is transformed into a composite system consisting of the charge and a certain number of magnetic fluxes. This composite system is said to be a system of generalized quons. We can see that the “effective” space for the particle is M E IR2 \ {ml, . .., mN}. We assume that the i-th singular point is characterized by real numbers qi, -1 5 qi I 1, i = l,... , N, and the initial statistics of charged particles moving around the i-th singular point. We also assume that ‘TVis an operator for 29r left rotation around this point. Let us denote by G N the group generated by all such rotations. It is in general a free group generated by (pi. It is obvious that two rotations aiaj and ajai yield the same phase. Hence we can assume that the group GN is abelian and it can be identified with N copies of the group of integers 2. In this way we have GNSZNSZ$ Now let us consider
. . . $2
(N summands).
(15)
the quantum Weyl algebra W(B, C) corresponding to the system of generalized quons. If we substitute the solution (12) into the definition of W(B, C), then we obtain the quantum Weyl algebra W&N) for quons. In this way W&N) is an algebra generated by xi and x*i (i, j = 1,. . . , N), and subject to
ON COMMUTATION
161
RELATIONS FOR QUONS
relations x*ixi
=
x*ixj = bjixjx*i
1 + qi SAIJ*~ for i = j,
xixj = bijxjxi
for all
X*jX*i
i, j,
for i # j, for all
= bjiX*iX*j
i, j,
(16)
(no summation) [22]. In our physical interpretation the generator xi describes a quantum state with the phase arising as a result of one left rotation around the i-th singular point. The state with the phase corresponding to two rotations around two different points mi and mj can be described by the monomial xixj. Similarly x*i corresponds to the phase arising as the result of one right rotation around the i-th singularity. This means that there is the so-called standard gradation for the algebra W&N) by the group GN, see [47] for details. For this gradation we have grade(x*i) = -oi. grade(xj) = gi, (17) Observe that there is a mapping b : F x r ---+ @ - (0) (called by Scheunert commutation factor) such that we have the following relations
b(a + P,r) = b(a,-/)b(P,71, b(a,PMP, a) = 1, b(a,a) = 1,
[63] a
(18)
for every (Y,p, y E G N. The mapping b is defined by b(cri, gj) := bij for the generators of the grading group. It is easy to see that there is a unique extension of this definition to all pairs of elements of the grading group. One can see that there is a quantum matrix algebra corresponding to our generalized quons. This algebra is denoted by M&N) and is generated by the formula i k zj ‘1 =
bikbjlCjkCilZ/Z;
for 2 = ($). The comultiplication the followmg formulae AZ; = c
(no
Summation)
(19)
A, the counit E and the antipode
z; 8 zj”,
E(Z$ = CY;,
S(zj) = (z-1);.
S are given by
(20)
k
Note that the quantum Weyl algebra W,+(N) is covariant with respect to a coaction of the above quantum matrix algebra. For the left and right coaction we have xi @ zh =
bijCikZb
@Xi,
x*i
@ 2;
=
bikCijZi
@ X*i.
(21)
It is known that, in general, we have the following formula for the commutation factor b on the group GN, bij = (-I)% q% , (22) where Sij are elements of an integer-valued symmetric matrix, Rij are elements of a skew-symmetric integer-valued matrix, and q is a complex parameter [64]. It follows
W. MARCINEK
162
from the condition bii E 1 that Sii = 0. One can see that if we have q = exp(y) for p > 3, then Sij = 0 and the grading group GN = ZN can be reduced to 2: = Z, $ * * . $ 2,. It is interesting that if q = fl, then the grading group GN can be reduced to the group GN c Zz” := Zz@...@z2
(N summands).
(23)
As an example let us consider a quantum Weyl algebra Wq,b(N) corresponding to an electron in a singular magnetic field. In this case we assume that qi = (-l)N+l, and the factor b is given by the relation (24) where q := exp(niN) G (-l)N, L’ij = -L’j, = 1 for i # j, L’ii = 0, i,j = 1,2,. . . , IV. This means that bij = fl and the grading group is G z ZZ $ +1. $ Zs. In this case every electron is transformed into the composite system containing elementary charge e and N magnetic fluxes. If the number N of fluxes is even, then the system is said to be a composite fermion. If N is odd, then we obtain a composite boson [55, 561. The commutation rules can be given in the following form,
x*ixi _
1,
for i = j,
2*W + q-l xjx*i = 0,
for i < j,
xixj + qxjxi = 0,
for i < j.
(__l)N+lxix*i
=
(25)
If q = exp(ri(N + 1)) and N is even, then q = 1 and we obtain the following relations for composite fermions x*izi
+ xi2*i
=
1
,
for i = j,
x*izj _ xjx*i = 0,
for i < j,
xixj - xjxi = 0,
for i < j.
(26)
If N is odd, then we obtain relations for composite bosons x*ixi
x*i5j XiXj
_
xix*i
+ xjz*i +
XjXi
=
1
,
for i = j,
= 0,
for i < j,
0,
for i < j.
=
(27)
4. The category of states In order to define the Fock-like representation for the quantum Weyl algebra W(B, C) we need a B-symmetric algebra A and a noncommutative contraction mapping c. We are going to use the concept of monoidal categories in order to describe these
ON COMMUTATION
RELATIONS FOR QUONS
163
objects. A monoidal carego~~ C s C(@, Ic) is, briefly speaking, a category C equipped with a monoidal associative operation (a bifunctor) @ : C x C --+ C which has a two-sided identity object k, see [26, 28, 30, 66-691 for details. A (right) *-operation in the monoidal category C is a contravariant functor (-)* in C such that (-)** = idc, (U @IV)* = V* @U*, where U and V are arbitrary objects of the category category C is a family of maps
(28)
C. A (right) pairing in the
satisfying some compatibility axioms, see [53]. Let C be a monoidal category equipped with a (right) *-operation (-)* and a (right) pairing g, then such category is said to be a category with (right) duality. One can introduce a (left) duality structure in a similar way. Note that the right and left duality in the category C can be understood in general as two independent structures. But it is possible to assume that these two structures are related. Such a relation can be established by a set of natural isomorphisms called a generalized cross (or twist) symmetry. Note that the generalized cross symmetry is in fact an equivalence for the right and left duality in C. Hence, in the category equipped with such generalized cross symmetry we can identify the right and left duality. The generalized cross symmetry can be given by a set of natural isomorphisms 9 = {!&*,v : u* @3v v @au*} (30) such that %AYW,W
= (RP,w
@idv)o(idu
@%,w), (31)
%A*,VBW
= Wv
@%P,w)o(%P,v
@idw),
for all objects U, V, W in C. Note that the generalized cross symmetry is in general not a braid symmetry. For the braid symmetry in the category with duality we need some additional transformations like !P~{!P~,~:u@~-vvu}
(32)
for arbitrary objects U, V in C and !P E (!Pu*,v* : u* @v*
-
v* 63 u*}
(33)
for arbitrary objects U*, V* in C*. We need also some new commutative diagrams for all these transformations and pairings. In fact, a family of natural isomorphisms
~={%cw:u~w--+w~u},
(34)
W. MARCINEK
164
such that we have the following relations RW,W
= (~~,w~idv)o(idv~~~,w),
Q%,VQW
= Wv
(35) @J%,w)o(%,v
@idw),
is said to be a braidingor a braid symmew on C. The corresponding category is said to be a braided monoidal category [26, 66-681. It is denoted by C _=C(@, Ic,S). If in addition !P$,v = id, (36) for all objects U, V E C, then !P is said to be a symmetry or tensor symmetry and the corresponding category C is called a symmetric monoidal or tensor category, see [45, 46, 691. The monoidal category with unique duality and braid symmetry is said to be rigid [26, 28, 301. For the quantum Weyl algebra W(B, C) there is a rigid braided monoidal category with duality C(B, C, g). Let us briefly recall the construction of such category, see [53] for more details. Note that the category C contains: the basic field k = @ of complex numbers, the space E and its complex dual E*, all tensor products of E and E*, all direct sums, and some quotients. The *-structure is given as the usual complex conjugation operation. The pairing (29) can be introduced as an extension of the initial pairing (2). We have here the following four linear mappings
(37)
where (B*)fl = B$. All above mappings can be extended to tensor products of spaces E and E* by formulae (35). In this way we obtain ~~*C3k,E@’
..-
CC’)
o . . . o c(1)
o . . . o C(“+‘-l)
o . . . o C(“-l)
o C(“+‘)
o
. . . 0 Cc”),
(38)
for the generalized twist symmetry, and
(39)
for the additional symmetry. In this way we obtain the braid symmetry in our category C. An algebra d G An(E) := TE/IB is said to be a B-symmetric algebra over E [59, 701 if it is generated by the relation m o (idE@E - B)(& 8 zj) = 0,
(40)
16.5
ON COMMUTATIONRELATIONSFOR QUONS where m is the multiplication
in the algebra, or equivalently Zisj _
c B$V k,l
= 0.
(41)
On the other hand we have a B-symmetric algebra A’ over the space E*. It is an algebra defined as the quotient A* 5 d&E*) := TE*/I*, such that we have the following relations ~;;a;Z*kg*l = 0. 2*+1 _ c *k,*l
Obviously, A and A* are quadratic, associative and unital algebras over E and E*, respectively, see Baez [69]. They are graded A = @&,v A”,
do G k,
A1 G E,
A* = @nENd*n,
No = k,
d*l E E*.
We denote by P : TE -
(42)
A and P* : TE* --+ A* the quotient mappings. We have
+ (B CZJ idEBk--a) o (idE @ B @IidEmk-3) + . . . , (43)
P; = idE*ak + B* @ idE*Bk-2
+ (B* ~3idE.Bk--2) o (idEe @ B* 8 idElk+) + . . . , for the k-th homogeneous
components
of P and P*, respectively. The multiplication
m in A can be given by the relation
_ m(f @ 9) = for f E E@“, tj E E@,
f = P&),
%+df
8
(44)
6)
and g = Pl(ij). For example we have
zizj=m(zi~~zj):=~i~ij+~~ Let us define a right contraction
skasl.
mapping c : E* @JTE --+ TE.
DEFINITION. We define the k-th component ck : E* QDEmk -+ contraction mapping c by the following formulae cl = (*I’),
Emkvl of the tight
ck := cl @ idEm”-, + (idE. @ c&l) o (C @ idEak-l).
(45)
The left contraction mapping can also be defined, see [57]. Note that the above contraction is the same as the evaluation mapping from the previous papers, see [51, 50, 581. It is easy to see that we have ck+l
on E* @ E@‘” @ E@.
:=
Ck Cl+ idEB’
+ (id E’ ‘%cl) 0 (*E*,E@” @ i&B!)
(46)
166
W. MARCINEK
LEMMA. The contraction mapping c ti braid commutative, i.e. we have (47)
Proof:We prove the lemma only for k =
1 = 1. We have
C2 [ib '8(i&c@ - B)] = [(cl@ id.!r) + (idE 63ci) o (G @ idE)] [idE* 8 (idEBE - B)]
= (ci @ idE) [idi+ @ ((idE@E + 6) o (id - B))] = 0, 0
where the relation (5) has been used.
Let us consider in particular the braided monoidal category corresponding to our quon systems. In this case we obtain a category of GN-graded spaces denoted by C(@, q, b, GN). If we substitute (12) into (37) we obtain the braid symmetry for quons. For example we have
The algebra A for the quon system is an algebra generated by {xi (i = 1, . . ., N} and by the relation zizj = bij& (no summation). (49) This algebra can be regarded as the algebra of all braid equivalent quantum states for left rotations. LEMMA. For the contraction mapping c corresponding to generalized quons we have the relations Ck_i(idE* ‘8 ck) [(idE*@E* - !&*,E*) ‘8 idAk] = 0 (50) on E*@E*@Ak. Proof: We prove the lemma only for k = 2. We calculate ci 0 (idE* ‘8 c~)(z*~ @ x*j @ xm @ 2”) = = =
(51)
On the other hand, we have cl 0 (idEe @ cl) o (B* 8 idE,s,,E)(%*i@ x*j @Jzrn @Ixn) = bij
c~(z*~ @(Pz*~
= bij@m@n
+ bmiSinZm))
+ bijbmajin@.
(52) 0
ON COMMUTATION
167
RELATIONS FOR QUONS
It follows from the above braid commutativity (50) that the contraction mapping c can be projected to the algebra A. Hence, the projected contraction is a mapping Q : E* 631A -+ A given by the relations 01 :=
Cl
E
ak(idE* ‘8 pk) := 4-1 We can prove that for the contraction rule [57]
(.I.),
(53)
o ck.
Q we have the following deformed
Qk+l 0 (idE* 8 m) = m 0 [(ok @ idAl) + (idA& 8 (wl)o (@E-,Ak
@
idAl)] ,
Leibniz
(54)
defined on E* @ dk 18Al. The braiding @E*,Ak is given by @E.,Ak (idE* @ pk) = (% @ idE-)!PE* ,A”.
(55)
It follows from (50) that for quons we also have the relation ol+r 0 (idEB @Jal) 0 [idE*mE=@A- (@P,E* @JidA)] = 0
(56)
on E* ~3E* 63A. We have for example os(z*i 8 &r’)
= @“xl + ck&“Zk.
(57)
5. The Fock-like representation We are going to construct now a representation a : W(B, C) ---+ End(d) of the quantum Weyl algebra W(B,C) on the algebra A. For the generators xi and x*j (i,j = l,..., N) we define azif - m(2 8 f),
!Y&*ifZ cX(x*i8 f)
(58)
for every f E A. We can extend this representation to the whole algebra W(B, C). Obviously, our commutation relations for the creation and anihilation operators should be consistent with the defining relations for the algebra W(B, C). One can also consider a representation of the quantum Weyl algebra W on the algebra A* in a similar way. Let us define a P-bracket by the relation [&, hj]p G [e, .],P(& 8 ;Lj) := c o (id - P)(&, &&), where ki stands for a,*., or a,., c is the composition
(59)
mapping and
P(a*i @ aj.j) := a*(?<@&),
(60)
where 2 stands for Zig or zi, and the braiding 9 on the right-hand side of the above formula is given by (37). We have here the following important theorem.
168
w. MARCINEK
THEOREM. On the algebra A we have the following commutation rdatiom for the representation of the quantum Weyl algebra W(B, C), [U,*i, U2J]P = 6ij, Proof:
[t&l, CS,ci]#= 0,
[&*i 7a,*Li]@ = 0,
(61)
Using (58) we obtain (a,*;
0
U,j)f
0 U,j
(U,i
E
QZ+I 0
(ib
)f - mo(idE
(U,*i OU,*j )f - a~_1 o
@rn)(~*~8 xj
@Im)(xi
@xj
CZJ f),
@ff),
(62)
(idEe @ CX~)(Z*~ 8 x*j 8 f),
for f E Al. We also have 0
C
!P(U,*i @ U,j)f
CO!P(U,i
@O&)f
c 0 !P(U& @ +j)f
E
m o (idE @ Q~+I) 0 (.C ~23 idA)(x*i @ xj C3f),
E m
o
(idE am> 0 (B @ idA)(xi @ xj ~3f),
(63)
= cyl+l o (idEe @ cul)(B* @IidA)(x*i @ x*j @ f).
It follows immediately from the definition (59) of the @-bracket that we have [;li,;lj3*f=
[&O&j
-CO!F(&,@iij)]f.
(64)
NOWwe can calculate the first relation (61) as follows
l!Pf = [qfl 0 (idE*
[Us-i, azi
= V-Lo (aI
@
f) @ m)(x*$ @ xj @If) - m o (idE @JCII)o (c @IidA)] (x*~ @ xj ~23
idA)(x*i 8 xj @ f) = bij f,
where f E A and the relation
(65)
(54) has been used. For the second relation (61) we
obtain [%i, %j]tP.f = m o (m ~3idA) [((idE@E - B) @ idA)(xi @ xj ~3f)] = 0,
(66)
where the associativity and B-commutativity of the algebra A have been used. Finally, we calculate the third commutation relation (61) [U,*i , a,.&f
= alp1 o (idEe @ a~) [idE*wE*BA - (B* @id/t)] (CC** C3x*j Q3f) = 0,
where the relation (56) has been used.
(67) 0
Let us consider the representation of the algebra Wq,b(N) corresponding to quons. In this case we have the following action for the creation and anihilation operators
(68)
ON COMMUTATION
[r&
= gg
RELATIONS FOR QUONS
(i = 1, .*., IV). For the commutation
relations we obtain
aiaj- bijajai = 0,
aaaj -Cijajaf = &jl,
169
ara; - bija;aa = 0,
(69)
where a: := a,.* and ai := a,$. We introduce here the following state vector nlv) = (xi)nl . . . (dy
I%,...,
(70)
up to some normalization factor. For the ground state vector (0) E Id we have as usual a ,*ilO) = 0. The scalar product can be given by the following formula (BJQc := s,,
(,_lP;i)
for 9 E E@‘” and f E EBn, where PA is an operator
(71) defined by induction
PL+l := (id KJPL) o R,,
where P: G id. The operator
R,
(72)
is given by the formula
R n := id + C(1) + @1)~(2)
+ . . . + c(l) . . . c(n-11,
(73)
where @ti) := idE @ . +. @ 0 8 . . . @ idE, c on the i-th place. If the operator bounded operator acting on some Hilbert space such that (i)
c is a
C = C*,
(ii) IICl] 5 1,
(74)
(iii) C? is Yang-Baxter, then-according to Boiejko and Speicher [7]-the scalar product given by (71) is positive definite. Note that the existence of a nontrivial kernel of the operator RI e id + 0 is essential for the nondegeneracy of the scalar product. If the kernel ker RI is nontrivial, then the scalar product (71) is degenerate. Hence we must remove this degeneracy by factoring the mentioned above scalar product by the kernel of RI. In this way we obtain the following product (#B,C
:= (@jc
(75)
for s = P,(g), t = P,(i). One can prove that this scalar product defined. For the quon system we obtain (xcl%)q,b
=
Xn(r)
c
(%I lzr(j,)) -** (xi, Ixr(j,)),
[52, 531 is well
(76)
?rES, where xa,xT Edn,
xa = xil-..xin,
X, = xjl...xjn,
and
(77) (i,j)EJ(r)
(i,j)EJ(a)
170
W. MARCINEK
J(T) := ((4 j) : 1 I 6 j I n}, r(i) > x(j), ni is the number of elements of the set K(r) := {(i,j) E J(T): i = j}. It follows from the mentioned above theorem of Boiejko and Speicher that the corresponding scalar product is positive definite if all qi are real, -1 5 qi < 1, and bij = bij. Acknowledgments The author would like to thank A. Borowiec and J. Lukierski for discussion and critical remarks, and C. Juszczak and R. Ralowski for any other help. REFERENCES 0. W. Greenberg: Phys. Rev. Lett. 64 (1990), 705. 0. W. Greenberg: Phys. Rev. D43 (1991), 4111. R. N. Mohapatra: Phys. Lett. B242 (1990), 407. D. I. Fivel: Phys. Rev. Lett. 65 (1990), 3361. [5] D. Zagier: Commun. Math. Phys. 147 (1992), 199. [6] M. Boiejko and R. Speicher: Interpolation between bosonic and fermionic relations given by generalized Rrownian motions, Preprint FSB 132-691, Heildelberg (1992). [7] M. Boiejko and R. Speicher: Math. Ann. 300 (1994), 97. [8] R. J. Finkenlstein: q-Generating functions and the q-oscilators, Proceedings of the XXI International Conference on Differential Geomem’c Methods in Theoretical Physics, China, 5-9 June 1992, ed. N. Yang, M. L. Ge and X.W. Zhon, pp. 201-205, World Scientific, Singapore 1993. [9] R. Scipioni: Phys. Lett. B327 (1994) 56. [lo] F. Wilczek: Phys. Rev Lett. 48 (1982), 114. [Ill F. Wilczek and A. Zee: Phys. Rev. Lett. 51 (1983), 2250. [12] S. Majid: Anyonic quantum groups, in Spinors, Twistors and Clifford algebras, ed. Z. Oziewicz et al, Kluwer, Dordrecht 1993, p. 327. [13] S. Majid: Czech. J. Phys. 44 (1994) 1073. [14] Yu Ting and Wu Zhao-Yan: Science in China A37 (1994), 1472. [15] S. Meljanac and A. Perica: Mod. Phys. Lett. A!J (1994), 3293. [16] Y. S. Wu: Phys. Rev. Lett. 52 (1984) 2103. [17] Y. S. Wu: Phys. Rev. Lett. 53 (1984), 111. [18] D. P. Arovas, R. Schrieffer, F. Wilczek and A. Zee: Nucl. Phys. B251 (1985), 117. [19] J. K. Kim, W. T. Kim and H. Ihin: J. Phys. A27 (1994) 6067. [20] V. Bardek, M. Doresic and S. Meljanac: Phys. Rev. D49 (1994), 3059. [21] Yue-lin Shen and Mo-lin Ge: Zamoiodchihov-Pad Algebra in 2-component Anyons, Nankai University Preprint, China, August 1995. 1221 W. Marcinek: On quantum Weyl algebras and generalized quons, in Proceedings of the symposium: Quantum Groups and Quantum Spaces, Warsaw, November 20-29, 1995, Poland, ed. R. Budzyiiski, W. Pusz and S. Zakrzewski, Banach Center Publications, Warsaw 1997. [23] Y. S. Wu: J. Math. Phys. 52 (1984), 2103. [24] T. D. Imbo and J. March-Russel: Phys. Lett. B252 (1990), 84. [25] S. Majid: Int. J. Mod. Phys. A5 (1990), 1. [26] S. Majid: J. Math. Phys. 34 (1993), 1176. [27] S. Majid: J. Math. Phys. 34 (1993), 4843. [28] S. Majid: J. Math. Phys. 34 (1993), 2045. [29] S. Majid: Algebras and Hopf Algebras in Braided Categories, in Advances in Hopf Algebras, Plenum 1993. [30] S. Majid: J. Geom. Phys. 13 (1994), 169. [31] S. Majid: AMS Cont. Math. 134 (1992), 219. [32] W. Marcinek: Rep. Math. Phys. 33 (1993), 117. [l] [2] [3] [4]
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05 098, May 1997. [55] W. Marcinek: On algebraic model of composite fermions and bosons, in Proceedings of the IXth Max Born Symposium, Kmpacz, September 25 - 28, 1996, Poland, ed J. Lukierski et al., PWN, Warszawa 1997. [56] J. K. Jain: Phys. Rev. Lett. 63 (1989) 199, Phys. Rev. B40 (1989), 8079; 41 (1990), 7653. [57] W. Marcinek: On *-Algebras and Noncommutative Geometry, Preprint KL-TH 95/19 (1995). [58] J. C. Baez: Adv. Math. 95 (1992), 61. [59] A. Borowiec, V. K. Kharchenko and Z. Oziewicz: Calculi on Clifford-Weyl and exterior algebras for Hecke braiding, Conferencia dictata por Zbigniew Oziewicz at Centro de Investigation en Matematicas, Guanajuato, Mexico, Seminario de Geometria, jueves 22 de abril de 1993, and at the Conference on Differential Geometric Methods, Ixtapa, Mexico, September 1993. [60] E. E. Demidov: Uspekhi Mat. Nauk 48 (1993) 39 (in Russian). [61] E. E. Mukhin: Commun. Alg. 22 (1994), 451. [62] A. Borowiec and V. K. Kharchenko: Coordinate calculi on associative algebras, in Quantum Groups, Formalism and Applications, Ed. J. Lukierski et al, PWN, 1994, pp. 231-272, q-alg/9501018. [63] M. Scheunert: J. Math. Phys. 20 (1979), 712. [64] Z. Oziewicz: Lie algebras for arbitrary grading group, in Differential Gebmehy and Its Applications, ed. J. Janyska and D. Krupka, World Scientific, Singapore 1990. [65] S. MacLane: Categories for Working Mathematician, Graduate Text in Mathematics 5, Springer, Berlin 1971.
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