Table Of ContentHopf Structure and Green Ansatz of
Deformed Parastatistics Algebras
6
0
0 Boyka Aneva1,2 Todor Popov1,3
2
n 1Institutefor NuclearResearch and Nuclear Energy,
a Bulgarian Academy of Sciences
J bld. Tsarigradsko chauss´ee 72, BG-1784 Sofia, Bulgaria
1
1 2 Fakult¨at fu¨r Physik, Ludwig-Maximilians-Universit¨at Mu¨nchen,
D-80333 Mu¨nchen, Germany
2
v 3 Laboratoire d’Informatique de l’Universit´e Paris-Nord,
6 CNRS (UMR 7030) and Universit´e Paris-Nord
1
99 av. J-B Cl´ement, F-93430 Villetaneuse, France
0
2
1
4 Abstract
0
Deformed parabose and parafermi algebras are revised and endowed with
/
h Hopf structure in a natural way. The noncocommutative coproduct allows for
p construction of parastatistics Fock-like representations, built out of the sim-
-
plest deformed bose and fermi representations. The construction gives rise to
h
t quadratic algebras of deformed anomalous commutation relations which define
a
the generalized Green ansatz.
m
:
v
i
X
1 Introduction
r
a
Wigner was the first to remark that the canonical quantization was not the most
generalquantizationscheme consistent with the Heisenberg equations of motions [1].
ParastatisticswasintroducedbyGreen[2]asageneralquantizationmethodofquan-
tum field theory different from the cannonical Bose and Fermi quantization. This
generalized statistics is based on two types of algebras with trilinear exchange rela-
tions, namely the parafermi and parabose algebras.
The representations of the parafermi and parabose algebras are labelled by a
non-negative integer p - the order of parastatistics. The simplest non-trivial repre-
sentations arise for p = 1 and coincide with the usual Bose(Fermi) Fock representa-
tions. The states in a Bose(Fermi) Fock space are totally symmetric(antisymmetric),
i.e., they transform according to the one dimensional representions of the symmet-
ric group. Fock-like representations of parastatistics of order p ≥ 2 correspond to
higher-dimensional representations of the symmetric group in the Hilbert space of
multicomponent fields.
1
At the core of the interest in generalized statistics is (twodimensional) statistical
mechanics of phenomena such as fractional Hall effect, high-T superconductivity.
c
TheexperimentsonquantumHalleffectconfirmthe existenceoffractionallycharged
excitations [3]. Models with fractional statistics and infinite statistics have been
explored, termed as anyon statistics [4] and quon statistics [5].
The attempts to developnonstandardquantumstatistics evolvednaturallyto the
study of deformed parastatistics algebras. The guiding principle in these develop-
ments is the isomorphism between the parabose algebra pB(n), parafermi algebra
pF(n) (with n degrees of freedom) and the universal enveloping algebra of the or-
thosymplectic algebra osp(1|2n), resp. orthogonal algebra so(2n+1). The quantum
countepartspB (n)andpF (n)weredefinedtobeisomorphicasalgebrastothe quan-
q q
tizeduniversalenvelopingalgebras(QUEA)U (osp(1|2n))[6],resp. U (so(2n+1))[7].
q q
In the presentworkwe write a complete basis of defining relationsof the algebras
pB (n) and pF (n)(see Theorem 1) extending what has been done in [6, 7, 8]. The
q q
novelty with respect to the known definition of deformed parastatistics is the system
of homogeneous relations (9, 10). They allow to continue the isomorphism of the
algebras as Hopf algebra morphism (see Theorem 2) which endows the deformed
parastatistics algebra at hand with natural Hopf structure. With the defined Hopf
structure the parastatistics algebraspB (n) and pF (n) become isomorphic as Hopf
q q
algebras to the QUEA U (osp(1|2n)) and U (so(2n+1)), respectively.
q q
The Green ansatz is intimately related to the coproduct on the parastatistics al-
gebras; it was realized that every parastatistics algebra representation of arbitrary
order p arises through the iterated coproduct [9](see also [10]). We make use of the
noncocommutativecoproductonthe HopfparastatisticsalgebraspB (n)andpF (n)
q q
to construct a quadratic algebra which is a deformation of the Green ansatz for the
classical algebras pB(n) and pF(n).
The paper is organized as follows. In section 2 we define the relations of the
quantized parastatistics. Section 3 is devoted to the analysis of the Hopf algebra
structure of the proposed quantized parastatistics algebras. In Section 4 we show
that the q-deformed bosonic (fermionic) oscillatoralgebra arises as the simplest non-
trivial representation of the deformed parastatistics. Further in Section 5 the Green
ansatz is generalized for the deformed parastatistics algebras pB (n) and pF (n).
q q
Throughout the text by an associative algebra we mean an associative algebra
with unit 1 over the complex numbers C.
2 Deformed Parastatistics Algebras
We first recall the definitions of the parastatistics algebras introduced by Green [2]
as a generalization of the Bose-Fermi alternative.
DEFINITION 1 The parafermi algebra pF(n) (parabose algebra pB(n)) is an as-
sociative (super)algebra generated by the creation a+ and annihilation a− operators
i i
for i=1,...,n subject to the relations
[[[[a+,a−]],a+]] = 2δ a+ [[[[a+,a+]],a+]] = 0
i j k jk i i j k
(1)
[[[[a+,a−]],a−]] = −2δ a− [[[[a−,a−]],a−]] = 0
i j k ik j i j k
2
where [[a,b]] = ab−(−1)deg(a)deg(b)ba is the supercommutator and all the generators
of pF(n)(pB(n)) are taken to be even deg(a±)=¯0 (odd deg(a±)=¯1). (1
j j
TheparafermialgebraisisomorphictotheuniversalenvelopingalgebraU(so(2n+
1)) of the orthogonal algebra so(2n+1), pF(n) ≃ U(so(2n+1))[11] while the para-
bose algebra is isomorphic to the universal enveloping algebra U(osp(1|2n)) of the
orthosymplectic superalgebra osp(1|2n), pB(n)≃U(osp(1|2n))[12].
Theideaofquantizationoftheparastatisticsalgebrasisto“quantize”theclassical
isomorphisms i.e., to deform the trilinear relations (1) in such a way that the aris-
ing deformed parafermi pF (n) and parabose pB (n) algebras are isomorphic to the
q q
quantizeduniversal enveloping algebra(QUEA)ofaLie(super)algebra[13,14,15,16]
pF (n)≃U (so(2n+1)) pB ≃U (osp(1|2n)). (2)
q q q q
The proofs of the algebra isomorphisms pB ≃ U (osp(1|2n))[6] and pF (n) ≃
q q q
U (so(2n +1))[7] has shown the equivalence of the paraoscillator definition of the
q
U (osp(1|2n)) and U (so(2n+1)) with their definition in terms of Chevalley gener-
q q
ators. In this way a minimal set of relations (a counterpart of the Chevalley-Serre
relations)isobtainedprovidinganalgebraic(butnotlinear)basisofthedefiningideal
of the QUEA at hand.
Weareinterestedinacompletedescriptionofthedefiningidealfortheparastatis-
tics algebras (i.e., the counterpartof the Cartan-Weyldefinition of the QUEA). This
is not only a question ofpure academic interest, our motivationcame fromthe study
of the Hopf algebraic structure on the parastatistics algebras which to the best of
our knowledge was studied only for some particular cases (see [17] for pB (2)). The
q
complete basis of relations is generated from the known algebraic one and allows for
endowing the pF (n) and pB (n) with a Hopf algebra structure. We now sketch the
q q
procedure of deriving the complete U -linear basis for the parastatistics algebras.
q
The Lie superalgebra osp(1|2n), denoted as B(0|n) in the Kac table [18] has the
sameCartanmatrixasthesimpleB algebraso(2n+1). TheChevalley-Serrerelations
n
of QUEAs Uq(so(2n + 1)) and Uq(osp(1|2n)) with generators q±Hi ≡ q±Hαi and
E±i ≡E±αi, corresponding to the simple roots αi, read
qHiq±Hj =q±HjqHi qHiE±jq−Hi =q±aijE±j 1≤i,j ≤n
[2][E ,E ]=δ [2H ] [[E ,E ]]=[2H ] 1≤i≤n−1
i −j ij i n −n n
[E ,E ]=0 |i−j|≥2
±i ±j
[E ,[E ,E ] ] =0 1≤i≤n−2
±(i+1) ±(i+1) ±i q q−1
[E ,[E ,E ] ] =0 = [[[[E ,E ] ,E ]],E ] 1≤i≤n−1
±i ±i ±(i+1) q q−1 ±(n−1) ±n q−1 ±n ±n q
(3)
where [x,y] = xy − qyx is the q-commutator, α is the only odd simple root of
q n
osp(1|2n) and a =(α ,α ) is the symmetrized Cartan matrix (same for both cases)
ij i j
given by a = 2δ − δ − δ − δ . The quantum bracket is chosen to be
ij ij in i+1j ij+1
[x]:= qx2−q−x2 .
q21−q−12
1Inthisdefinitiononlythelinearlyindependentrelationsarewritten,otherrelationsfollowfrom
the(super-)Jacobi identities
3
The essential point in the proof of the isomorphism is the change of basis for
the QUEA by choosing the orthogonal system of roots ε as an alternative of the
i
simple roots. The ladder operators E+εi and E−εi related to the roots εi are the
parastatistics creation and annihilation operators a+ and a− [6, 8] and the change
i j
ε = n α implies
i Pk=i k
a+ = [E ,[E ,...[E ,E ] ...] ]
i i i+1 n−1 n q−1 q−1 q−1 (4)
a− = [[...[E ,E ] ...,E ] ,E ]
i −n −n+1 q −(i+1) q −i q
With the help of the inverse change α = ε −ε , i < n, and α = ε the cor-
i i i+1 n n
responding change of basis on the Cartan subalgebra reads H = h −h , i < n,
i i i+1
Hn = hn. By construction qhiqhj = qhjqhi. The inverse change of basis allows to
express the Chevalley ladder generators as
Ei = [21]q−hi+1[[a+i ,a−i+1]] E−i = [21][[a+i+1,a−i ]]qhi+1 i<n
(5)
E = a+ E = a−
n n −n n
The complete basis (over C(q)) of relations defining the deformed parastatistics
algebras,i.e., the analog of (1) is given by the following
THEOREM 1 The deformed parafermionic pF (n) (parabosonic pB (n)) algebra is
q q
the associative (super)algebra generated by the creation and annihilation operators a±
i
and Cartan generators q±hi for i=1,...,n subject to the relations
qhi −q−hi
qhia±j q−hi =q±δija±j [[a+i ,a−i ]]= q12 −q−12 =[2hi] (6)
[[[[a+i ,a−j ]],a+k]]q−δikσj,k = [2]δjka+i qσi,jhj +(q−q−1)θi,j;ka+i [[a+k,a−j ]], (7)
[[[[a+i ,a−j ]],a−k]]q−δjkσi,k = − [2]δika−j q−σi,jhi −(q−q−1)θj,i;k[[a+i ,a−k]]a−j (8)
together with the analogues of the Serre relations
[[[[a±,a±]],a±]] +q[[[[a±,a±]],a±]]=0 i <i ≤i (9)
i1 i3 i2 q2 i1 i2 i3 1 2 3
[[a±,[[a±,a±]]]] +q[[a±,[[a±,a±]]]]=0 i ≤i <i (10)
i2 i1 i3 q2 i1 i2 i3 1 2 3
where all the generators a± are taken to be even, deg(a±) = ¯0 (odd, deg(a±) = ¯1)
i i i
and the symbols θ , σ stay for θ = 1ǫ ǫ (ǫ −ǫ ), σ = ǫ +δ or
i,j;k i,j i,j;k 2 ij ijk jk ik i,j ij ij
σ =ǫ −δ . (2
i,j ij ij
The deformed parastatistics algebras admit an anti-involution ∗
(a±)∗ =a∓ (q±hi)∗ =q∓hi (q)∗ =q−1 (11)
i i
induced by the anti-involution on the Chevalley basis (E )∗ =E , H∗ =H .
±i ∓i i i
2) ǫ is the Levi-Civita symbol with ǫij = 1 for i < j. The tensor θi,j;k = −θj,i;k is vanishing
exceptfori<k<j andi>k>j whenittakes values+1and−1,respectively.
4
To prove the theorem we make use of the R-matrix FRT-formalism for QUEA
U (g) of a simple (super-)Lie algebra g (see [15],[16]), introducing the L-functionals
q
for U (g) in the form of upper (lower)-triangular matrices L(+) (L(−))
q
R(+)L(±)L(±) =L(±)L(±)R(+) R(+)L(+)L(−) =L(−)L(+)R(+) (12)
1 2 2 1 1 2 2 1
where L(±) = L(±) ⊗1, L(±) = 1⊗L(±) and R(+) = PRP is the corresponding
1 2
R-matrix for U (g).
q
The (n+1)×(n+1)minorL(+),1≤i,j ≤n+1ofthe (2n+1)×(2n+1)matrix
ij
L(+) for the QUEA U (so(2n+1)) and U (osp(1|2n)) is very simple when expressed
q q
in terms of the generators a±
i
qh1 ω[[a+,a−]] ω[[a+,a−]] ... ω[[a+,a−]] ca+
1 2 1 3 1 n 1
0 qh2 ω[[a+,a−]] ... ω[[a+,a−]] ca+
2 3 2 n 2
(+) 0 0 qh3 ... ω[[a+,a−]] ca+
L = 3 n 3 (13)
(cid:16) ij (cid:17)1≤i,j≤n+1 ... ... ... ... ... ...
0 0 0 ... qhn ca+
n
0 0 0 ... 0 1
where ω =q21 −q−12. The coefficient c=q−21(q−q−1). One has (L(ij+))∗ =Lj(−i).
(±)
The relations (6), (7), (8) involving the entries of the minors of L (13) for
ij
1 ≤ i,j ≤ n+1 follow directly from the RLL-relations(12) with the corresponding
R-matrix upon restricting the indices from 1 to n+1. The restrictionis possible due
to the ice condition[19].
We label the LHS (up to scalars in C(q)) of the homogeneous relations (9,10) by
Λi1,i3 = [[[[a+,a+]],a+]] +q[[[[a+,a+]],a+]] with i <i <i
i2 i1 i3 i2 q2 i1 i2 i3 1 2 3
Λi1,i2 = [[[[a+,a+]],a+]] with i <i
i2 i1 i2 i2 q 1 2 (14)
Λ˜i1,i2 = [[a+,[[a+,a+]]]] +q[[a+,[[a+,a+]]]] with i <i <i
i3 i2 i1 i3 q2 i1 i2 i3 1 2 3
Λ˜i2,i2 = [[a+,[[a+,a+]]]] with i <i
i3 i2 i2 i3 q 2 3
The QUEA U (gl ) has a natural inclusion in U (so(2n+1)) and U (osp(1|2n))
q n q q
beinggeneratedbytheChevalleygeneratorsE±i,1≤i≤n−1andq±hi. (associated
with the A subdiagram in the B Dynkin diagram). The inclusions U (gl ) ֒→
n−1 n q n
pF (n)andU (gl )֒→pB (n)defineanadjointU (gl )-actiononpF (n)andpB (n)
q q n q q n q q
(for i≤n−1)
adEia+j =[Ei,a+j ]qδij−δi+1j =δi+1ja+i adE−ia+j =[E−i,a+j ]qHi =δija+i+1
Let L denote the space of states Λ and Λ˜ where by states we mean the cubic
polynomials determined from (14) up to multiplication with scalars C(q). The ho-
mogeneous relations (9,10) are U (gl )-covariant with respect to the adjoint action.
q n
More precisely one has the following
LEMMA 1 The space L is an irreducible finite-dimensional U (gl )-module with
q n
lowest weight Λn−1,n.(For the proof see the appendix.)
n
5
The distinguished state Λn−1,n expressed in terms of the Chevalley basis E is the
n i
last Serre relation in (3)
Λn−1,n =[[[[E ,E ] ,E ]],E ] =0
n n−1 n q−1 n n q
and thus Λn−1,n has to be set to zero in pF (n) and pB (n). Hence the whole
n q q
representationLbuiltthroughtheU (gl )-adjointactiononthelowestweightΛn−1,n
q n n
istrivialwhichprovesthehomogeneousrelations(9,10)fora+. Theonesfora−follow
i i
by conjugation.
3 Hopf structure on parastatistics algebras
The QUE algebrasU (so(2n+1)) and U (osp(1|2n)) (3) endowed with the Drinfeld-
q q
Jimbo coalgebraic structure [13], [14]
∆H = H ⊗1+1⊗H S(H ) = −H ǫ(H )=0
i i i i i i
∆Ei = Ei⊗1+qHi ⊗Ei S(Ei) = −q−HiEi ǫ(Ei)=0 (15)
∆E−i = E−i⊗q−Hi +1⊗E−i S(E−i) = −E−iqHi ǫ(E−i)=0
becomeHopfalgebraandHopfsuperalgebra,respectively. (3 OnehasS(x∗)=S(x)∗.
The isomorphism(2) between the QUEA U (so(2n+1)) (U (osp(1|2n))) and the
q q
deformedparastatisticsalgebrainducesastructureofaHopf(super)algebraonpF (n)
q
(pB (n)). One can formulate the following
q
THEOREM 2 ThedeformedparafermionicalgebrapF (n),thedeformedparabosonic
q
algebrapB (n)isaHopfalgebra, aHopfsuperalgebra, respectivelywhenendowedwith
q
(i) a coproduct ∆ defined on the generators by ∆q±hi =q±hi ⊗q±hi
∆a+ = a+⊗1+qhi ⊗a++ω [[a+,a−]]⊗a+ (16)
i i i X i j j
i<j≤n
∆a− = a−⊗q−hi +1⊗a−−ω a−⊗[[a+,a−]] (17)
i i i X j j i
i<j≤n
(ii) a counit ǫ defined on the generators by ǫ(q±hi)=1, ǫ(a±)=0
i
(iii) an antipode S defined on the generators by S(q±hi)=q∓hi,
n−i
S(a+) = −q−hia+− (−ω)s W+i W+j1...W+js−1q−hjsa+ (18)
i i X X j1 j2 js js
s=1 i<j1<...<js≤n
n−i
S(a−) = −a−qhi − (ω)s a−qhjsW−js ...W−j2W−j1 (19)
i i X X js js−1 j1 i
s=1 n≥js>...>j1>i
where W+ij =q−hi[[a+i ,a−j ]], W−ji =[[a+j ,a−i ]]qhi and ω =q12 −q−12.
3)ForsuperalgebrasS isagradedantihomomorphism,S(ab)=(−1)deg(a)deg(b)S(b)S(a).
6
Proof: The Hopf structure on the elements of L(+) and L(−) compatible with
the Drinfeldstructure(15)(definedonthe Chevalleybasis)isgivenbythe coproduct
∆L±, the counit ǫ(L(±)) and the antipode S(L(±)) [15]
∆L(±) = L(±)⊗L(±) ǫ(L(±))=δ L(±)S(L(±))=δ (20)
ik Xj ij jk ik ik Xj ij jk ik
(i) For the diagonal elements L(+) =qhi the coproduct formula in (20) yields
ii
∆(L(+))= L(+)⊗L(+) =L(+)⊗L(+) =q±hi ⊗q±hi. (21)
ii X ij ji ii ii
1≤j≤2n+1
(+)
The coproduct of the elements L when 1≤i≤n has the form
in+1
∆L(+) = L(+)⊗L(+) =L(+) ⊗1+ L(+)⊗L(+) (22)
in+1 X ij jn+1 in+1 X ij jn+1
1≤j≤2n+1 i≤j≤n
wherewehaveusedthetriangularityofL(+) andL(+) =1. Insertingintoeq.(22)
n+1n+1
the values L(+) =ca+ (13) and abridging the constant c we get
in+1 i
∆a+ =a+⊗1+ L(+)⊗a+ (23)
i i X ij j
i≤j≤n
which completes the proof of (16) in view of (13). Then ∆a− =(∆a+)∗.
i i
(ii) It follows from the definition of the counit in (20).
(iii) For the diagonal elements the antipode formula in (20) implies S(L(+)) =
ii
(L(+))−1henceS(q±hi)=q∓hi. Forthenondiagonalelementsduetothetriangularity
ii
of L(+) the antipode formula (20) gives rise to the following system of equations
L(+)S(L(+) )=δ =⇒ L(+)S(L(+) )=−L(+) . (24)
X ij jn+1 in+1 X ij jn+1 in+1
i≤j≤n+1 i≤j≤n
Here we have made use of S(L(+) ) = S(1) = 1. In view of S(L(+) ) = cS(a+)
n+1n+1 in+1 i
this is a linear triangular system for S(a+) which after normalisation takes the form
i
S(a+)+ω Wi+S(a+) = −q−hia+ where Wi+ =q−hi[[a+,a−]] (25)
i X j j i j i j
i<j≤n
andthe solutionofthissystemyieldseq.(18). TheantipodesS(a−)(19)areobtained
i
through the conjugation, S(a−)=(S(a+))∗. (cid:3)
i i
This theorem is interesting in its own because it defines the Hopf structure on
anotherbasis of generatorsfor QUEAofthe algebraso(2n+1)and the superalgebra
osp(1|2n).
4 The oscillator representations
Theunitaryrepresentationsπ oftheparastatisticsalgebraspB(n)andpF(n)(eq. 1)
p
with unique vacuum state are indexed by a non-negative integer p [20] (see also [21]
7
and references therein). The representation π is the lowest weight representation
p
with a unique vacuum state |0i annihilated by all a− and labelled by the order of
i
parastatistics p
π (a−)|0i=0 π (a−)π (a+)|0i=pδ |0i, π (x)|0i=ǫ(x)|0i (26)
p i p i p j ij 0
where the vacuum representation, i.e., the trivial one corresponds to the counit ǫ of
the Hopf parastatistics algebra. In the representation π (26) of the nondeformed
p
parastatistics algebras (1) the hamiltonian H = h = 1[a+,a−] and the number
i i 2 i i ∓
operator N =a+a− associated to the i-th paraoscillator are related by
i i i
p
H =h =N ∓ (27)
i i i
2
where the upper (lower) sign is for parafermions (parabosons).
Inthe representationπ ofthedeformedparastatisticsalgebrasthe quantumana-
p
logue of the relation (27) holds
[a+,a−] =[2]H =[2h ]=[2N ∓p]
i i ∓ i i i
which implies the deformed analogue of the π defining condition (26)
p
π (a−)π (a+)|0i=[p]δ |0i, π (x)|0i=ǫ(x)|0i (28)
p i p j ij 0
The constant∓[p]/[2]plays the roleof energyofthe vacuum as the constant∓p/2in
(27) for the nondeformed algebras.
Thealgebraoftheq-deformedfermionic(bosonic)oscillatorsF (n)(B (n))arises
q q
as a representation π of order p=1 of the pF (n)(pB (n))
q q
a+a+±q∓ǫija+a+ = 0 a−a−±q∓ǫija−a− = 0
i j j i i j j i
a−i a+i ±qa+j a−i = q±Ni a−i a+i ±q−1a+i a−i = q∓Ni (29)
a+a−±q∓ǫjia−a+ = 0 a−a+±q∓ǫjia+a− = 0 i6=j
i j j i i j j i
We have adopted the notaion π(x)=x and use N =h ∓ 1.
i i 2
Theanalysis[22]ofthepositivityofthenormforthepB (n)andpF (n)represen-
q q
tations in the simplest case p = 1 shows that such unitary representations (realized
as finite dimensional factor representions) exist only for q being a root of unity.
Remark. Unlike the case ofpB (n)and pF (n), the deformedrelationsofbosonic
q q
and fermionic oscillator algebras (29) do not define Hopf ideals.
5 Green Ansatz
The Green ansantz was introduced by Green in the same paper [2] in which he de-
fined parastatistics. We briefly recall it and then bring it in a form convenient for
deformation.
Let us consider a system with n degrees of freedom quantized in accordance with
theparafermiorparabosestatisticsoforderp,i.e.,asystemofnparaoscilatorswhich
8
isaparticularrepresentationπ (oforderp)oftheparastatisticsalgebrawithtrilinear
p
exchange relations (1).
The Green ansatz states that the parafermi (parabose) oscillators a+ and a− can
i i
be represented as sums of p fermi (bose) oscillators
p
π (a±)= a±(r) (30)
p i X i
r=1
satisfyingquadratic commutationrelationsof the same type (i.e., fermifor parafermi
and bose for parabose) for equal indices (r)
[a−(r), a+(r)] =δ , [a−(r),a−(r)] =[a+(r),a+(r)] =0, (31)
i k ± ik i k ± i k ±
and of the opposite type for the different indices
[a−(r),a−(s)] =[a+(r),a+(s)] =[a−(r), a+(s)] =0 r 6=s. (32)
i k ∓ i k ∓ i k ∓
The upper (lower) signs stay for the parafermi (parabose) case.
The coproduct endows the tensor product of A-modules of the Hopf algebra A
with the structure of an A-module. Thus one can use the coproductfor constructing
arepresentationoutofsimpleones. Thesimplestrepresentationsoftheparastatistics
algebras are the oscillator representations π (with p= 1). Higher representations π
p
of parastatistics of order p≥2 arise through the iterated coproduct [9].
Let us denote the (p-fold) iteration of the coproduct by (4
∆(0) =ǫ, ∆(1) =id, ∆(2) =∆, ... ∆(p) =(∆⊗1⊗...⊗1)◦∆(p−1) (33)
p−1
| {z }
andπdenotestheprojectionfromthe(deformed)parafermiandparabosealgebraonto
the(deformed)fermionicF(F )andbosonicB(B )Fockrepresentation,respectively.
q q
PROPOSITION 1 The Green ansatz is equivalent to the commutativity of the fol-
lowing diagrams
pF(n) −∆→(p) pF(n)⊗p pB(n) −∆→(p) pB(n)⊗p
π ց ↓π⊗p π ց ↓π⊗p (34)
p p
F(n)⊗p B(n)⊗p
Proof: Using the coproduct of the Theorem 2 for q = 1 and projecting on the
Fock representation we can choose the components of the Green ansatz to be the
summands in the expressions
p p
π⊗p◦∆(p)(a±)= 1⊗...⊗1⊗π(a±)⊗1⊗...⊗1:= a±(r) (35)
i X i X i
r=1 r−1 p−r r=1
| {z } | {z }
ThecheckthattheGreencomponentsa±(r)satisfythebilinearcommutationrelations
i
(31)and(32)isdirect,howeveronehastokeepinmindthatthetensorproductisZ -
2
gradedintheparabosecaseandnon-gradedintheparafermicase,whichexplainswhy
4)Thedefinitionof∆(p) extended withthecounitǫisconsistentwithπ0=ǫ
9
theanomalouscommutationrelations(32)appear. Weemphasizethatthegradingof
the tensor product turns out to be the opposite to the (independent) grading of the
bose or fermi algebra which appears on each site (r).
The diagrams (34) are commutative if and only if
π (a±)=π⊗p◦∆(p)(a±) (36)
p i i
which is exactly the statement of the Green ansatz (30). (cid:3)
We are now in a position to extend the Green ansatz to the deformed parafermi
pF (n) and parabose pB (n) algebras. The simplest representation of pF (n) and
q q q
pB (n) of parastatistics order p = 1, are the deformed fermionic F and bosonic B
q q q
Fock representations, respectively and let π be the projection on these Fock spaces.
DEFINITION 2 The system of quadratic exchange relations stemming from the
commutativity of the diagrams
pF (n) −∆→(p) pF (n)⊗p pB (n) −∆→(p) pB (n)⊗p
q q q q
π ց ↓π⊗p π ց ↓π⊗p (37)
p p
F (n)⊗p B (n)⊗p
q q
is the deformed Green ansatz of parastatistics of order p. Here ∆(p) stays for the p-
fold non-cocommutative coproduct (33) on the Hopf algebras pF (n) and pB (n) (see
q q
Theorem 2).
Letusshowtheconsistencyofthecondition(28)withthedeformedGreenansatz.
The vacuum state |0i(p) of the representation π is to be identified with the tensor
p
powerofthe oscillator(p=1)vacuum,|0i(p) =|0i⊗p. Evaluatingthe iteratedgraded
commutator (6)
(qhi)⊗p−(q−hi)⊗p
∆(p)[[a+, a−]]=[[∆(p)a+, ∆(p)a−]]= (38)
i i i i q12 −q−12
on the vacuum state |0i⊗p in the oscillator representations π⊗p we get the defining
condition (28) of the deformed π
p
∓π⊗p◦∆(p)[[a+, a−]]|0i(p) =π (a−)π (a+)|0i(p) =[p]|0i(p) (= qp2 −q−p2 |0i(p))
i i p i p i q12 −q−12
since π(qhi)=qNi∓21, which proves the consistency.
The Green components a±(r) in a pF (n) or pB (n) representation π of paras-
i q q p
tatistics of order p will be chosen to be
a+(r) = π⊗p◦∆(r−1)⊗1⊗∆(p−r) n L(+)⊗a+⊗1
i (cid:16)Pk=1 ik k (cid:17) (39)
a−(r) = π⊗p◦∆(r−1)⊗1⊗∆(p−r) n 1⊗a−⊗L(−)
i (cid:16)Pk=1 k ki (cid:17)
Note that the conjugation ∗ acts as reflection on the Green indices (r)
(a±(r))∗ =a∓(r∗) r∗ =p−r+1.
i i
10
