Noncommutative Superspaces Covariant Under $OSp_q(1|2)$ Algebra

Abstract

Using the corepresentation of the quantum group $ SL_q(2)$ a general method for constructing noncommutative spaces covariant under its coaction is developed. The method allows us to treat the quantum plane and Podle\'s' quantum spheres in a unified way and to construct higher dimensional noncommutative spaces systematically. Furthermore, we extend the method to the quantum supergroup $ OSp_q(1|2).$ In particular, a one-parameter family of covariant algebras, which may be interpreted as noncommutative superspheres, is constructed.

Full text

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arXiv:math/0511436v1 [math.QA] 17 Nov 2005
Lie Theory and
Its Applications in Physics VI
ed. V.K. Dobrev et al, Heron Press, Sofia, 2006
Noncommutative Superspaces Covariant
Under OSpq(1|2) Algebra
N. Aizawa1, R. Chakarabarti2,
1Department of Mathematics and Information Sciences, Graduate School of Sci-
ence, Osaka Prefecture University, Daisen Campus, Sakai, Osaka 590-0035,
Japan
2Department of Theoretical Physics, University of Madras, Guindy Campus,
Chennai 600 025, India
Abstract
Using the corepresentation of the quantum group SLq(2) a general method for con-
structing noncommutative spaces covariant under its coaction is developed. The method
allows us to treat the quantum plane and Podle´ s’ quantum spheres in a unified way and
to construct higher dimensional noncommutative spaces systematically. Furthermore, we
extend the method to the quantum supergroup OSpq(1|2). In particular, a one-parameter
family of covariant algebras, which may be interpreted as noncommutative superspheres,
is constructed.
1Introduction
Quantum groups provide a very powerful tool for investigations of noncommu-
tative geometry, since they may be regarded as a noncommutative extension of
linear Lie groups. Pioneering works by Manin [1], Woronowicz [2], Wess and
Zumino [3] are followed by hundreds of publications (see for example [4, 5]
and references therein). One way of using quantum groups for noncommutative
geometry is, by regarding them as an example of noncommutative manifolds,
to develop a harmonic analysis on quantum groups. Another way, which may
be more familiar to large class of physicists, is to regard a quantum group as
a transformation matrix of vectors. Because of the noncommutative nature of
quantum groups, the vectors transformed by a quantum group are a priori non-
commutative. Namely, the components of the vectors have nontrivial commu-
tation relations. In order to fit such vectors in theories of physics, we require
covariance, that is, the commutation relations are preserved by quantum group
transformations. Noncommutative vectors obeying a covariant algebra may be
given geometrical interpretation.
Let us consider SUq(2) as an example. If we consider a covariant algebra
transformed by the fundamental representation, it may be interpreted as a non-
commutative analogue of two dimensional flat space [1]. While if we take a
1

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2Noncommutative Covariant Spaces of OSpq(1|2)
covariant algebra for adjoint representation, it may be regarded as a noncommu-
tative extension of 3-sphere [6]. Higher dimensional representations will give
higher dimensional noncommutative spaces. However, no such work has been
done because mainly of computational difficulty. Furthermore, there are only a
few works on noncommutativeanalogues of superspaces in the context of quan-
tum groups despite the fact that supersymmetry is one of the most important
notions of theoretical physics.
In the present work, using the corepresentations of quantum groups, we de-
velopa generalmethodfor constructingnoncommutativespaces for the simplest
and the most important quantum (super) groups SLq(2) and OSpq(1|2). In the
first part of this paper (§2 and §3), the case SLq(2) is considered. It will be seen
that, by our method, the quantum plane and the quantum spheres are treated on
the same footing and that the higher dimensional noncommutative spaces may
be constructed systematically. In the second part (§4), we extend the method
to OSpq(1|2). As an application of our method, noncommutative superspace
and a one-parameterfamily of noncommutativesuperspheres are explicitly con-
structed. Finally §5 is devoted to concluding remarks.
2
SLq(2) and its corepresentations
This section is a brief review of the definitions and representations for the quan-
tum group SLq(2) and the quantum algebra Uq[sl(2)] that is dual to SLq(2).
There are several goodtextbookson this topics. Readers may refer, for example,
to [4,5] and references therein.
The quantum group SLq(2) is generated by four elements a,b,c and d sub-
ject to the relations
ab = qba, ac = qca,bd = qdb,
ad − da = (q − q−1)bc,cd = qdc,
ad − qbc = da − q−1bc = 1.
bc = cb,
(1)
As is well-known,the coproduct(∆), the counit (ǫ) and the antipode (S) defined
as follows make SLq(2) a Hopf algebra:
∆
?a
?a
b
dc
?
?
=
?a
?
b
dc
?
·
⊗
?ab
dc
?
,ǫ
?ab
dc
?
=
?10
10
?
,
S
b
dc
=
d−q−1b
a−qc
?
.
(2)
With the Hopf algebra mappings, we define a corepresentation of a quantum
group. A vector space V is called a right SLq(2)-comodule if there exists a
linear mapping ϕR: V → V ⊗ SLq(2) satisfying
(ϕR⊗ id) ◦ ϕR= (id ⊗ ∆) ◦ ϕR,
Similarly, the left SLq(2)-comodule is defined as a vector space V equipped
with a linear mapping ϕL: V → SLq(2) ⊗ V such that
(id ⊗ ϕL) ◦ ϕL= (∆ ⊗ id) ◦ ϕL,
(id ⊗ ǫ) ◦ ϕR= id.
(3)
(ǫ ⊗ id) ◦ ϕL= id.
(4)

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N. Aizawa and R. Chakrabarti3
The mapping ϕR(ϕL) is called a corepresentation,or, equivalently,a right (left)
coaction of SLq(2) on V . It is known that each irreducible corepresentation
of SLq(2) is, as classical SL(2), specified by the highest weight j which takes
any nonnegative integral or half-integral values. Let V(j)be a right SLq(2)-
comodule with the highest weight j and {ej
basis:
ϕR(ej
?
The corepresentations of SLq(2) have been obtained explicitly [7–9]. We here
give j = 1/2 and j = 1 corepresentation matrices as an example:
m, m = j,j − 1,··· ,−j} be its
T(j)
m′m∈ SLq(2)
m) =
m′
ej
m′ ⊗ T(j)
m′m,
(5)
T(1/2)=
?ab
dc
?
,
(6)
T(1)=


a2
(1 + q−2)1/2ab
1 + (q + q−1)bc
(1 + q−2)1/2cd
b2
(1 + q−2)1/2ac
c2
(1 + q−2)1/2bd
d2

.
(7)
A comoduleof a quantumgroupis, in general, a modulei.e. a representation
space of the dual quantum algebra. We define the action of Uq[sl(2)] on V(j)by
Xej
m= ((id ⊗ X) ◦ ϕR)(ej
m) =
?
m′
ej
m′
?
X,T(j)
m′m
?
,X ∈ Uq[sl(2)] (8)
where ? , ? : Uq[sl(2)] ⊗ SLq(2) → C is the duality pairing of two Hopf
algebras. Thenit maybeverifiedthatthematrix
?
X,T(j)
m′m
?
givesanirreducible
representationofUq[sl(2)] with the highestweight j. The productspace V(j1)⊗
V(j2)is, in general, reducible and is decomposed into irreducible spaces as
j1⊗ j2= j1+ j2⊕ j1+ j2− 1 ⊕ ··· ⊕ |j1− j2|.
The decomposition is carried out by the Clebsch-Gordan coefficients (CGC)
(9)
eJ
M(j1,j2) =
?
m1,m2
Cj1j2J
m1m2Mej1
m1⊗ ej2
m2.
(10)
The CGC satisfy the following orthogonality relations
?
?
j,m
Cj1j2j
m1m2mCj1j2j
m′
1m′
2m′= δm1m′
1δm2m′
2,
(11)
m1,m2
Cj1j2j
m1m2mCj1j2j′
m1m2m′ = δjj′δmm′.
(12)
Two corepresentations are also coupled by CGC. The coupling is given by the
formula that is called the Wigner’s product law [10]:
δjj′T(j)
mm′ =
?
m1,m2
m′
1,m′
2
Cj1j2j
m1m2mCj1j2j′
m′
1m′
2m′T(j1)
m1m′
1T(j2)
m2m′
2.
(13)

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4Noncommutative Covariant Spaces of OSpq(1|2)
3Covariant algebras of SLq(2)
3.1General prescription
In this section, we give a general prescription to construct SLq(2)-covariant
algebras. By covariant algebras, we mean algebras whose defining relations
are preserved under the right coaction of SLq(2) defined by (5). Probably, the
simplest way to find such an algebra is to introduce an algebraic structure on the
comodule V(j). Let µ be a product in V(j), i.e., µ(f ⊗ g) = fg, f,g ∈ V(j).
We specifically consider the following composite object
µ(eJ
M(j,j)) =
?
m1,m2
CjjJ
m1m2Mej
m1ej
m2.
(14)
The right coaction on (14) is shown to be
ϕR◦ µ(eJ
M(j,j)) =
?
M′
µ(eJ
M′(j,j)) ⊗ T(J)
M′M.
(15)
The proof may be done in a straightforward way by inverting the relation (14)
ej
m1ej
m2=
?
JM
CjjJ
m1m2Mµ(eJ
M(j,j)),
(16)
and subsequently using the product law (13)
ϕR◦ µ(eJ
=
M(j,j))
?
?
2
m1m2
CjjJ
m1m2MϕR(ej
m1)ϕR(ej
m2)
=
m1m2
m′
1m′
CjjJ
m1m2Mej
m′
1ej
m′
2⊗ T(j)
m′
1m1T(j)
m′
2m2
(16)
=
?
m1m2
m′
1m′
2
?
J′M′
CjjJ
m1m2MCjjJ′
m′
1m′
2M′µ(eJ′
M′(j,j)) ⊗ T(j)
m′
1m1T(j)
m′
2m2
(13)
=
?
M′
µ(eJ
M′(j,j)) ⊗ T(J)
M′M.
Employing (15) we now extract a set of covariant relations under ϕR. The
J = 0 relation ϕR◦ µ(e0
scalar under the right coaction. It may be equated to a constant parameter r
0(j,j)) = µ(e0
0(j,j)) signifies that µ(e0
0(j,j)) is a
µ(e0
0(j,j)) =
?
mtransform identically under ϕR. Therefore
m. It may be noted that the following
m1,m2
Cjj 0
m1m20ej
m1ej
m2= r.
(17)
If J = j, then µ(ej
µ(ej
relations are covariant
m(j,j)) and ej
m(j,j)) is, in general, proportionalto ej
µ(ej
m(j,j)) =
?
m1,m2
Cjjj
m1m2mej
m1ej
m2= ξej
m,
(18)

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N. Aizawa and R. Chakrabarti5
where the proportionality constant ξ → 0 as q → 1. For J ?= 0,j, the ele-
ment µ(eJ
differently. The relevant covariant relations are, therefore, of the form
M(j,j)) can not be a scalar, nor proportional to eJ
Mas they transform
µ(eJ
M(j,j)) =
?
m1,m2
CjjJ
m1m2Mej
m1ej
m2= 0.
(19)
As will be seen from the examples given in the next subsection, the simul-
taneous use of all relations from (17) to (19) gives an inconsistent result, since
some of them do not have correct classical limits. In order to obtain a consis-
tent covariant algebra, we have to make a choice regarding the relations to be
used for defining the algebra. Then the consistency has to be verified. As it
is clear from the above discussion, the covariant algebras can have at most two
more parameters (r,ξ) in addition to the deformation parameter q. It is empha-
sised that the origin of the parameters is clearly explained in the framework of
the representation theory. We have formulated a method to construct SLq(2)-
covariant algebras with respect to the right coaction. It is possible to repeat the
same discussion for the left coaction.
3.2Quantum plane and quantum spheres
We apply the general prescriptionin the previoussubsection to j = 1/2 and j =
1 corepresentations. As will be seen, the obtained covariantalgebras correspond
to the quantum plane of Manin for j = 1/2 and the quantum spheres of Podle´ s
for j = 1.
Let us start with j = 1/2 case where the relevant tensor product decom-
position is given by 1/2 ⊗ 1/2 = 1 ⊕ 0. We denote the basis of V(1/2)by
(x,y) = (e1/2
by (6). Using explicit formula of CGC given in [4,5], we obtain from (17) for
J = 0
xy − qyx = r.
If we set r = 0, then (20) is reduced to the quantum plane relation. For J = 1,
we obtain, from (19), unacceptablerelations such as x2= y2= 0. Thus we take
only (20) as defining relations of our covariant algebra.
We next investigate j = 1 case, namely, the adjoint corepresentation of
SLq(2). Since the adjoint corepresentation of the classical SL(2) corresponds
to the fundamental corepresentation of SO(3), the covariant algebra may be
interpreted as a sphere. The relevant tensor product decomposition is 1 ⊗ 1 =
2⊕1⊕0.The basis of V(1), on which the quantummatrix (7) coacts, is denoted
by xm= e1
1/2,e1/2
−1/2). The quantum matrix which coacts on this basis is given
(20)
m. The covariant relation for J = 0 is obtained via (17)
x2
0− qx1x−1− q−1x−1x1= r.
(21)
Explicit constructions for the J = 1 case are obtained via (18)
(1 − q2)x2
x−1x0− q2x0x−1= ξx−1,
0+ qx−1x1− qx1x−1= ξx0,
x0x1− q2x1x0= ξx1.
(22)

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6Noncommutative Covariant Spaces of OSpq(1|2)
For J = 2, we obtain, from (19), unacceptable relations such as x2
we take (21) and (22) as defining relations of our covariant algebra. We need to
check the following conditions in order to verify whether or not the algebra is
well-defined:
(a) The constant r commutes with all generators
±1= 0. Thus
(b) Product of three generators, say x1x0x−1, has two ways of reversing its
ordering:
x1x0x−1
−→x1x−1x0
րց
x0x1x−1
x−1x1x0.
ցր
x0x−1x1
−→x−1x0x1
These two ways give the same result.
It is straightforward to verify that the conditions (a) and (b) are satisfied.
The covariant algebra defined by (21) and (22) was first introduced by Podle´ s
and interpreted as a noncommutative extension of 2-sphere [6]. The constant
r in (21) may be regarded as square of radius. While the parameter ξ in (22),
which goes to zero in the classical limit, does not exist in a commutative sphere.
We thus obtained a one-parameter family of noncommutative2-spheres.
By taking the higher dimensional corepresentations,one may systematically
obtain higher dimensional noncommutativespaces covariant under SLq(2).
4Covariant superspaces of OSpq(1|2)
4.1 General prescription
In this section, the discussions in the preceding sections are extended to a quan-
tum supergroup in order to obtain noncommutative superspaces. Since the rep-
resentation theories of quantum algebra Uq[sl(2)] and quantum superalgebra
Uq[osp(1|2)] are quite parallel, one can establish a prescription for construct-
ing OSpq(1|2)-covariant algebras similar to the one for SLq(2) by repeating
the same discussion as §3. We give our results without proofs, since the results
in this section have already published in [11]. Readers may refer to [11] for
details.
The universal enveloping algebra U = Uq[osp(1|2)] is generated by the two
even K±1, and the two odd elements v±satisfying the commutation properties
[12]
KK−1= K−1K = 1,
{v+,v−} = −K2− K−2
Kv±= q±1/2v±K,
q4− q−4.
(23)
Each irreducible representation (finite dimensional) of the algebra U is specified
by a nonnegative integer ℓ and the corresponding (2ℓ + 1) dimensional repre-
sentation space V(ℓ)is also Z2graded. Let { eℓ
m(λ) | m = ℓ,ℓ − 1,··· ,−ℓ }

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N. Aizawa and R. Chakrabarti7
be a basis of V(ℓ), where each basis vector has a definite parity. The index
λ = 0,1 specifies the parity of the highest weight vector eℓ
eℓ
Tensor product of two irreducible representations of U has been discussed
in [12,13]. It is, in general, reducible and decomposed into a direct sum of ir-
reducible representations. The rule of decomposition is identical to the classical
case:
ℓ(λ). The parity of
on eℓ
m(λ) equals ℓ − m + λ, as it is obtained by the application of vℓ−m
−
ℓ(λ).
ℓ1⊗ ℓ2= ℓ1+ ℓ2⊕ ℓ1+ ℓ2⊕ ℓ1+ ℓ2− 1 ⊕ ··· ⊕ |ℓ1− ℓ2|.
The irreducible basis of the tensor product representations is obtained by using
the CGC:
(24)
eℓ
m(ℓ1,ℓ2,Λ) =
?
m1,m2
Cℓ1 ℓ2 ℓ
m1m2meℓ1
m1(λ) ⊗ eℓ2
m2(λ),
(25)
where m = m1+ m2, and Λ = ℓ1+ ℓ2+ ℓ (mod 2) is the parity of the high-
est weight vector eℓ
in [11,13] and the orthogonality relations similar to (11), (12) have also been
obtained.
On the contrary to SLq(2), explicit expressions of corepresentationfor A =
OSpq(1/2) have not known yet. Employing the duality of the algebras U and
A, one can obtain the hithertounknowncorepresentationsof A from the already
knownirreduciblerepresentationsofU. LetDℓ(X;λ) bearepresentationmatrix
of X ∈ U on V(ℓ)
Xeℓ
m(λ) =
?
We define a corepresentation matrix T(ℓ)(λ) of A via the duality relation
ℓ(ℓ1,ℓ2,Λ). The CGC for the algebra U has been computed
m′
eℓ
m′(λ)Dℓ
m′m(X;λ).
(26)
Dℓ
m′m(X;λ) = (−1)ˆ
X(ℓ−m′+λ)?
X,T
(ℓ)
m′m(λ)
?
,
(27)
and the parity may be assigned as
?
T
(ℓ)
m′m(λ) = m′+ m (mod 2).
(28)
With this corepresentationmatrix, one can show that V(ℓ)is a right comodule of
A and that T(ℓ)(λ) satisfies the product law similar to (13). It is not difficult to
find T(ℓ)(λ) for lower values of ℓ from (27). For ℓ = 1, we obtain
T(1)(0) =


a
γ
c
α
e
δ
b
β
d

,T(1)(1) =


a−α
e
−δ
b
−γ
c
−β
d

,
(29)
where the entries in latin (greek) characters are of even (odd) parity. For ℓ = 2,

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8Noncommutative Covariant Spaces of OSpq(1|2)
the entries of corepresentation matrix are quadratic in ℓ = 1 entries
T(2)(0) =





a2
κ1aακ3abκ1αbb2
κ1aγ
κ3ac
κ1γc
c2
ae + q−1γα
κ2(aδ + cα)
γδ + q−1ce
κ1cδ
κ2(aβ + q−1γb)
ad + q−1[2]αδ + q−2bc
κ2(γd + q−1cβ)
κ3cd
−αβ + q−1eb
κ2(αd + δb)
ed + q−1βδ
κ1δd
κ1bβ
κ3bd
κ1βd
d2



(30)

,
where
κ1=
?
[4]
q[2],κ2=
?
q−1[3],κ3= κ1κ2,[n] =q−n/2− (−1)nqn/2
q−1/2+ q1/2
.
(31)
We are ready to discuss covariant algebras for the quantum supergroup A.
Following the arguments for SLq(2), we define the composite object
EL
M≡ µ(eL
M(ℓ,ℓ,Λ)) =
?
m1,m2
Cℓ ℓL
m1m2Meℓ
m1(λ)eℓ
m2(λ),
(32)
where Λ = L (mod 2). Then the right coaction on EL
Mis shown to be
ϕR(EL
M) =
?
M′
EL
M′ ⊗ T
(L)
M′M(Λ).
(33)
A following set of covariant relations are extracted from (33) depending on the
values of L
E0
0(0) =
?
?
m1,m2
Cℓ ℓ0
m1m20eℓ
m1(λ)eℓ
m2(λ) = r,(L = 0)
(34)
Eℓ
m(λ) =
m1,m2
Cℓ ℓ ℓ
m1m2meℓ
m1(λ)eℓ
m2(λ) = ξeℓ
m(λ),(L = ℓ)(35)
EL
M=
?
m1,m2
Cℓ ℓL
m1m2Meℓ
m1(λ)eℓ
m2(λ) = 0,(L ?= 0,ℓ)
(36)
In (35), the proportionality constant ξ is of even parity if λ = ℓ (mod 2), or odd
parity if λ ?= ℓ (mod 2). As alreadyseen in the case of SLq(2), the simultaneous
use of all relations from (34) to (36) gives an inconsistent result. We have to
make a choice of appropriate relations defining a covariant algebra. Then we
should check the consistency conditions (a) and (b) given in §3.2.
4.2Quantum superspace and quantum superspheres
As an application of the prescription given in the previous subsection, let us
examine the covariant algebras corresponding to ℓ = 1,2 with λ = 0. The
covariant algebra for ℓ = 1 is identified with the quantum superspace. The one
for ℓ = 2 is interpreted as a noncommutative extension of supersphere.

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N. Aizawa and R. Chakrabarti9
We start with the case of ℓ = 1, where the relevant tensor product decompo-
sition is givenby 1⊗1 = 2⊕1⊕0.We denote the basis of V(1)by zm= e1
on which the quantum supermatrix T(1)(0) in (29) coacts. Thus z±1are parity
even and z0is parity odd. Using the CGC given in [11], we obtain from (34) for
L = 0
q1/2z−1z1+ z2
m(0)
0− q−1/2z1z−1= r.
(37)
For L = 1, we have Λ ?= λ, and, therefore, the parameter ξ is a Grassmann
number:
−q1/2z0z1+ q−1/2z1z0= ξz1,
z−1z1+ (q−1/2+ q1/2)z2
q1/2z−1z0− q−1/2z0z−1= ξz−1.
0− z1z−1= ξz0,
(38)
For L = 2, we obtain, using (36), unacceptable relations such as z2
thus take (37) and (38) as defining relations of our covariant algebra. It is not
difficult to see that the consistency condition (a) is satisfied, while the condi-
tion (b) requires setting ξ = 0. Therefore, we define our covariant algebra by
combining relations (37) and (38), while maintaining ξ = 0:
1= 0. We
z1z0= qz0z1,
z1z−1= q2z−1z1− q(q−1/2+ q1/2)r,
z2
z0z−1= qz−1z0,
(39)
0= −q−1[2]z1z−1− q−1r.
This may be interpreted as the most general form of a quantum superspace. The
simplest quantum superspace corresponds to the choice of r = 0.
We next investigate a covariant algebra for ℓ = 2. This may be interpreted
as a supersymmetric extension of a noncommutative sphere, since ℓ = 2 corre-
sponds to the adjoint representation of the algebra A. The quantum supersphere
may have applications in integrable quantum field theories. Some models of
integrable field theory which has osp(1|2) symmetry and in which supersphere
appearsas atargetspacehavebeenconsidered[14]. Ifanextensionofsuchmod-
els having quantum algebra symmetry is considered, the quantum supersphere
will also appear as a target space.
Letus denotethe basis of V(2)byYm= e2
Y0,Y±2areofevenparity,andY±1areofodd. We seekacovariantalgebraunder
the right coaction of the quantumsupermatrix T(2)(0) in (30). In order to regard
the obtained covariant algebra as a noncommutativeextension of a supersphere,
we needone relation definingthe radius of the supersphereand ten commutation
relations ofsuperspherecoordinatesYm. In additionto those relations, two more
relationswhichrelateY2
may loose their nilpotency at the quantum level. We thus have to find thirteen
relations to define the quantum supersphere.
The relation for radius is obtained via (34), i.e. L = 0
m(0), wherem = 0,±1,±2.Here
±1toothercoordinateareneeded,sincetheoddelements
q−1Y2Y−2− q−1/2Y1Y−1− Y2
0+ q1/2Y−1Y1+ qY−2Y2= r,
(40)

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10Noncommutative Covariant Spaces of OSpq(1|2)
where r is a constant correspondingto the square of radius. As commutation re-
lations of the coordinates Ym, we admit the sets of covariant relations for L = 2
and L = 3 obtained via (35) and (36). Each of them contains five and seven
relations, respectively. We now have obtained the required number of commu-
tation relations and it is easy to verify that their classical limit coincide with the
commutative supersphere. To test whether they consistently define an algebra,
we need to check for the conditions (a) and (b) mentioned in (3.2). It may be
provedby direct computationthat the said conditions are, however,not satisfied.
In order to make the algebra well-defined, we incorporate the L = 1 relations.
With the aid of L = 1 relations, one can verify that the consistency conditions
are satisfied. The remaining L = 4 relations can not be incorporated, since they
contain unacceptable equations such as Y2
As a result, we have sixteen relations. As all the relations are covariant by
construction, their linear combinations are also covariant. Taking linear combi-
nations,therelationswhichdefinedthequantumsuperspherecovariantunderthe
coactionofthealgebraAaresummarizedasfollows: theradiusrelation(40),ten
commutation relations, two relations for Y2
cal limit of three constraints are not required in the commutativecase. However,
we need the constraints to make our algebra well-defined. The relations for Y2
show that the odd coordinate of commutative spheres are no longer nilpotent in
the noncommutative setting. Some of the defining relations contains one addi-
tional parameter ξ originated in (35). We thus have obtained a one-parameter
family of noncommutative superspheres. Explicit expressions of the defining
relations of the quantum supersphere are found in [11].
Before closing this section, we briefly mention some properties of the quan-
tum superspheres obtained above. They enjoy two realizations: The first one
is the realization by U-covariant oscillator introduced in [15]. This realiza-
tion allows us, via realizing the covariant oscillator in terms of conventional
q-oscillator, to obtain a infinite dimensional matrix representation of our quan-
tum supersphere. In the second realization, the coordinates Ymof quantum su-
perespheres are expressed in terms of the elements of A. More precisely, Ykis
a linear combination of the entries of kth column of the adjoint corepresentation
matrix (30). Therefore, the quantum supersphere can be regarded as a subal-
gebra of A. This subalgebra is specified by the infinitesimal characterization
which was first developed for SLq(2) [16]. The infinitesimal characterization
tells us that amongst subalgebras of A, the quantum supersphere is the one an-
nihilated by a linear combination of the twisted primitive elements of U. An
element u ∈ U possessing a coproduct structure ∆(u) = g ⊗ u + u ⊗ g−1with
g ∈ U being a group-like element is said to be twisted primitive with respect to
g. There exist three twisted primitive elements in U, that is, v±and K − K−1.
We now define an action of an element of u ∈ U on a ∈ A by
a ⊙ u = (−1)ˆ aˆ u(u ⊗ id)(∆(a)) =
±2= 0.
±1and three constraints. The classi-
±1
?
(−1)ˆ aˆ u?u,a(1)
?a(2),
(41)
where Sweedler’s notation for coproduct, ∆(a) =?a(1)⊗ a(2), is used and
ˆ a, ˆ u denote the parity of the elements u, a. For a twisted primitive element u, it

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N. Aizawa and R. Chakrabarti11
is straightforward to verify that
a ⊙ u = 0andb ⊙ u = 0⇒(ab) ⊙ u = 0.
(42)
Thus a set of elements of A annihilated by a twisted primitive element u form a
subalgebra of A. Indeed, the quantum supersphere realized in terms of T(2)in
(30) is a subalgebra of A that is annihilated by the twisted primitive element PR
PR= −√g3v++√g1v−,
Yk⊙ PR= 0,
PRconsists of onlyoddtwisted primitiveelements. This is a differencefromthe
quantum sphere for SLq(2). In that example, all the twisted primitive elements
contribute to the annihilation operator of quantum sphere.
(43)
(44)
k = ±2, ±1, 0.
5 Concluding remarks
We have developed a common general prescription for constructing noncom-
mutative covariant spaces of SLq(2) and OSpq(1|2). By this construction, it
is possible to obtain covariant algebras for a given representation of SLq(2) or
OSpq(1|2). Indeed, the known noncommutative spaces, namely Manin’s quan-
tum plane and Podl´ es’ quantum spheres, were recovered for SLq(2) and novel
ones, i.e. their extensions to OSpq(1|2), were obtained. We note a difference of
the present work from others [1,18,19]. In order to obtain higher dimensional
noncommutativespaces, we use a higher dimensionalrepresentationof the fixed
quantum group SLq(2) (or OSpq(1|2)), while higher rank quantum groups are
used in [1,18,19].
We believe that the results of this work are useful for making progress in
constructing supersymmetric versions of noncommutative geometry. For in-
stance, we construct noncommutative superspace, say quantum supersphere, by
our method. Then we may consider differential calculi on the space. It allows us
to compute its curvature, metric and so on based on the framework of Ref. [17].
It may also be possible to extend our method to higher rank quantum (super)
groups by taking into account the multiplicity of irreducible decomposition of
tensor product representations.
Acknowledgments
The work of N.A. is partially supported by the grants-in-aid from JSPS, Japan
(Contract No. 15540132). The other author (R.C.) is partially supported by the
grant DAE/2001/37/12/BRNS, Government of India.
References
[1] Yu. I. Manin, Quantum groups and non-commutative geometry, Montre´ eal Preprint,
CRM-1561 (1988).
[2] S. L. Woronowicz, Publ. RIMS, Kyoto Univ. 23, 117 (1987).

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12Noncommutative Covariant Spaces of OSpq(1|2)
[3] J. Wess and B. Zumino, Nuclear Physics B (Proc. Suppl.) 18B, 302 (1990).
[4] A. Klimyk and K. Schm¨ udgen, Quantum Groups and Their Representations,
Springer-Verlag (1997).
[5] L. C. Biedenharn and M. A. Lohe, Quantum Group Symmetry and q-Tensor Analy-
sis, World Scientific (1995).
[6] P. Podle´ s, Lett. Math. Phys. 14, 193 (1987).
[7] T. Masuda et al, J. Funct. Anal. 99, 357 (1991).
[8] L. L. Vaksman and Ya. S. Soibel’man, Func. Anal. Appl. 22, 170 (1988).
[9] T. H. Koornwinder, Indag. Math. 51, 97 (1989).
[10] V. A. Groza, I. I. Kacurik and A. U. Klimyk, J. Math. Phys. 31, 2769 (1990).
[11] N. Aizawa and R. Chakrabarti, J. Math. Phys. 46, 103510 (2005).
[12] P.P. Kulish and N.Yu. Reshetikhin, Lett. Math. Phys. 18, 143 (1989).
[13] P. Minnaert and M. Mozrzymas, J. Math. Phys. 35, 3132 (1994); J. Phys. A:Math.
Gen. 28, 669 (1995).
[14] H. Saleur and B. Wehefritz-Kaufmann, Nucl. Phys. B628, 407 (2002); ibid. B663,
443 (2003).
[15] F. Thuillier and J.-C. Wallet, Phys. Lett. B323, 153 (1994).
[16] M.S. Dijkhuizen and T. Koornwinder, Geom. Dedicata 52, 291 (1994).
[17] N. Aizawa and R. Chakrabarti, J. Math. Phys. 45, 1623 (2004).
[18] Yu. I. Manin, Comm. Math. Phys. 123, 163 (1989).
[19] L. L. Vaksman and Ya. S. Soibel’man, Leningrad Math. J. 2, 1023 (1991).