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Uniqueness of Uq(N) as a quantum gauge group

arXiv:hep-th/9305176v2 22 Jun 1993SMI-4-93Uniqueness of Uq(N) as a quantum gauge groupandrepresentations of its differential algebraI.Ya.Aref’evaandG.E.Arutyunov ∗†Steklov Mathematical Institute, Russian Academy of Sciences,Vavilov st.42, GSP-1,117966, Moscow, RussiaJune 22, 2018AbstractTo construct a quantum group gauge theory one needs an algebra which isinvariant under gauge transformations. The existence of this invariant algebra isclosely related with the existence of a differential algebra δHGq compatible withthe Hopf algebra structure.

It is shown that δHGq exists only for the quantumgroup Uq(N) and that the quantum group SUq(N) as a quantum gauge group isnot allowed.The representations of the algebra δHGq are constructed. The operators corre-sponding to the differentials are realized via derivations on the space of all irreducible∗-representations of Uq(2).

With the help of this construction infinitesimal gaugetransformations in two-dimensional classical space-time are described.∗E-mail:arefeva@qft.mian.su, arut@qft.mian.su†Supported in part by RFFR under grant N93-011-1471

1IntroductionRecently a construction of quantum group gauge theory (QGGT), i.e. the gauge theorywith a quantum group playing the role of the gauge group, has been initiated [1]-[10].In spite of the impressive successes of applying gauge theory to the description ofall known physical interactions the natural question about the possibilities of extendingthe strict frames of gauge theories arises.One can think that an enlargement of therigid framework of gauge theory would help to solve fundamental theoretical problems ofspontaneous symmetry breaking and quark confinement.

The theory of quantum groupslooks rather attractive as the mathematical foundation of a new theory since the generalrequirements of symmetries of a physical system can be formulated on the language ofquantum groups [11, 2].At the last two years attempts where made to understand the algebraic structure ofQGGT [1]-[10]. The main efforts have been to keep the classical form of gauge transfor-mations for the gauge potential A:A →A′ = TAT −1 + dTT −1.

(1)Roughly speaking, the problem is in the following. Assume that T is an element of aquantum group.

What differential calculus should we consider and from what algebra Ashould be taken a gauge potential A to guarantee that A′ also belongs to the algebra A?One of suitable resolutions of this problem has been recently found by Isaev and Popovicz[10]. In their scheme T and dT are realized as generators of the differential extensionδHGq of a quantum group Gq compatible with the Hopf algebra structure.It is natural to try to find δH-extension for SUq(N) that would lead to the algebraicformulation of the SUq(N) gauge theory.

It is known that the δH extension of GLq(N)does exist [12, 10, 13]. In this paper we deal with the analogous construction for thequantum group Uq(N).

The quantum group Uq(N) is one of the real forms of GLq(N) andmay be obtained from GLq(N) by introducing ∗-involution operation. The δH extensionof GLq(N) also admits ∗-involution and we get δH(Uq(N)).

To obtain the δH-extensionof SUq(N) one needs to fix the quantum determinant equal to unity. However, at thispoint one obstacle arises.

Namely, the quantum determinant is not a central element ofδH(Uq(N)) and therefore cannot be fixed equal to unity.Summing up, it is possible to present an algebraic construction of the quantum groupgauge potential for Uq(N) but the quantum group SUq(N) as a gauge group is not allowed.Thus, if one believes that the ordinary gauge theory is obtained as the classical limit,q →1, of some QGGT then one can speculate that QGGT predicts the group U(2).From this point of view the fact that electroweak group is U(2) = SU(2) ⊗U(1) looksrather promising.Fields defined on the classical space-time and taking value on a quantum group orquantum algebra should be the natural object of the QGGT [1]. However, just at thispoint there is the problem for a straightforward application of the standard approachto quantum groups.The ordinary local gauge theory is based on the existence of asufficiently wide class of differentiable maps from space-time into a group.

This classcan be easily constructed since a Lie group is a smooth manifold and so it is possibleto regard its coordinates as functions on space-time. In the standard quantum groupapproach the space of c-number parameters numerating points of a quantum group is notavailable.

Usually the theory of quantum groups is formulated in terms of the function2

algebra Fun(Gq) on a quantum group Gq. Adopting this view and trying to describeQGGT one can expect that ordinary gauge theory may be formulated in terms of thefunction algebra Fun(Gq), i.e.

on the dual language. However, it is not suitable for fieldtheory applications.

So it is clear that one cannot build QGGT in the framework of thestandard quantum group approach. We need to extend the usual content of quantumgroups by introducing in the theory new objects.

In other words, to consider a map ofthe classical space-time R4 into a quantum group we need a more liberated treatment ofa quantum group or quantum plane than the ordinary theory offers. Such an approachwas suggested in [1].

For consideration of this problem see also [15, 16, 17, 18, 19]. Anexample showing the necessity of introduction new objects is given by the analogue of theexponential map for quantum groups.

It turns out that in addition to a quantum groupone should introduce a set of generators taking value in so-called ”quantum superplane”[1].In this paper we present an explicit realization of a differentiable local map from theclassical space-time R2 into the quantum group Uq(2) supplied with ∗-involution and com-patible with the bicovariant differential calculus on Uq(2). We will see that for this purposeit is suitable to consider a quantum group as the set of all its irreducible unitary repre-sentations and think of parameters numerating these representations as ”coordinates” ona quantum group.

Note that this consideration is in the line of the approach [1] (see also[16]) and has an implicit support in a definition of integral on a quantum group proposedin [20]1. By the compatibility of a map R2 →Uq(2) with the bicovariant differential calcu-lus on Uq(2) we mean that exterior derivative d acting on elements of the quantum groupcan be decomposed over a basis {εi} of ordinary differential forms on R2: d = εi ⊗∂i.To get representations for the derivatives ∂i we start from a construction of representa-tions for (T, L)-pair, where L is a quantum gauge field with zero curvature L = dTT −1 = ω0ω+ω−ω1!.

The simple consideration shows that it is impossible to realize the opera-tors ω0, ω+, ω−, ω1 in the space of an irreducible representation of the algebra Fun(Uq(2)).This means that we have to extend the space of representation. We deal with a direct in-tegral of Hilbert spaces over parameters labelled irreducible representations of Uq(2).

Wefind a simple formulae for the operators ω0, ω+, ω−, ω1 and then get the representationsfor dT. It turns out that there are two types of representations of dT corresponding todifferentiations over two parameters specified irreducible unitary representations of Uq(2).For these two different representations we use notations ∂1T and ∂2T.

Thus we realize thederivatives ∂i as differentiations over parameters of representations of a quantum groupitself, i.e. as differentiations over ”coordinates” on a quantum group.Now having at hand an explicit form of derivatives of Uq(2) elements one can locallyconstruct a differentiable map R2 →Uq(2) as following:T(x) = a(x)b(x)c(x)d(x)!= a + xi∂iab + xi∂ibc + xi∂icd + xi∂id!, i = 1, 2.

(2)Differentiation of operators a(x), . .

. , d(x) with respect to x1, x2 gives the derivatives com-patible with the bicovariant differential calculus.

In this construction we have to limitourselves by considering the two-dimensional classical space-time since the space of pa-rameters of infinite dimensional unitary representations of Uq(2) is two-dimensional.1This approach to integral on quantum group is used to define a lattice QGGT [21]3

Therefore we have for the bicovariant differential calculus on Uq(2) a usual ”classical”picture: if the quantum group is a set of its irreducible unitary representations then thequantum group derivatives are indeed derivatives with respect to the coordinates on Uq(2).The paper is organized as follows. In section 2 we describe an algebraic approach toconstructing QGGT.

In section 3 we introduce ∗-involution for the δH-extension of thealgebra Fun(GLq(N)) and prove a no-go theorem that only δH(Uq(N)) exists and the δH-extension of SUq(N) is not allowed. In sections 4 and 5 two inequivalent ∗-representationsof the δH-extension of Uq(2) in the Hilbert space are constructed.

We use them in section6 to write out an explicit form of a two-dimensional map into the quantum group Uq(2).2Algebraic Scheme of QGGTLet T belongs to a quantum matrix group G, i.e. T is the subject of the relations:R12T1T2 = T2T1R12.

(3)Here R12 is a quantum R-matrix and T1 = T ⊗I, T2 = I ⊗T (see [22, 23] for details).Recall that due to the existence of the Hopf algebra structure for quantum groups theproduct gT of two elements being the subjects of equation (5) satisfiesR12(gT)1(gT)2 = (gT)2(gT)1R12,(4)if the entries of the matrices g and T mutually commute.To construct QGGT one can start with the consideration of gauge fields having zerocurvature [14]:L = dTT −1. (5)In order to give a meaning to (5) we need to specify the differential dT on a quantumgroup, i.e.

to determine the differential calculus. Differential calculi on quantum groupswere developed in [22, 20, 12, 24, 13].

The operator of exterior derivative is supposed tohave the usual properties:d2 = 0,d(AB) = (dA)B + A(dB). (6)If T and dT are understood as matrices with non-commutative entries then L is alsoa matrix with non-commutative entries.

It is interesting to know if there are permutationrelations between the entries of L that can be written in terms of R-matrix. For the caseof GLq(N) the answer is yes if one considers a special differential calculus.

It can beformulated by introducing the set of generators dTij that satisfy the relations [10, 12, 14,24]:R12(dT)1T2 = T2(dT)1R−121 ,(7)R12(dT)1(dT)2 = −(dT)2(dT)1R−121 . (8)The relations (3),(7) and the definition (5) yield the quadratic algebra for L:R12L1R21L2 + L2R12L1R−112 = 0.

(9)The differential calculus [12, 10] is compatible with the Hopf algebra structure in the sensethat the following equation is satisfiedR12d(gT)1(gT)2 = d(gT)2(gT)1R−121(10)4

(for more precise definition see section 3). For ordinary groups the field L is transformedunder gauge transformationsT →gT(11)as follows:L →L′ = gLg−1 + dgg−1.

(12)It is remarkable that equations (4) and (10) are enough to guarantee the invariance ofthe algebra (9) under transformations (12). Thus to construct QGGT we have to extendour quantum group to the δH Hopf algebra.The next nontrivial step is to postulate for the gauge potential A of the general formthe same quadratic algebra as for L:R12A1R21A2 + A2R12A1R−112 = 0.

(13)The relations (4),(10) are again enough for invariance of (13) under gauge transformations(1). Note that as in the case of zero curvature potential we assume that the entries of thematrices A and g are mutually commutative.

In what follows we will regard the quadraticalgebra (13) as the algebra A of quantum group gauge potentials. The correspondingcurvature has the formF = dA −A2and it is transformed under gauge transformations asF →gFg−1.

(14)To conclude the brief presentation of the formal algebraic QGGT construction one hasto mention that the action should be taken in the form trqF 2, since the q-trace [22, 3] isinvariant under (14). Note that F also belongs to a quadratic algebra [10] defined by thereflection equations [25].3*-involution for the δH extension of Fun(Uq(2))The δH extension δH(GLq(N)) of the Hopf algebra Fun(GLq(N)) is the Hopf algebra [3]itself with the comultiplication ∆, the counity ǫ and the antipod S which are defined by∆(T) = T ⊗T ,ǫ(T) = 1 ,S(T) = T −1 ,∆(dT) = dT ⊗T + T ⊗dT ,ǫ(dT) = 0 ,S(dT) = −T −1dTT −1 .

(15)Now we are going to show that δH(GLq(N)) admits ∗-involution.Recall the definition of ∗-involution of a Hopf algebra. An involution ∗of a Hopfalgebra A is a map A →A which is the algebra antiautomorphism and the coalgebraautomorphism obeying two conditions:1.

(a∗)∗= a2. S(S(a∗)∗) = a for any a ∈A .5

Let q be real. Supposing the existence of the ∗-involution for the Hopf algebra A andapplying ∗to equation (7) one finds:R12(T ∗)2(dT ∗)1 = (dT ∗)1(T ∗)2R−121(16)Using the Hopf algebra structure of δH(GLq(N)) (15) and defining relations (3)-(7) onecan deduce that S(dT) obeys equation:R12(S(T)t)2(S(dT)t)1 = (S(dT)t)1(S(T)t)2R−121 ,(17)where t means the matrix transposition.

Comparing (16) and (17) we see that it is possibleto make the identification:T ∗= S(T)tand(dT)∗= S(dT)t = −(T −1)t(dT)t(T −1)t . (18)One can check that the operation ∗introduced in the last equation is the involution ofthe Hopf algebra δH(GLq(N)).

The ∗-Hopf algebra arising in such a way is nothing butthe δH extension δH(Uq(N)) of the algebra Fun(Uq(N)).Let us show that the quantum determinant D for the GLq(N) is not a central elementof the extended algebra δH(GLq(N)). It is well known ([22]) that D can be written in theform:D = trP −(T ⊗T)= trP −T1T2P −,(19)where P −is a projector:P −= −PR + qIq + 1q,(20)that can be treated as the quantum analog of symmetrizator in Cn ⊗Cn.

Then we have:(dT)1P −23T2T3P23 = P −23R−112 R−113 T2T3(dT)1R−131 R−121 P23 . (21)For the projector P −we have the relation that follows from definition (20):P −23R13R12 = qP −23 .

(22)Therefore (21) reduces to1qP −23T2T3(dT)1R−131 R−121 P −23 .The transposition of (22) givesR−131 R−121 P −23 = 1qP −23 ,where the fact was used that (P −23)t = P −23. Now equation (21) takes the form:(dT)1P −23T2T3P −23 = 1q2P −23T2T3P −23(dT)1 .Finally taking the trace we obtain:dTD = 1q2DdT .

(23)Thus D is not a central element of δH(GLq(N)). Applying the involution to D we findthat D is the unitary element of Fun(Uq(N)):D∗D = DD∗= I.These two facts play a crucial role in constructing ∗-representations of Fun(Uq(N)) in aseparable Hilbert space.6

4Type I representation of the Hopf algebraδH(Fun(Uq(2)))Let us consider the question about ∗-representations of the Hopf algebraδH(Fun(Uq(2)))in the separable Hilbert space H. Since Fun(Uq(2)) has the completion that is a C∗-algebra the elements Tij can be realized as bounded operators in H.Then ∂Tij areunbounded operators. It can be proved by using (23).

Let us suppose that ∂Tij is abounded operator and q < 1. Then the norm ||∂Tij|| is defined andD†∂TijD = 1q2∂Tij .This allows one to write||D†∂TijD|| = ||∂TijD|| ≤||∂Tij||||D|| = ||∂Tij||.So we obtain1q2||∂Tij|| ≤||∂Tij||and therefore 1/q2 ≤1 that contradicts to q < 1.It is difficult to begin with constructing representations of the algebra (3), (7), (8)since the involution condition for ∂Tij is rather complicated (we will come back to thisquestion in the next section).

But this difficulty can be solved by introducing the new setof generators L = ∂TT −1 on which the action of the involution is simple. From (3), (7)one can deduce the defining relation for T and L:R12L1R21T2 = T2L1 .

(24)The pair (T, L) is called the (T, L)-pair [1]. The involution property for L isL† = −L .Now we are going to construct the special representation of the (T, L)-pair in a Hilbertspace for the case T ∈Fun(Uq(2)).

Let T = ||tij|| be the matrix of the form:T = abcd!,the inverse of which isT −1 = D−1 d−1qb−qca!.Here D = ad −qbc is the quantum determinant. Taking into account the involution onecan write: a∗= D−1d and c∗= −1qD−1b.

L is the matrix:L = ω0ω+ω−ω1!,where the entries obey the following involution relations ω∗0 = −ω0, ω∗+ = −ω−, ω∗1 = −ω1,i.e. ω0 and ω1 are antihermitian.

The explicit form of the permutation relations for the(T, L)-pair isaω0 = q2ω0a,cω0 = ω0c,7

bω0 = q2ω0b,dω0 = ω0d,(25)aω+ = qω+a,cω+ = qω+c + µω0a,bω+ = qω+b,dω+ = qω+d + µω0b,(26)cω−= qω−c,aω−= qω−a + µω0c,dω−= qω−d,bω−= qω−b + µω0d,(27)aω1 = ω1a + µcω+,bω1 = ω1b + µdω+,cω1 = q2ω1c + qµω−a,dω1 = q2ω1d + qµω−b,(28)where µ = q −1q.Let π be a ∗-representation of the algebra Fun(Uq(N)) in the separable Hilbert spacel2. The operators π(tij) are supposed to be continuous ones.

In [26, 27] it was provedthat every irreducible ∗-representation π of Fun(Uq(N)) is unitary equivalent to the oneof the following two series:1. One dimensional representations ξψ given by the formulae:ξψ(a) = eiψ,ξψ(c) = 0 ,ψ ∈R/2πZ .2.

Infinite dimensional representations ρφ,θ in a Hilbert space with orthonormal basis{en}∞n=0:ρφ,θ(a)e0 = 0 ,ρφ,θ(a)en = ei(θ+φ)q1 −e−2nhen−1 ,ρφ,θ(c)en = eiθe−nhenρφ,θ(d)en =q1 −e−2(n+1)hen+1 ,ρφ,θ(b)en = −eiφe−(n+1)hen(29)Here θ, φ ∈[0, 2π), q = e−h.Thus for Fun(Uq(N)) the set ˆF of equivalence classes of irreducible unitary repre-sentations consists of two separate components each of these is numerated by continuousparameters θ, φ ∈T 2 = S1 × S1 playing the role of coordinates. We shall concentrateour attention on the infinite-dimensional component since only a trivial representation ofdifferentials corresponds to one-dimensional representations of Uq(2).The straightforward algebraic consideration shows that it is impossible to realisethe operators ω0, ω+, ω−, ω1 in the space of an irreducible representation of the algebraFun(Uq(2)).

This means that one have to extend the space of representation or in otherwords to work with reducible representations of Fun(Uq(2)). It turns out that a suitableconstruction deals with a direct integral of Hilbert spaces.

Let us consider the Hilbertspace H of functions on a circle taking value in l2. It is known [28] that there exists thecanonical isomorphism H = l2 ⊗L2(S1) andH =ZS1 H(φ)dφ,8

where L2(S1) is the space of square integrable functions on a circle obeying the condition:Z π−π |f(φ)|2dφ < ∞,for any f ∈L2(S1) and H(φ) = l2. Now the reducible representation of Fun(Uq(2)) in Hcan be defined in the following manner:ˆa(e0 ⊗f) = 0,ˆa(en ⊗f) =q1 −e−2nhen−1 ⊗ei(θ+φ)f,ˆd(en ⊗f) =q1 −e−2(n+1)hen+1 ⊗f,(30)ˆb(en ⊗f) = −e−(n+1)hen ⊗eiφf,ˆc(en ⊗f) = e−nhen ⊗eiθf,where the operators ˆa,ˆb, ˆc, ˆd, correspond to a, b, c, d. On putting φ equal to some valueφ0 the irreducible representation πθ,φ0 of Fun(Uq(N)) stands out.The scalar product in H is given by< (en ⊗f), (em ⊗g) >= (en, em)Z π−π f¯gdφ .

(31)Introduce the following hermitian operator K defined on a dense region in L2(S1):(Kf) (θ, φ) =Xnanmq2nγn =e−2ih ∂∂φf(θ, φ),(32)where f = Pn anγn is an arbitrary element of H, γ = eiφ.Now let us take ω0 to be the operator in H:ω0(en ⊗f) = ien ⊗e−2ih ddφf. (33)Then ω0 is antihermitian as it is required.The permutation relations of ω0 with theoperators ˆa, .

. .

, ˆd are precisely (25). For the operator ω+ one can choose the realization:ω+(en ⊗f) = −ie(n+2)hq1 −e−2nhen−1 ⊗eiφe−2ih ddφf.

(34)Introducing formally the inverse operators ˆb−1 and ˆc−1:ˆb−1(en ⊗f) = −e(n+1)hen ⊗e−iφf,ˆc−1(en ⊗f) = enhen ⊗e−iθf,it is possible to rewrite ω+ in terms of ˆa, . .

. , ˆc−1:ω+ = −q−3 dc−1ˆae−2ih ddφ = −q−3ˆc−1ˆaω0.

(35)By straightforward calculations one can check the fulfilment of the permutation relations(26) on a dense region where all operators coming in (35) are defined. Taking ω−to bethe hermitian conjugation of ω+ with respect to the scalar product (31):ω−= ˆdˆb−1ω0,(36)9

we find that the relations (27) are satisfied.Now the question arises how to find the operator ω1 that must be antihermitian andobey (28). We choose for ω1 the following ansatz:ω1 = P(ˆa, .

. .

, ˆc−1)ω0,(37)where P(ˆa, . .

. , ˆc−1) is a polynomial in ˆa, .

. .

, ˆc−1. Then the first line in (28) reads:ˆaP = 1q2Pˆa −µq3ˆa , ˆbP = 1q2Pˆb −µq3 ˆdˆc−1ˆa,(38)and the second one is achieved from the first by hermitian conjugation.

Equations (38)have the simple solution P = −1q2ˆb−1 ˆdˆc−1ˆa. Therefore ω1 takes the form:ω1 = −1q2ˆb−1 ˆdˆc−1ˆa(39)The found operators ω0, ω+, ω−, ω1 combined with (30) give ∗-representation of the (T, L)-pair.

Coming back to the derivatives ∂T = LT we see that∂a = 0,∂b = −1q3c−1Dω0,∂c = 0,∂d = −1q3d(bc)−1Dω0. (40)This means that our differentials can be treated as elements of the algebra Fun(Uq(2)which is extended by adding the new element ω0 provided that the elements b and c areinvertible.

The equalities ∂a = 0 and ∂c = 0 seem rather restrictive and give a hint thatother ∗-representations for which ∂a ̸= 0, ∂c ̸= 0 should also exist. In the next section wewill construct one of such examples.5Type II ∗-representation of δH(Fun(Uq(2)))The permutation relations between the elements a, b, c, d and their derivatives follow from(7).

In particular we have:a(∂a) = q2(∂a)a,c(∂a) = q(∂a)c,b(∂a) = q(∂a)b,d(∂a) = (∂a)d.(41)Let us consider now the Hilbert space of squire integrable functions on a torus T 2 = S1×S1taking value in l2. The scalar product in H has the form:(f, g) =ZT 2 < f(θ, φ), g(θ, φ) > dθdφ,(42)where <, > is a scalar product in l2.

As in the previous section the reducible representationof Fun(Uq(2)) in H can be defined by the formulas (30) where θ is no longer a parameterand f = f(θ, φ) ∈H.Note that equations (41) are compatible with the condition ∂a = (∂a)∗. This allowsone to choose for ∂a the simple realization by the hermitian unbounded operator:(∂ˆa) (θ, φ) =Xnmanmqn+mγn1 γm2 =e−ih ∂∂φ −ih ∂∂θ f(θ, φ),(43)10

where f = Pnm anmγn1 γm2 is an arbitrary element of H, γ1 = eiθ, γ2 = eiφ and ∂ˆa is theoperator that corresponds to a.One can go further and require the condition ∂a = (∂a)∗to be consistent with theinvolution (18) whose explicit form is(∂a)∗= −q2(D−1) q2d2(∂a) −bd(∂c) −qdc(∂b) + 1qbc(∂d)!,(44)(∂d)∗= −q2(D−1)(q2ad −D)(∂a) −qab(∂c) −qac(∂b) + a2(∂d),(45)(∂b)∗= −q2(D−1)−q3cd(∂a) + qad(∂c) + q2c2(∂b) −ac(∂d),(46)(∂c)∗= −q2(D−1) −q2db(∂a) + b2(∂c) + 1qad(∂b) −1qba(∂d)!. (47)I this way we can express ∂d in terms of ∂a, ∂b and ∂c:∂d = q(bc)−1 −(q2d2 + 1q2D2)(∂a) + bd(∂c) + qdc(∂b)!.

(48)Therefore to specify ∂d we need the explicit realization of ∂b and ∂c that are compatiblewith (46), (47). The permutation relations between ∂b, ∂c and the elements of Fun(Uq(2))area(∂c) = q2(∂c)a + qµ(∂c),c(∂c) = q2(∂c)c,b(∂c) = (∂c)b + µ(∂a)d,d(∂c) = (∂c)d,(49)a(∂b) = q(∂b)a + µb(∂a),c(∂b) = (∂b)c + µd(∂a),b(∂b) = q2(∂b)b,d(∂b) = q(∂b)d.(50)To solve equations (49) and (50) we take as in the previous section:∂b = I∂a,∂c = J ∂awhere I and J are polynomials in ˆa, .

. .

, ˆd,ˆb−1, ˆc−1 that should be defined. By simplecomputations we find:∂ˆb = qˆc−1 ˆd∂a,∂ˆc = ˆdˆb−1∂ˆa.

(51)Then (48) reduces to the form:∂ˆd = (ˆbˆc)−1(q3 ˆd2 −1qD2)∂ˆa. (52)Using the found representation of the derivatives (51),(52) one can show that the permu-tation relations for ∂d:a(∂d) = (∂d)a + µ(∂b)c + µb(∂c),c(∂d) = q(∂d)c + µd(∂c),b(∂d) = q(∂d)b + µq(∂b)d,d(∂d) = q2(∂d)d.(53)are satisfied.

The conjugation of the operators ∂b, ∂c, ∂d with respect to the scalar product(42) gives(∂b)∗= −qab−1(∂a),(∂c)∗= −1q2c−1(a∂a),(∂d)∗= q(a2 −1)(bc)−1(∂a). (54)The consistency of equations (54) with (44)-(47) can be easily checked.

Hence the formulae(43),(51),(52) give the other ∗-representation of the δH-extension of Fun(Uq(2)). We referto this representation as the type II.11

6Two-dimensional Local Gauge TransformationsNow having at hand an explicit representations of the δH- extension of Uq(N) we are ableto construct two-dimensional infinitezimal local gauge transformations. As was mentionedin Introduction it seems reasonable to identify the representations of types I and II withderivations in two linear independent directions of space-time.

To construct an explicitrealization of two-dimensional differentials of Uq(2) elements it is convenient to introducethe operators K1 and K2 acting on H in the following manner:(K1f)(θ, ϕ) =Xnmanmq2nγn1 γm2 =e−2ih ∂∂ϕf(θ, ϕ),(55)(K2f)(θ, ϕ) =Xnmanmqn+mγn1 γm2 =e−ih ∂∂ϕ−ih ∂∂θ f(θ, ϕ)(56)where γ1 = eiθ, γ2 = eiφ and f ∈H. Then according to our conjecture we may regard theformulae (40) as the derivatives of the elements of Uq(2) with respect to the coordinatex1 on R2, and the formulae (43),(51),(52) as the derivatives in the x2 direction.

Thus thedecomposition of differentials on Uq(2) over the basis {dxi} of differential forms on R2 isδa = K2 ⊗dx2,δb = −1q3c−1DK1 ⊗dx1 + qc−1dK2 ⊗dx2,(57)δc = db−1K2 ⊗dx2,δd = −1q3d(bc)−1DK1 ⊗dx1 + (bc)−1 q3d2 −1qD2!K2 ⊗dx2. (We use δ for exterior derivative rather than d to avoid misunderstanding with the elementd).Let us make a comment about equation (8).

Formally this equation follows from (7)and the identity d2 = 0. However, we have not define in the above construction an actionof the operator d on dT.

Nevertheless, dT = δaδbδcδd!does satisfy the relation (8).This is non-trivial since the right hand side of (8) contains two differentials dx1 and dx2and is the result of direct calculations.The meaning of the formulae (57) is that locally we have a two-parameter map R2 →Uq(2) compatible with the bicovariant differential calculus on Uq(2). For example, onecan writeb(x1, x2) = b −1q3c−1DK1 ⊗x1 + qc−1dK2 ⊗x2,then the usual derivations of b with respect to x1 or x2 give the operators on H withdesirable properties (7),(8).

It would be interesting to construct a global map R2 →Uq(2)and this is the subject of further investigations.ACKNOWLEDGMENTThe authors are grateful to P.B.Medvedev and I.Volovich for interesting discussions.12

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