Representations of quantum SU(2) operators on a local chart

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1 Journal of Physics: Conference Series PAPER OPEN ACCESS Representations of quantum SU() operators on a local chart To cite this article: Elmar Wagner 016 J. Phys.: Conf. Ser View the article online for updates and enhancements. Related content - The cyclic representations of the quantum algebra U q (osp(,1)) in terms of the Z n - algebra Mo-Lin Ge, Xu-Feng Liu and Chang-Pu Sun - q-analogues of the parabose and parafermi oscillators and representations of quantum algebras R Floreanini and L Vinet - A q-deformed osp(1,) superalgebra and its two-component coherent state representations Le-Man Kuang, Gao-Jian Zeng and Fa-Bo Wang This content was downloaded from IP address on 04/1/018 at 15:5

2 Representations of quantum SU() operators on a local chart Elmar Wagner Instituto de Física y Matemáticas, Universidad Michoacana, Edificio C-3, Cd. Universitaria, Morelia, Mich., Mexico elmar@ifm.umich.mx Abstract. Hilbert space representations of quantum SU() by multiplication operators on a local chart are constructed, where the local chart is given by tensor products of square integrable functions on a quantum disc and on the classical unit circle. The actions of generators of quantum SU(), generators of the opposite algebra, and noncommutative partial derivatives are computed on a Hilbert space basis. 1. Introduction In [4], a twisted Dirac operator for quantum SU() is constructed by using so-called disc coordinates. The starting point of this construction is a representation of quantum SU() by multiplication operators on the tensor product of L (S 1 ) with the quantum disc algebra, where the inner product on the latter space is defined by a positive weighted trace. The construction of the Dirac operator uses noncommutative first order differential operators satisfying a twisted Leibniz rule. Only this twisted Leibniz rule is needed to prove that the Dirac operator has bounded twisted commutators with differentiable functions, thus avoiding an explicit description of the Hilbert space actions of the involved operators. On the other hand, an explicit description of the Hilbert space representation of the quantum algebra of differentiable functions and the action of the Dirac operator is crucial for determining the analytic properties of the twisted spectral triple, for instance the spectrum of the Dirac operator, its K-homology class, its eigenvectors, and the C*-closure of the quantum algebra. The purpose of the present work is to describe the actions of all operators occurring in [4] on a Hilbert space basis. Unfortunately, a detailed analysis of the twisted spectral triple from [4] is behind the scope of this paper.. Quantum SU() in disc coordinates Let q (0, 1). The coordinate ring O(SU q ()) of quantum SU() is the *-algebra generated by c and d satisfying cd = qdc, c d = qdc, cc = c c, (1) d d + q cc = 1, dd + cc = 1. () Classically, i. e. q = 1, the universal C*-algebra generated by c and d is isomorphic to C(S 3 ). The universal C*-closure C(SU q ()) of O(SU q ()) has been studied in [6] and [8]. Here we will Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1

3 use the fact that it is generated by the operators c, d B(l (N) l (Z)) given by c(e n b k ) = q n e n b k+1, d(e n b k ) = 1 q (n+1) e n+1 b k, (3) where {e n } n N and {b k } k Z are orthonormal bases for l (N) and l (Z), respectively. Note that c and d from (3) do indeed satisfy the relations (1) and (). Moreover, the representation of O(SU q ()) defined by (3) is faithful. Let u : S 1 S 1, u(e it ) = e it, denote the unitary generator of C(S 1 ). On the orthonormal basis {b k := 1 π e ikt : k Z} for L (S 1 ) = l (Z), multiplication by u becomes the bilateral shift Consider the bounded operators z and z on l (N) given by z e n := Obviously, z and z satisfy the quantum disc relation ub k = b k+1, k Z. (4) 1 q (n+1) e n+1, z e n := 1 q n e n 1. (5) z z q z z = 1 q. (6) The polynomial *-algebra generated by z and z will be denoted by O( ). Here and subsequently, D := {z C : z < 1} and D := {z C : z 1} stand for the open and closed unit discs, respectively, and O( ) will be called quantum disc algebra. As shown in [], the C*-closure C( ) of O( ) is isomorphic to the Toeplitz algebra. Moreover, (5) determines the unique (up to unitary equivalence) faithful irreducible *-representation of C( ), see e. g. [5]. Defining y B(l (N)) by y e n := q n e n, n N, (7) we have y = 1 zz, yz = qzy, z y = qyz. (8) In particular, y = 1 zz C( ). Finally, with u, z and y given in (4), (5) and (7), the operators c and d in (3) can be written c = y u = 1 zz u, d = z 1. (9) The representation (9) corresponds to the classical parametrization ψ : D [ π, π] S 3, ψ(z, t) := (z, 1 z z e it ), The restriction of ψ to the open set D ( π, π) defines a dense coordinate chart for S 3 = SU() which is compatible with the standard differential structure on S 3. Moreover, the generators c and d correspond in the classical limit q 1 to the coordinate functions c(z, 1 z z e it ) = 1 z z e it, d(z, 1 z z e it ) = z. We call them disc coordinates because the discs ψ t ( D) := {ψ(z, t) : z D} for fixed t ( π, π] correspond to the -dimensional symplectic leaves associated to a known Poisson group structure on SU() [1].

4 3. Differential calculus One of the differential operators occurring in [4] is the classical partial derivative i t C (1) (S 1 ). On the basis vectors b k from (4), one obviously has acting on i t b k = kb k, k Z. (10) Our next aim is to describe noncommutative partial derivatives z and z on the quantum disc algebra. As in [7], consider the first order differential *-calculus d : O( ) Ω( ) given by Ω( ) = dz O( ) + dz O( ) with O( )-bimodule structure z dz = q dz z, z dz = q dz z, z dz = q dz z, z dz = q dz z, satisfying the Leibniz rule d(fg) = d(f)g + f d(g) for all f, g O( ). We define the partial derivatives z and z by d(f) = dz z (f) + dz z (f), f O(). On monomials, one gets d(z n z k ) = n 1 j=0 q j dz z n 1 z k + k 1 l=0 q n+l dz z n z k 1, therefore z (zn z k (n 1) 1 qn ) = q z n 1 z k, 1 q z (zn z k n 1 qk ) = q z n z k 1. 1 q From (6), it follows that the monomials z n z k span the linear space O( ). Using the facts that commutators satisfy the Leibniz rule, [z, z] = [z, z 1 ] = 0, [z, z] = y, and y commutes 1 q with z and z in the same way as dz and dz do, one readily verifies that z (f) = 1 y [z, f], 1 q z (f) = 1 1 q y [z, f], f O( ). (11) In order extend the partial derivatives to an algebra of differentiable functions, we first observe that, by (6) and (8), each f O( ) can be written f = N z n p n (y ) + p n (y )z n, N, M N, (1) with polynomials p n in one variable. formulas in (11) together with (8) give Furthermore, for all functions f C(spec{y }), the where z f(y ) = 1 y (z f(y ) f(y )z ) = f(y ) f(q y ) 1 q y q y z = q f(y )z, z f(y ) = 1 y (zf(y ) f(y )z) = z f(y ) f(q y ) 1 q y q y = z q f(y ), q f(x) := f(x) f(q x) x q x denotes the q -difference operator and the operators f(y ) are defined by the spectral calculus of the self-adjoint operator y. The reason why we prefer to consider functions f = f(y ) instead of f = f(y) is because z 1 z z is a differentiable function on D but z 1 z z is not. Note that spec(y ) = {q n : n N} {0} 3

5 f(q and lim k ) f(q k+ ) ( k = lim 1 f(q k ) f(0) q k q k+ k q f(q k+ ) f(0)) 1 q q k 1 q q = f (0) if the derivative of f in 0 exists. In that case, q ψ(y ) defines a bounded operator on l (N). This k+ together with (1) motivates the following definition of an algebra of differentiable functions on the quantum disc: { N } F (1) ( ) := z n f n (y ) + f n (y )z n : M, N N, f j C (1) (spec(y )). (13) Observe that differentiability of f C (1) (spec(y )) amounts to the differentiability of f in 0 which is the only accumulation point of spec(y ). Using that f(y) F (1) ( ) if and only if f(q k y) F (1) ( ) for any k Z, it is easily seen that F (1) ( ) is a *-algebra of bounded operators on l (N) containing O( ). It follows from the (twisted) Leibniz rules y [ζ, z n f(y)] = y [ζ, z n ]f(y) + q n z n y [ζ, f(y)], y [ζ, f(y)z n ] = y [ζ, f(y)]z n + f(y)y [ζ, z n ], f C (1) (spec(y )), ζ {z, z }, n N, that the actions of z and z given by the expressions in (11) are well defined on F (1) ( ). 4. Noncommutative integration Let s denote the unilateral shift operator acting on l (N) by se e = e n+1. Using the relations z = s 1 q y, z = s 1 y, f(y)s = sf(qy), s f(y) = f(qy)s, (14) where f L (spec(y)) := {g : spec(y) C : bounded}, one sees that each ψ F (1) ( ) can be written ψ = N sn g n (y ) + M g n(y )s n with continuous (but not necessarily differentiable) functions g j C(spec(y )). For a noncommutative integration theory on the quantum disc, it happens to be more convenient to work with the *-algebra of bounded operators { N } F( ) := s n g n (y) + g n (y)s n : M, N N, g j L (spec(y)). (15) According to the above remark, F (1) ( ) F( ). We follow [4, 5, 7] and define for α > 0 : F( ) C, ψ := (1 q) Tr l (N)(ψ y α ). (16) Since y α is a positive compact operator for all α > 0, the weighted trace is a well-defined positive functional on F( ). As the trace over weighted shift operators vanishes, we have ψ = ( N ) s n g n (y) + g n (y)s n = (1 q) g 0 (q n )q αn. (17) n N In terms of the Jackson integral 1 0 f(y)d qy = (1 q) n N f(qn )q n, the functional written may be ( N ) s n g n (y) + g n (y)s n = 1 π 0 π ( N ) e inθ g n (y) + g n (y)e inθ dθ y α 1 d q y. 4

6 By (14), the commutation relation between y α and functions from F( ) can be expressed by the automorphism σ α : F( ) F( ), σ α( N s n g n (y) + g n (y)s n) = N (q α s) n g n (y) + g n (y)(q α s) n. (18) Then hy α = y α σ α (h) and hence, by the trace property, gh = (1 q) Tr l (N)(ghy α ) = (1 q) Tr l (N)(σ α (h)gy α ) = for all f, g F( ). Note also that (σ α (h)) = σ α (h ), h F( ), where σ α denotes the inverse of σ α. Using Tr l (N)(s n s k f(y)y) = 0 if k n, one easily verifies that f, g := σ α (h)g (19) is faithful. Therefore f g, f, g F( ), (0) defines an inner product on F( ). Its Hilbert space closure of will be denoted by L ( ). 5. Representations on the quantum disc The left multiplication of O( ) on F( ) defines a bounded *-representation of O( ) on L ( ) since xf, g = f x g = f, x g, f, g F( ), x O( ). In [4], representations of the opposite algebra O( ) op play also a crucial role in the definition of the Dirac operator. Recall that the opposite algebra A op of an algebra A is the vector space A with the opposite multiplication a op b op = (ba) op. A op remains to be a *-algebra if A is one. The right multiplication of O( ) on F( ) defines a representation of O( ) op which is not a *-representation because x op f, g = fx, g = x f g = f g σ α (x ) = f, σ α (x ) op g (1) by (19). Our next aim is to construct an orthonormal basis for L ( ). To this end, we introduce the following notation: As customary, the symbol δ nm denotes the Kronecker delta. For k Z, let δ q k : R {0, 1} be defined by δ q k(t) := If a is a densely defined operator, we set { 1, t = q k, 0, t q k. a #k := { a k, k 0, a k, k < 0. () 5

7 Note that δ q n is continuous on spec(y) and δ q n(y) is the orthogonal projection onto the onedimensional subspace span{e n } l (N). For any (measurable) function f : spec(y) C, we have f(y)δ q n(y) = δ q n(y)f(y) = f(q n )δ q n(y). (3) In particular, δ q m(y)δ q n(y) = δ mn δ q n(y). Furthermore, it follows from (14) that sδ q n(y) = δ q n(q 1 y)s = δ q n+1(y)s, s δ q n(y) = δ q n(q y)s = δ q n 1(y)s. (4) As a consequence, for all k, n N such that n < k, s k δ q n(y) = δ q n k(y)s k = 0, δ q n(y)s k = s k δ q n k(y) = 0, (5) since δ q n k(t) = 0 on spec(y). From ss = 1 δ q 0(y), we get ss k δ q n(y) = (1 δ q 0(y))s k 1 δ q n(y) = s k 1 (1 δ q k 1(y))δ q n(y) = s k 1 δ q n(y) (6) as we may assume that n > k 1 by (5). Note that s #k s #l = s #k+l does not hold in general, for instance s #1 s # 1 = 1 δ q 0(y) s #0. However, from (6) and s s = 1, we conclude that s #l s #k δ q n(y) = s #l+k δ q n(y) for all k, l Z, n N. (7) Using (0), (7), (19), (3) and (17), we compute s #k δ q n(y), s #l δ q m(y) = δ q n(y)s # k s #l δ q m(y) = s #l k δ q m(y)δ q n(y) = (1 q)q αn δ nm δ kl. The last equation is the key to constructing an orthonormal basis for L ( ). Proposition 1. For n N and k Z such that k n, define (8) η nk := q αn/ 1 q s #k δ q n(y). (9) Then {η nk : n N, k Z, k n} is an orthonormal basis for L ( ). Proof. It follows from (8) that the set defined in the proposition is orthonormal. Therefore it only remains to show that it is complete. As L ( ) is the closure of F( ), it suffices to show that the elements s k f(y) and f(y)s k can be expanded in this basis, where f L (spec(y)) and k N. Since δ q n(y) is the orthogonal projection onto the eigenspace corresponding to the eigenvalue q n of the self-adjoint operator y, the spectral theorem gives f(y) = n N f(qn )δ q n(y). Thus s k f(y) = 1 q n N qαn/ f(q n )η nk and the series converges in L ( ) since f is bounded and {q αn/ } n N l (N). Similarly, by (5), f(y)s k = n N f(q n )δ q n(y)s k = n N f(q n )s k δ q n+k(y) = 1 q q αk/ n N q αn/ f(q n )η n+k,k, where the sequence of coefficients {q αn/ f(q n )} n N belongs to l (N). We will now compute the actions of our noncommutative multiplication and partial differential operators on this basis. To begin, y η nk = q αn/ 1 q y s #k δ q n(y) = q k q αn/ 1 q s #k n+k q αn/ y δ q n(y) = q 1 q s #k δ q n(y) = q n+k η nk, (30) 6

8 where we used (14) in the second equality and (3) in the third. Similarly, y op η nk = η nk y = q αn/ 1 q s #k δ q n(y)y = q n q αn/ 1 q s #k δ q n(y) = q n η nk. (31) Writing z = s 1 q y, it follows from (30) and (7) that z η nk = q αn/ 1 q s 1 q y s #k δ q n(y) = = 1 q (n+k+1) q αn/ 1 q s #k+1 δ q n(y) 1 q (n+k+1) η n,k+1, (3) and analogously, for z = s 1 y, z η nk = q αn/ 1 q s 1 y s #k δ q n(y) = = 1 q (n+k) q αn/ 1 q s #k 1 δ q n(y) 1 q (n+k) η n,k 1, (33) The actions of z op and z op are calculated by applying (4), (3) and (9): z op η nk = η nk z = q αn/ 1 q s #k δ q n(y)s 1 q y = q αn/ 1 q s #k+1 δ q n 1(y) 1 q y = q α/ 1 q n q α(n 1)/ 1 q s #k+1 δ q n 1(y) = q α/ 1 q n η n 1,k+1, (34) z op η nk = η nk z = q αn/ 1 q s #k δ q n(y)s 1 y = q αn/ 1 q s #k 1 δ q n+1(y) 1 y = q α/ 1 q (n+1) q α(n+1)/ 1 q s #k+1 δ q n+1(y) = q α/ 1 q (n+1) η n+1,k 1. (35) Note that (z op ) = q α z op = (σ α (z )) op as observed in (1). By Equation (13) and the remark following it, the basis vectors η nk belong to F (1) ( ). As eigenvectors of the self-adjoint operator y, they also belong to the domain of the unbounded operator y. We will use the formulas in (11) to compute the actions of z and z on η nk. From (30), (33) and (35), it follows that z η nk = 1 y (z η 1 q nk η nk z ) = 1 y (z η 1 q nk z op η nk ) = q (n+k) 1 q ( q Further, by (30), (3) and (34), z η nk = 1 y (z η 1 q nk η nk z) = ( = q (n+k) 1 q q 1 q (n+k) η n,k 1 q α/ 1 q (n+1) η n+1,k 1 ). (36) 1 1 q y (z η nk z op η nk ) 1 q (n+k+1) η n,k+1 q α/ 1 q n η n 1,k+1 ). (37) To compare the representations of z and z with the irreducible ones from (5), it is convenient to change the basis. For n, k N, set e nk := η n,k n. As an immediate consequence of Proposition 1, {e nk : n, k N} is an orthonormal basis for L ( ) and the change of basis is given by the unitary operator U : L ( ) L ( ), U e nk = η n,k n, U η nk = e n,k+n. 7

9 In the new basis, we have z e nk = U z η n,k n = y e nk = U y η n,k n = q k U η n,k n = q k e nk, 1 q (k+1) U η n,k n+1 = 1 q (k+1) e n,k+1, (38) z e nk = U z η n,k n = 1 q k U η n,k n 1 = 1 q k e n,k 1. (39) In particular, on each Hilbert space H n := span{e nk : k N} = l (N), (40) we recover the unique irreducible *-representations (5) of the quantum disc O( ). Further, for the opposite operators, one gets y op e nk = U y op η n,k n = q n U η n,k n = q n e nk, (41) z op e nk = U z op η n,k n = q α/ 1 q n U η n 1,k n+1 = q α/ 1 q n e n 1,k, (4) z op e nk = U z op η n,k n = q α/ 1 q (n+1) U η n+1,k n 1 = q α/ 1 q (n+1) e n+1,k. (43) Now, setting ζ op := q α/ z op, ζ op := q α/ z op and H op k := span{e nk : n N} = l (N), (44) we obtain an irreducible *-representation of O( ) op on H op k, that is, (ζop ) = ζ op and Note that we have of course ζ op ζ op q ζ op ζ op = 1 q. (45) [f, g op ] = f g op g op f = 0 for all f O( ), g op O( ) op, (46) since the elements of O( ) act only on the second index of e nk and the elements of O( ) op act only on the first. We compute the actions of z and z on e nk by using the commutator representation (11) again. Thus z e nk = 1 y (z e 1 q nk z op ( e nk ) = q k 1 q q 1 q k e n,k 1 q 1 α/ ) q (n+1) e n+1,k, z e nk = 1 y (z e 1 q nk z op ( e nk ) = q k 1 q q 1 q (k+1) e n,k+1 q α/ ) 1 q n e n 1,k. Finally recall that (3) defines a faithful *-representation of C(SU q ()), and consider the Hilbert space L ( ) L (S 1 ) with orthonormal basis {e nkl : n, k N, l Z}, where e nkl := e nk b l = q αn/ 1 q s #k n δ q n(y) 1 π e ilt. (47) As in (9), we define a *-representation of O(SU q ()) on L ( ) L (S 1 ) by setting c := y u = 1 zz u, d := z 1. (48) 8

10 It follows from (38) (40) that L ( ) L (S 1 ) decomposes into the direct sum L ( ) L (S 1 ) = n N H n L (S 1 ) = n N l (N) l (Z), and on each copy of l (N) l (Z), we recover the representation from (3). Hence the representation (48) decomposes into the infinite orthogonal sum of *-representations which are all unitarily equivalent to (3). In particular, (48) defines a faithful *-representation of O(SU q ()) and the C*-closure of the *-algebra generated by c and d from (48) is isomorphic to C(SU q ()). To obtain *-representations of the opposite algebras O(SU q ()) op and C(SU q ()) op, the generators c op and d op have to satisfy the opposite relations of (1) and (). Analogous to d = z 1 and c = 1 zz u with z satisfying (6) and u being a unitary generator of C(S 1 ), a *-representation of O(SU q ()) op is given by replacing z by ζ op satisfying (45) (the opposite relation of (6)) and u by a unitary operator u op with the same properties as u. For u op we may take u which generates the same C*-algebra C(S 1 ) as u does. The adjoint of u is chosen for consistency because then c and d act as forward shifts, and c op and d op act as backward shifts. For ζ op = q α/ z op and ζ op = q α/ z op, as defined in the paragraph after Equation (43), we get 1 ζ op ζ op = 1 z op z op = y op. Thus, setting c op := y op u = 1 ζ op ζ op u, d op := ζ op 1, (49) yields a *-representation of O(SU q ()) op on L ( ) L (S 1 ). Moreover, with H op k (44), we have the Hilbert space decomposition defined in L ( ) L (S 1 ) = k N H op k L (S 1 ) = k N l (N) l (Z), and, by (41) (43) and (4), c op and d op act on each H op k 0 L (S 1 ) = span{e nk0 b l : n N, l Z} by c op (e nk0 b l ) = q n e nk0 b l 1, d op (e nk0 b l ) = 1 q (n 1) e n 1,k0 b l. (50) Note that (50) are the adjoint relations of (3), and that taking the opposite relations of (1) and () amounts to interchanging c and d with their adjoints c and d. Therefore, as much as (3) defines a faithful *-representation of O(SU q ()) and C(SU q ()), (50) yields a faithful *-representation of O(SU q ()) op and C(SU q ()) op and so does their direct sum representation on L ( ) L (S 1 ). Finally we remark that the operators c and d from (48) commute with the operators c op and d op from (49) since u and u belong to the commutative C*-algebra C(S 1 ), and the operators in the left factor of the tensor products commute by (46). As a consequence, the representations of C(SU q ()) and C(SU q ()) op on L ( ) L (S 1 ) commute. For the convenience of the reader, we finish the paper by collecting the most important formulas in a final theorem. Theorem. Let z, z B(l (N)) denote the generators of the quantum disc O( ) given in (5) and let y = 1 z z. With F( ) defined in (15), the *-algebra F( ) L (S 1 ) is considered an algebra of bounded functions on a local chart for quantum SU() in the following sense: Define an inner product on F( ) L (S 1 ) by f φ, g ψ := f g φψ dλ = (1 q) Trl (N)(f g y α ) S 1 φψ dλ, S 1 α > 0, where λ stands for the Lebesgue measure on S 1. The Hilbert space completion of F( ) L (S 1 ) will be denoted by L ( ) L (S 1 ). Furthermore, with F (1) ( ) described in (13), the *-subalgebra F (1) ( ) C (1) (S 1 ) of F( ) L (S 1 ) is viewed as an algebra of differentiable functions 9

11 on the local chart. The Hilbert space L ( ) L (S 1 ) has an orthonormal basis of differentiable functions {e nkl : n, k N, l Z} F (1) ( ) C (1) (S 1 ), where e nkl = q αn/ 1 q s #k n δ q n(y) 1 π e ilt. The left multiplication x (f ψ) := xf ψ, x O( ), f ψ F( ) L (S 1 ), defines a bounded *-representation of O( ) on L ( ) L (S 1 ). On the above orthonormal basis, the operators z, z and y act by z e nkl = 1 q (k+1) e n,k+1,l, z e nkl = 1 q k e n,k 1,l, y e nkl = q k e nkl. The right multiplication x op (f ψ) := fx ψ, x O( ) op, f ψ F( ) L (S 1 ), defines a bounded representation of O( ) op on L ( ) L (S 1 ) which is not a *-representation. The actions of z op, z op = q α (z op ) and y op = (y op ) are given by z op e nkl = q α/ 1 q n e n 1,kl, z op e nkl = q α/ 1 q (n+1) e n+1,kl, y op e nkl = q n e nkl. Let u C(S 1 ), u(e it ) = e it, denote the unitary generator of C(S 1 ). Setting c := y u = 1 zz u, d := z 1, c op := y op u = 1 z op z op u, d op := q α/ z op 1, yields commuting *-representations of C(SU q ()) and C(SU q ()) op on L ( ) L (S 1 ) given by left and right multiplication, respectively. On basis vectors, these representations read ce nkl = q k e nk,l+1, de nkl = 1 q (k+1) e n,k+1,l, c op e nkl = q n e nk,l 1, d op e nkl = 1 q n e n 1,kl. The partial derivatives z, z and t act on differentiable functions g φ F (1) ( ) C (1) (S 1 ) by z (g φ) = z g φ, z (g φ) = z g φ and t (g φ) = g tφ. On basis elements, one obtains z e nkl = q k ( 1 q q 1 q k η n,k 1,l q 1 α/ ) q (n+1) e n+1,kl, 1 q (k+1) η n,k+1,l q α/ ) 1 q n e n 1,kl, z e ( nkl = q k 1 q q t e nkl = ile nkl. Note that the representations of C(SU q ()) and C(SU q ()) op do not depend on α but the actions of z and z do so. Acknowledgments The author thanks Ulrich Krähmer and Andrzej Sitarz for interesting discussions on the subject, and acknowledges financial support from CIC-UMSNH and the Polish Government Grant 01/06/M/ST1/ References [1] Chari V and Pressley A 1994 Quantum Groups (Cambridge: Cambridge Univ. Press) [] Klimec S and Lesniewski A 1993 J. Funct. Anal [3] Klimyk A U and Schmüdgen K 1998 Quantum Groups and their Representations (New York: Springer) [4] Krähmer K and Wagner E 015 Twisted Dirac operator on quantum SU() in disc coordinates (in preparation) [5] Kürsten K-D and Wagner E 007 Publ. RIMS Kyoto Univ [6] Masuda T, Nakagami Y and Watanabe J 1990 K-Theory [7] Shklyarov D L, Sinel shchikov S D, Vaksman L L 1997 Mat. Fiz. Anal [8] Woronowicz S L 1987 Publ. RIMS, Kyoto Univ