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Proceedings of the RIMS Research Project 91 on Infinite Analysis

arXiv:hep-th/9212064v1 10 Dec 1992Proceedings of the RIMS Research Project 91 on Infinite Analysisc⃝World Scientific Publishing CompanyCLEBSCH-GORDAN AND RACAH-WIGNERCOEFFICIENTS FOR Uq(SU(1, 1))N.A. LISKOVA and A.N.

KIRILLOVSteklov Mathematical InstituteFontanka 27, Leningrad 191011, USSRReceivedOctober 21, 1991ABSTRACTThe Clebsch-Gordan and Racah-Wigner coefficients for the positive (or negative)discrete series of irreducible representations for the noncompact form Uq(SU(1, 1)) ofthe algebra Uq(sl(2)) are computed.n◦0. IntroductionNow it is well known the great significance which the Clebsch-Gordan and Racah-Wigner coefficients for the algebra Uq(SU(2)) has in the conformal field theory,topological field theory, low-dimensional topology, in the theory of a q-special func-tions.

In this note we compute the Clebsch-Gordan and Racah-Wigner coefficientsfor the non-compact form Uq(SU(1, 1)) of the Hopf algebra Uq(sl(2)) in the casecorresponding to the tensor product of the irreducible representations of positive(or negative) discrete series. Our main result consists of two parts.

At first weobtain the formula for the Clebsch-Gordan and Racah-Wigner coefficients in thecase mentioned above as an analytical continuation of corresponding formula forthe algebra Uq(sl(2)) in the region of negative values of parameters. At the secondwe find the simple substitutions which transforms the corresponding formula forUq(sl(2)) into ones for Uq(SU(1, 1)) and vice versa.Acknowledgements.The authors thank F.A.

Smirnov, L.A. Takhtajan and L.L.Vaksman for interesting discussions and remarks. We would like to express gratitudeto the organizers of the RIMS 91 Project “Infinite Analysis” for the invitation totake participation in the workshop of this Project and the secretaries of RIMS forthe various assistance and the help in preparing the manuscript to publication.

2N.A. Liskova and A.N.

Kirillovn◦1. Algebra Uq(sl(2)) and it compact forms.The algebra Uq(sl(2)), [1,2], is generated by elements {K, K−1, X±} with thecommutation relations:K · K−1 = K−1 · K = 1,KX±K−1 = q±1X±;X+X−−X−X+ =K −K−1q1/2 −q−1/2 .

(1)The following formula for the comultiplication [3], the antipode and counit on thegenerators define the structure of a Hopf algebra on Uq(sl(2)):∆(X±) = X± ⊗K1/2 + K−1/2 ⊗X±,∆(K) = K ⊗K;(2)S(X±) = −q±1/2X±,S(K) = K−1;(3)ε(K) = 1,ε(X±) = 0. (4)We denote this Hopf algebra by Uq := (Uq(sl(2)), ∆, S, ε).

The maps ∆′ = σ ◦∆,S′ = S−1, where σ is the permutation in Uq(sl(2))⊗2, i.e. σ(a ⊗b) = b ⊗a, alsodefine the structure of Hopf algebra on Uq(sl(2)).

From (2) and (3) it follows thatUq−1 :=Uq(sl(2)), ∆′, S′, ε≃Uq−1(sl(2)), ∆, S, ε(5)as the Hopf algebras. It is well known (e.g.

[6]) that Uq1(sl(2)) and Uq2(sl(2)) areisomorphical as a Hopf algebras if q1 = q2 or q1q2 = 1. Remark that the square ofthe antipode S2(K) = K, S2(X±) = q±1X± does not coincide with identity map.Comultiplications ∆and ∆′ are connected in Uq(sl(2))⊗2 by the following au-tomorphism [5]∆′(a) = R∆(a)R−1,a ∈Uq,(6)where R ∈Uq(sl(2))⊗2.

The element R is called the universal R-matrix. It satisfiesthe relations(∆⊗id)R = R13R23(id ⊗∆)R = R13R12(s ⊗id)R = R−1(7)where the indices show the embeddings R into Uq(sl(2))⊗3.The center of thealgebra Uq(sl(2)) (if q is not equal to a root of unity) is generated by the q-analogof Casimir’s element [2,3]2c = (q1/2 + q−1/2)K1/2 −K−1/2q1/2 −q−1/22+ X+X−+ X−X+.

(8)Let us give now some useful formulas∆(Xm± ) =mXl=0hmliXl±Kl−m2⊗Xm−l±Kl/2;[Xn+, Xm−]=min(n,m)Xl=1[l]!hmli hnliXm−l−Xn−l+lYj=1q−12 (m−n+l−j) · K −q12 (m−n+l−j) · K−1q1/2 −q−1/2,(9)

Clebsch-Gordan and Racah-Wigner coefficients for Uq(SU(1, 1))3where we use the following notations[m] = qm2 −q−m2q1/2 −q−1/2 ,[m]! =mYj=1[j],[0]!

= 1,hmli=[m]![l]! [m −l]!,if0 ≤l ≤m.n◦2.

The real forms of a Hopf algebra A = (A, m, ∆, S, ε).First, let us recall the definition of ∗-antiinvolution of Hopf algebra A (e.g. [6,7]).It is a map ∗: A →A such that the following diagrams are commutative1)A ⊗Am−−−→A∗−−−→AσxmA ⊗A∗⊗∗−−−→A ⊗A(antiautomorphism of algebra);2)A∗−−−→A∆−−−→A ⊗A∆∗⊗∗A ⊗A(automorphism of co-algebra);3)∗2 = idA(involution);4)A∗−−−→AySyS′A∗−−−→A, i.e.

(∗◦S)2 = idA;5)ε(a∗) = ε(a),a ∈A.Two antiinvolutions ∗1 and ∗2 are called to be equivalent if there exists automor-phism ϕ of the Hopf algebra A such that the diagramA∗1−−−→AϕyyϕA∗2−−−→Ais commutative.The real form of the Hopf algebra A is by definition the pair(A, ∗) consisting of the Hopf algebra A and the class of antiinvolutions, which areequivalent to ∗. The real forms of Uq(sl(n)) are classified in [4] and for the caseUq(sl(2)) in [6,7].Proposition 1 ([4,6,7]).The real forms of Uq(sl(2)) are exhausted by thefollowing types:a)Uq(SU(2)), −1 < q < 1, q ̸= 0(a compact real form),K∗= K,X∗± = X∓;

4N.A. Liskova and A.N.

Kirillovb)Uq(SU(1, 1)), −1 < q < 1, q ̸= 0(a non compact real form),K∗= K,X∗± = −X∓;c)Uq(sl(2, R)), |q| = 1(a non compact real form),K∗= K,X∗± = −X±.We note that the real Lie algebras SU(1, 1) and sl(2, R) are equivalent (via theCayley transformation) in the classical case (q = 1), but in the quantum case thesetwo real forms are not equivalent.It is an interesting problem to quantize theirreducible unitary representations of the Lie algebras sl(2, R) and sl(2, C) (see e.g.[12]).n◦3. Irreducible unitary representations of Uq(SU(1, 1)), 0 < q < 1.Let us remind that the left Uq(SU(1, 1))-module V is called unitary Uq(SU(1, 1))representation if there exists a positive definite Hermitian scalar product ( , ) on Vsuch that(ax, y) = (x, a∗y),x, y ∈V,a ∈Uq(SU(1, 1)),(10)where the antiinvolution ∗defines the real form Uq(SU(1, 1)).

The Casimir operator(see the formula (8)) acts on an irreducible unitary representation V of Uq(SU(1, 1))as a scalar: C|V = cV · IdV and (−cV ) ∈R+. Before to formulate the result (e.g.

[8],[6]) concerning the classification of unitary irreducible representations of thealgebra Uq(SU(1, 1)) let us introduce some notations.Let us fix q = exp(−h),h ∈R∗+ and take ε = 0, 1/2. Let Hε be a complex Hilbert space with orthonormalbases{em m = ε + n, n ∈Z}.

(11)For any complex number j consider the following representation V jε of the algebraUq(SU(1, 1)) in the space HεX±ejm = ±[m ± j][m ∓j ± 1]1/2ejm±1,(12)Kejm = qmejm,where we use notations ejm for the bases of Hε instead of em.The irreducible unitary representations of Uq(SU(1, 1)), 0 < q < 1, are classified(up to the unitary equivalence) by the following types:I.Continuous (or principal) seriesV jε ,j = 12 −iσ,0 < σ < πh.II.Strange seriesV jε ,j = 12 −πih −s,s > 0.III.Complementary seriesV j0 ,0 < j < 12.

Clebsch-Gordan and Racah-Wigner coefficients for Uq(SU(1, 1))5IV.Discrete seriesa) Positive j ∈Z+ or j ∈12 + Z+,V j+ = {ejm m −j ∈Z+}.b) Negative j ∈Z+ or j ∈12 + Z+,V j−= {ejm m + j ∈Z−}.V.Exceptional representations j = 12,V12 ,+ = {ejm m ∈−12 + Z+},V12 ,−= {ejm m ∈12 + Z−}.The action of the generators of Uq(SU(1, 1)) in the cases IV and V are given by(12).Note that the continuous series we have−cj =1q1/4 + q−1/42+ 2 sin σh2q1/2 −q−1/2!2> 0,[m ± j][m ∓j ± 1] =hm ± 12i2+ 2 sin σh2q1/2 −q−1/2!2> 0,and for the strange series−cj =1q1/4 −q−1/42+ [s]2 > 0,[m ± j][m ∓j ± 1] =hm ± 12i2+qs/2 + q−s/2q1/2 −q−1/22> 0.The same inequalities are correct in all other cases.n◦4. Quantum Clebsch-Gordan coefficients for Uq(SU(1, 1)).We study the decomposition of the tensor product of two irreducible represen-tations of positive (or negative) discrete series for the algebra Uq(SU(1, 1)) andthe corresponding quantum q −3j symbols.

Our approach follows to the papers[9,10,11]. In the sequel we use notation V j := V j+.Theorem 1 (Clebsch-Gordan series for Uq(SU(1, 1))).We have the followingdecompositionV j1 ⊗V j2 =Mj≥j1+j2V j,j −j1 −j2 ∈Z+.

6N.A. Liskova and A.N.

KirillovProof.We will construct the lowest vectors in every irreducible componentV j ֒→V j1 ⊗V j2. For this aim let us consider a vectorej1j2jj=Xm1+m2=jam1,m2 ej1m1 ⊗ej2m2 ∈V j1 ⊗V j2.

(13)It is easy to see that ej1j2jj∈V j. This vector is a lowest vector in the componentV j if ∆(X−)ej1j2jj= 0.

So we obtain the recurrence relation on the coefficientsam1,m2, namelyam1+1,m2[m1 −j1 + 1][m1 + j1]1/2qm22+ am1,m2+1[m2 −j2 + 1][m2 + j2]1/2q−m12 = 0,(14)where j1 ≤m1 ≤j −j2 and m1 −j1 ∈Z. It is easy to find the solution of (14).

Wehaveaj1+k,j−j1−k =a0(−1)kq−k2 (j−1)[j −j1 −j2]! [j −j1 + j2 −1]!

[2j1 −1]![k]! [j −j1 −j2 −k]!

[j −j1 + j2 −k −1]! [2j1 + k −1]!1/2.

(15)The initial constant a0 may be found from the condition that vector (13) have thenorm equals to 1∥ej1j2jj∥2 =a20j−j1−j2Xk=0q−k(j−1)[j −j1 −j2]! [j −j1 + j2 −1]!

[2j1 −1]![k]! [j −j1 −j2 −k]!

[j −j1 + j2 −k −1]! [2j1 + k −1]!.

(16)Now we use the identity (e.g. [11])Xk≥0q−ak21[k]!

[b −k]! [c −k]!

[a −b −c + k]! = q−bc[a]![b]![c]!

[a −b]! [a −c]!.In our case we have a = 2j −2, b = j −j1 −j2, c = j −j1 + j2 −1.

Consequentlya0 = q14 (j−j1−j2)(j−j1+j2−1)[j −j2 + j1 −1]! [j + j1 + j2 −2]!

[2j −2]! [2j1 −1]!1/2.After substitution this expression into (15) and (13) we finally obtain the exactformula for (13)ej1j2jj=Xm1,m2j1j2jm1m2jSU(1,1)qej1m1 ⊗ej2m2,

Clebsch-Gordan and Racah-Wigner coefficients for Uq(SU(1, 1))7wherej1j2jm1m2jSU(1,1)q= δm1+m2,j · (−1)m1−j1q14 (j(j−1)+j1(j1−1)−j2(j2−1))−m1(j−1)2·· [j −j1 −j2]! [j −j1 + j2 −1]!

[j −j2 + j1 −1]! [j + j1 + j2 −2]!

[m1 −j1]! [j −j2 −m1]!

[j + j2 −m1 −1]! [j1 + m1 −1]!

[2j −2]!1/2. (17)We obtain the expression (17) for the lowest vector in the irreducible componentV j ֒→V j1 ⊗V j2.

Also we see that it is unique up to multiplication on nonzerocomplex number. In order to find the others weight vectors let us apply to thevector ej1j2jjthe raising operators ∆(Xm+ ).

After some normalization we obtainthe orthonormal bases ej1j2jmin V j. Let us define the quantum Clebsch-Gordancoefficients for the ∗-algebra Uq(SU(1, 1)) from the decompositionej1j2jm=Xm1,m2j1j2jm1m2mSU(1,1)qej1m1 ⊗ej2m2.

(18)From the formula (9) and (17), (18) we deduceTheorem 2 (Formula for the Clebsch-Gordan coefficients).j1j2jm1m2mSU(1,1)q= δm1+m2,m · (−1)j1−m1q14 (cj+cj1 −cj2)−m1(m−1)2˜∆(j1j2j)·[2j −1][m −j]! [m1 −j1]!

[m1 + j1 −1]! [m2 −j2]!

[m2 + j2 −1]! [m + j −1]!1/2(19)Xr≥0(−1)rqr2 (m+j−1) ·1[r]!

[m −j −r]! [m1 −j1 −r]!

[m1 + j1 −r −1]!·1[j −j2 −m1 + r]! [j + j2 −m1 + r −1]!,wherecj =j(j −1),˜∆(j1j2j) ={[j −j1 −j2]!

[j −j1 + j2 −1]! [j + j1 −j2 −1]!

[j + j1 + j2 −2]! }1/2.Note that formula (19) may be obtained from [10], formula (3.4), by the formalreplacements m 7→−m, j 7→−j, jα 7→−jα, mα 7→−mα (α = 1, 2), q →q−1 and[−n]!

→1[n−1]! if n ≥0.

Let us recall that the summation in (19) is taken only oversuch r that all factorials in the denominator are nonnegative. On the other side,from Theorem 2 it is easy to see that there exists the following relation between theClebsch-Gordan coefficients for Uq(SU(2)) and Uq(SU(1, 1)).

8N.A. Liskova and A.N.

KirillovTheorem 3. We havej1j2jm1m2mSU(2)q=ϕ1ϕ2ϕn1n2nSU(1,1)q−1,(20)wherej1 = 12(n1 + n2 + ϕ2 −ϕ1 −1),j2 = 12(n1 + n2 + ϕ1 −ϕ2 −1),m1 = 12(n2 −n1 + ϕ1 + ϕ2 −1),m2 = 12(n1 −n2 + ϕ1 + ϕ2 −1),j = ϕ −1,m = ϕ1 + ϕ2 −1,n1 = 12(j1 + j2 −m1 + m2 + 1),n2 = 12(j1 + j2 + m1 −m2 + 1),ϕ1 = 12(j2 −j1 + m1 + m2 + 1),ϕ2 = 12(j1 −j2 + m1 + m2 + 1),ϕ = j + 1,n = j1 + j2 + 1.The theorem 3 is the quantum analog of the corresponding classical result (forq = 1), see e.g.

[13].The symmetry’s properties for the Clebsch-Gordan coefficients of the algebraUq(SU(1, 1)) follows according to the Theorem 3 from the corresponding ones forthe algebra Uq(SU(2)) (e.g. [11]).

Here we mention only one.Corollary 4.We havej1j2jm1m2mSU(1,1)q= (−1)j1+j2−jj2j1jm2m1mSU(1,1)q−1.Similarly, it is possible to defind the decomposition into irreducible componentof the tensor product of the irreducible representations for the negative discreteseries and to compute the corresponding Clebsch-Gordan coefficients. We give onlythe answer.Theorem 5.

Assume that V j1 and V j2 lies in the negative discrete series forUq(SU(1, 1)). Thena)V j1 ⊗V j2 =⊕j≤j1+j2V j, j −j1 −j2 ∈Z−;b)j1j2jm1m2mSU(1,1)q= δm1+m2,m(−1)j1−m1 ˜∆(−j1, −j2, −j)q14 (cj+cj1 −cj2)+ m1(m+1)2·[−2j −1][j −m]!

[j1 −m1]! [−m1 −j1 −1]!

[j2 −m2]! [−m2 −j2 −1]!

[−m −j −1]!1/2(21)·X(−1)rqr2 (m+j+1)1[r]! [j −m −r]!

[j1 −m1 −r]! [−m1 −j1 −r −1]!·1[j2 −j + m1 + r]!

[−j −j2 + m1 + r −1]!.Remark.The formula (21) may be obtained from (19) by the replacementsmα, jα, m, j, q on −mα, −jα, −m, −j, q−1 (α = 1, 2).

Clebsch-Gordan and Racah-Wigner coefficients for Uq(SU(1, 1))9n◦5. Quantum Racah-Wigner coefficients for Uq(SU(1, 1)).In this section we consider the q-analog of a 6j-symbols for the tensor productV j1 ⊗V j2 ⊗V j3 of three irreducible representations of the positive discrete seriesfor Uq(SU(1, 1)).

As in the case of the algebra Uq(SU(2)), there are two ways toobtain an irreducible components in this tensor product. One is to decompose firstV j1 ⊗V j2 = ⊕j12V j12 and then to take an irreducible submodules in V j12 ⊗V j3.The other is to decompose first V j2 ⊗V j3 = ⊕j23V j23 and then V j1 ⊗V j23.

Thesetwo ways give two complete orthogonal bases in V j1 ⊗V j2 ⊗V j3:ej12jm (j1 j2 | j3) =Xm1,m2,m3j12j3jm12m3mSU(1,1)qj1j2j12m1m2m12SU(1,1)q· ej1m1 ⊗ej2m2 ⊗ej3m3;(22)ej23jm (j1 | j2 j3) =Xm1,m2,m3j1j23jm1m23mSU(1,1)qj2j3j23m2m3m23SU(1,1)q· ej1m1 ⊗ej2m2 ⊗ej3m3. (23)The matrix elements of the matrix, connecting these bases will be called SU(1, 1) q−6j-symbols:ej2jm (j1 j2 | j3) =Xj23j1j2j12j3jj23SU(1,1)qej23jm (j1 | j2 j3).

(24)Using the graphical technique (e.g. [10]) we may rewrite the definition (24) ofq-6j-symbols in the formj1j2j3j=Xj23j1j2j12j3jj23SU(1,1)qj1j2j3jActing by the same way as in the case of Hopf algebra Uq(sl(2)) (e.g.

[10]), we mayfind the formula for q-6j-symbols for an irreducible representations of the positivediscrete series for Uq(SU(1, 1)). The answer may be obtained from [10], formula(5.7) by the replacements all j’s on −j’s and [−n]!

on1[n−1]!, if n > 0. However, itis possible to receive for q-6j-symbols the result of the type (20).Theorem 6.abedcfSU(1,1)q=αβεδγϕSU(2)q,

10N.A. Liskova and A.N.

Kirillovwhereα = a + b + c + d2−1;β = c −a −b −d2;γ = a + c + d −b2−1;δ = b + c + d −a2−1;ε = e −1;ϕ = f −1;a = α −β + γ −δ + 12;b = α −β −γ + δ2;c = α + β + γ + δ + 12;d = γ + δ −α −β2;e = ε + 1;f = ϕ + 1;So, it is easy to see that q-6j-symbolsna b ed c foq satisfies the orthogonality re-lation, the Racah identity, the Biedenharn-Elliot identity and the face variant ofquantum Yang-Baxter equation, e.g. see identities (6.16)-(6.19) from [10].After the completion of this note the authors were known about the work of Y.Shibukawa [14] which also contains the calculation of the Clebsh-Gordan coefficientsfor the positive discrete series of the algebra Uq(SU(1, 1)).References1.

Kulish, P.P., Reshetikhin N.Yu, Quantum linear problem for the Sine-Gordon equa-tion and higher representations, Zap. Nauch.

Semin. LOMI 101 (1980) 101-110 (inRussian).2.

Jimbo, M., A q-difference analog of U(g) and the Yang-Baxter equation, Lett. Math.Phys, 10 (1985), 63–69.3.

Sklyanin E.K., Uspehi Mat. Nauk, 40 (1985), N2, 214 (in Russian).4.

Faddeev, L., Reshetikhin, N.Yu., and Takhtajan, L.A., Quantization of Lie algebrasand Lie groups, Algebra i Analiz 1 (1989), N1 (in Russian).5. Drinfeld V.G., Quantum groups, Proc.

ICM 1 (Berkeley Academic Press, 1986), 798–820.6. Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M., Saburi, Y., and Ueno, K., Unitaryrepresentations of the quantum group SUq(1, 1) I, II, Lett.

Math. Phys.

19 (1990),187–204.7. Soybelman, Y.L, and Vaksman, L.L., Algebra functions on quantum group SU(2),Funk Anal.

i appl. 22 (1988) N3, 1-14 (in Russian).8.

Vaksman, L.L., and Korogodsky, L.I., Harmonic analysis on quantum hyperbolids,Preprint, 1990 (in Russian).9. Kirillov, A.N., and Reshetikhin, N.Yu., Representations of the algebra Uq(sl(2)), q-orthogonal polynomials and invariants of links, LOMI Preprint E-9-88, Leningrad,1988.10., Representations of the algebra Uq(sl(2)), q-orthogonal polynomials and in-variants of links, In: Kac V.G.

(ed), Infinite dimensional Lie algebras and groups.Proc. CIRM, 1988.

Advanced Series in Mathematical Physics, 7 285-339, Singapore,New Jersey, London: World Scientific, 1989.11. Kirillov, A.N., Quantum Clebsch-Gordon coeffficients, Zap.

Nauch. Semin.

LOMI, 168(1988), 67-84 (in Russian).

Clebsch-Gordan and Racah-Wigner coefficients for Uq(SU(1, 1))1112. Podleˇc, P., Complex quontum groups and their real representations, RIMS Preprint,Kyoto Univ., 754, May 1991.13.

Barut, A., and Raczka, R., Theory of group representations and applications, (WarszawaPWN-Polish Scientific Publishers, 1980), 717p.14. Shibukawa Y., Clebsch-Gordan coefficients for Uq(SU(1, 1)) and Uq(sl(2)), and lin-earization formula of matrix elements.

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