Clebsch-gordan and Racah-wigner Coefficients for U Q (su (1, 1))

The Clebsch-Gordan and Racah-Wigner coefficients for the positive (or negative) discrete series of irreducible representations for the noncompact form Uq(SU(1, 1)) of the algebra Uq(sl(2)) are computed. n0. Introduction Now it is well known the great significance which the Clebsch-Gordan and RacahWigner 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 functions. In this note we compute the Clebsch-Gordan and Racah-Wigner coefficients for the non-compact form Uq(SU(1, 1)) of the Hopf algebra Uq(sl(2)) in the case corresponding to the tensor product of the irreducible representations of positive (or negative) discrete series. Our main result consists of two parts. At first we obtain the formula for the Clebsch-Gordan and Racah-Wigner coefficients in the case mentioned above as an analytical continuation of corresponding formula for the algebra Uq(sl(2)) in the region of negative values of parameters. At the second we find the simple substitutions which transforms the corresponding formula for Uq(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 gratitude to the organizers of the RIMS 91 Project “Infinite Analysis” for the invitation to take participation in the workshop of this Project and the secretaries of RIMS for the various assistance and the help in preparing the manuscript to publication. 2 N.A. Liskova and A.N. Kirillov n1. Algebra Uq(sl(2)) and it compact forms. The algebra Uq(sl(2)), [1,2], is generated by elements {K,K , X±} with the commutation relations: K ·K = K ·K = 1, KX±K −1 = qX±; X+X− −X−X+ = K −K q1/2 − q−1/2 . (1) The following formula for the comultiplication [3], the antipode and counit on the generators define the structure of a Hopf algebra on Uq(sl(2)): ∆(X±) = X± ⊗K 1/2 +K ⊗X±, ∆(K) = K ⊗K; (2) S(X±) = −q X±, S(K) = K ; (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, where σ is the permutation in Uq(sl(2)) , i.e. σ(a ⊗ b) = b ⊗ a, also define the structure of Hopf algebra on Uq(sl(2)). From (2) and (3) it follows that Uq−1 := ( Uq(sl(2)),∆ , S, ε )