## Representations of the algebra $$U_ q(sl(2))$$, $$q$$-orthogonal polynomials and invariants of links.(English)Zbl 0742.17018

Infinite dimensional Lie algebras and groups, Proc. Conf., Marseille/Fr. 1988, Adv. Ser. Math. Phys. 7, 285-337 (1989).
[For the entire collection see Zbl 0741.00064.]
The authors discuss the $$q$$-deformation $$U_ q(\hbox{sl}(2))$$ and its real form $$U_ q(\hbox{su}(2))$$ as a Hopf algebra, its irreducible representations and finite-dimensional $$R$$-matrices. The decomposition of the tensor product of two irreducible finite-dimensional representations is given by a $$q$$-analogue of Clebsch-Gordan coefficients. There are relations between $$R$$-matrices and $$q$$-Clebsch-Gordan coefficients (section 3). Using the connections between quantized universal algebras and link invariants introduced by the second author, a graphical representation of the connections between $$R$$-matrices and $$q$$-Clebsch- Gordan coefficients is proposed (section 4). The $$q$$-analogues of the Wigner-Racah $$6j$$-symbols are defined and related to $$q$$-hypergeometric functions (section 5). Graphical representations of the $$q$$-$$6j$$-symbols are introduced and the Racah-Wilson polynomials are shown to be orthogonal (section 6). Based on the graphical representation of the $$q$$- $$6j$$-symbols a model for link invariants corresponding to higher representations of $$U_ q(\hbox{su}(2))$$ is obtained (section 7). Evidently the authors are unaware of the publications and results of T. Koornwinder.

### MSC:

 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, $$p$$-adic groups, Hecke algebras, and related topics 57M25 Knots and links in the $$3$$-sphere (MSC2010)

Zbl 0741.00064