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.


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