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**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.

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.

Reviewer: H.Boseck (Greifswald)

### 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) |