##
**Compact quantum groups and \(q\)-special functions.**
*(English)*
Zbl 0821.17015

Baldoni, Velleda (ed.) et al., Representations of Lie groups and quantum groups. Proceedings of the session of the European School of Group Theory, held in Trento, Italy, July 19-30, 1993. Harlow: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 311, 46-128 (1994).

The purpose of this article is to give an introduction to Hopf algebras and compact quantum groups and to basic hypergeometric functions and related \(q\)-orthogonal polynomials. The article is written on the base of the author’s lectures at the European School of Group Theory (Trento, 1993).

The author presents the basic theory of Hopf algebras and their corepresentations. The corresponding proofs are either given in full detail or they are sketched such that the reader can easily fill in the gaps. Then compact quantum groups are described. At the beginning, the author avoids \(C^*\)-algebras in the definition and in the proofs, but formulates everything on the Hopf \(*\)-algebra level. Then the \(C^*\)- algebra completion appears as a final observation instead of an essential part of the definition. The Haar functional plays a crucial role.

The second part of the paper is devoted to an introduction to basic hypergeometric functions. The more elementary \(q\)-special functions such as \(q\)-exponential and \(q\)-binomial series are treated in a self- contained way. For higher \(q\)-hypergeometric functions some identities are given without proofs. The author gives here an overview of the main \(q\)-orthogonal polynomials. The Askey tableau of classical orthogonal polynomials (for which \(q = 1\)) is discussed. The \(q\)-Hahn tableau is described. A self-contained introduction to the Askey-Wilson polynomials is given. The relations of compact quantum groups to \(q\)-special functions are not described explicitly.

For the entire collection see [Zbl 0809.00015].

The author presents the basic theory of Hopf algebras and their corepresentations. The corresponding proofs are either given in full detail or they are sketched such that the reader can easily fill in the gaps. Then compact quantum groups are described. At the beginning, the author avoids \(C^*\)-algebras in the definition and in the proofs, but formulates everything on the Hopf \(*\)-algebra level. Then the \(C^*\)- algebra completion appears as a final observation instead of an essential part of the definition. The Haar functional plays a crucial role.

The second part of the paper is devoted to an introduction to basic hypergeometric functions. The more elementary \(q\)-special functions such as \(q\)-exponential and \(q\)-binomial series are treated in a self- contained way. For higher \(q\)-hypergeometric functions some identities are given without proofs. The author gives here an overview of the main \(q\)-orthogonal polynomials. The Askey tableau of classical orthogonal polynomials (for which \(q = 1\)) is discussed. The \(q\)-Hahn tableau is described. A self-contained introduction to the Askey-Wilson polynomials is given. The relations of compact quantum groups to \(q\)-special functions are not described explicitly.

For the entire collection see [Zbl 0809.00015].

Reviewer: A.Klimyk (Kiev)

### MSC:

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |

33D45 | Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) |

46L89 | Other “noncommutative” mathematics based on \(C^*\)-algebra theory |