The classical and quantum \(6j\)-symbols.

*(English)*Zbl 0851.17001
Mathematical Notes (Princeton). 43. Princeton, NJ: Princeton Univ. Press. ix, 164 p. (1996).

This book discusses the representation theory of classical and quantum \(U(\text{sl} (2))\) with an eye towards topological applications of the latter. Some general notions of the representation theory of \(U(\text{sl} (2))\) are given. To facilitate computations related to tensor product decompositions, diagrammatic techniques based upon spin networks are introduced. In this context, the Temperley-Lieb algebra is defined, and Clebsch-Gordan coefficients are computed explicitly. Next, \(6j\)-coefficients are defined; orthogonality and the Elliott-Biedenharn identity are proved diagrammatically. Further properties and an explicit single sum expression are deduced. The diagrammatic techniques are essentially the same as those developed by A. P. Yutsis, I. B. Levinson and V. V. Vanagas [Mathematical apparatus of the theory of angular momentum, Israel Program for Scientific Translations, Jerusalem (1962; Zbl 0111.42704)], and familiar to some physicists working in that field.

In the next chapter, representations of the quantum enveloping algebra \(U_q (\text{sl} (2))\) are considered. Concrete representations are constructed via the braid group, leading again to diagrammatic techniques. The quantum Clebsch-Gordan theory and recoupling theory (quantum \(6j\)-coefficients) are developed. For the root of unity case, the quantum trace and color representations are analysed. Finally, it is explained how to use the \(6j\)-symbol to give the definition of the Turaev-Viro invariants of 3-manifolds.

The book gives a self-contained treatment of the subject. Most of the results of the chapter on \(U(\text{sl} (2))\) are known in the physics literature, but here they are deduced in a mathematically rigorous way. The book gives a fine treatment and overview of topological results related to quantum \(U_q (\text{sl} (2))\). In an attempt of being self-contained however, some of the intermediate results have an elaborate combinatorial proof whereas they can be shown in a few lines using known results (e.g. Lemma 3.7.3 is a restatement of the \(q\)-Vandermonde theorem for basic hypergeometric series).

In the next chapter, representations of the quantum enveloping algebra \(U_q (\text{sl} (2))\) are considered. Concrete representations are constructed via the braid group, leading again to diagrammatic techniques. The quantum Clebsch-Gordan theory and recoupling theory (quantum \(6j\)-coefficients) are developed. For the root of unity case, the quantum trace and color representations are analysed. Finally, it is explained how to use the \(6j\)-symbol to give the definition of the Turaev-Viro invariants of 3-manifolds.

The book gives a self-contained treatment of the subject. Most of the results of the chapter on \(U(\text{sl} (2))\) are known in the physics literature, but here they are deduced in a mathematically rigorous way. The book gives a fine treatment and overview of topological results related to quantum \(U_q (\text{sl} (2))\). In an attempt of being self-contained however, some of the intermediate results have an elaborate combinatorial proof whereas they can be shown in a few lines using known results (e.g. Lemma 3.7.3 is a restatement of the \(q\)-Vandermonde theorem for basic hypergeometric series).

Reviewer: J.Van der Jeugt (Gent)

##### MSC:

17-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras |

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

17B35 | Universal enveloping (super)algebras |

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) |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

81-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory |