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**Ribbon graphs and their invariants derived from quantum groups.**
*(English)*
Zbl 0768.57003

The authors construct the generalization of the Jones polynomial of links to the case of graphs in \(R^ 3\). They introduce the so-called coloured ribbon graphs in \(R^ 2 \times [0,1]\) and define for them Jones-type isotopy invariants. The approach to colouring is based on Drinfeld’s notion of a quasitriangular Hopf algebra. For each quasitriangular Hopf algebra \(A\) the authors define \(A\)-coloured (ribbon) graphs. The colour of an edge is an \(A\)-module. The colour of a vertex is an \(A\)-linear homomorphism intertwining the modules which correspond to edges incident to this vertex. The category of an \(A\)-coloured ribbon graph is a compact braided strict monoidal category introduced by A. Joyal and R. Street [Braided monoidal categories, Macquarie Math. Reports, Report No. 860081 (1986)]. If \(A\) satisfies a minor additional condition, then the authors construct a canonical covariant functor from the category of \(A\)- coloured ribbon graphs into the category of \(A\)-modules. In the case of \(A\) being the quantized universal enveloping algebra of \(sl_ 2\) this functor generalizes the Jones polynomial of links. If \(A=U_ h(sl_ n(C))\) this generalizes the Jones-Conway (Thomflyp) polynomial and for \(A=U_ hG\), \(G=so(n)\), \(sp(2k)\) the Kauffman polynomial. The paper under review was followed by the authors’ paper “Invariants of 3-manifolds via link polynomials and quantum groups” [Invent. Math. 103, 547-597 (1991; Zbl 0725.57007)]in which the authors construct new topological invariants of compact oriented manifolds (employing the methods of the paper under review). The construction was partially inspired by ideas of E. Witten [Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121, 351-399 (1989; Zbl 0667.57005)]who considered quantum field theory defined by the nonabelian Chern-Simon action and applied it to the study of 3-manifolds (on physical level of rigor).

Reviewer: J.H.Przytycki (Riverside)

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |

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

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

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

### Keywords:

generalization of the Jones polynomial; graphs in \(R^ 3\); coloured ribbon graphs; quasitriangular Hopf algebra; quantized universal enveloping algebra of \(\text{sl}_ 2\); Kauffman polynomial; quantum field theory
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\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)

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### References:

[1] | [Br] Brejxxxxxxxxxxxer, M.R., Moody, R.V., Patera, J.: Tables of dominant weight multiplicites for representation of simple Lie groups. New York: Dekker 1984. |

[2] | [D] Deling, P., Milne, J.S.: Tannakian Categories in Hodge cicles motives and simura varieties. Lecture Notes ion Mathematics, vol 900. Berlin, Heidelberg, New York: Springer 1982 |

[3] | [Dr1] Drinfeld, V.G.: Proceedings of the International Congres of Mathematics, vol. 1, p. 798–820. Berkeley, California: New York: Academic Press 1986) |

[4] | [Dr2] Drinfeld, V.G.: Quasicommutative Hopf algebras. Algebra and Anal.1 (1989) |

[5] | [FRT] Faddeev, L., Reshetikhin, N., Takhtajan, L.: Quantized Lie Groups and Lie algebras. LOMI-preprint, E-14-87, 1987, Leningrad |

[6] | [FY] Freyd, P.J., Yetter, D.N.: Braided compact closed categories with applications to low dimensional topology. Preprint (1987) |

[7] | [JS] Joyal, A., Street, R.: Braied Manoidal categories. Macquarie Math. Reports. Report No. 860081 (1986) |

[8] | [JO] Jones, V.F.R.: A polynomial invariant of Knots via von Neumann algebras. Bull. Am. Math. Soc.12, 103–111 (1985) · Zbl 0564.57006 · doi:10.1090/S0273-0979-1985-15304-2 |

[9] | [KT] Tsuchiya, A., Kanlie, Y.: Vertex operators in the conformal field theory of onP1 and monodromy representations of the braid group. In: Conformal field theory and solvable lattice models. Lett. Math. Phys.13, 303 (1987) |

[10] | [KV] Kauffman, L.H., Vogel, P.: Link polynomials and a graphical calculus. Preprint (1987) |

[11] | [K] Kirillov, A.N.:q-analog of Clebsch-Gordon coeffcients for slz. Zap. Nauch. Semin. LOMI,170, 1988 |

[12] | [KR] Kirillov, A.N., Reshetikhin, N.Yu.: Representation of the algebraU q(sl 2),q-orthogonal polynomials and invariants of links. LOMI-preprint, E-9-88, Leningrad 1988 |

[13] | [L] Luztig, G.: Quantum derformations of certain simple modules over enveloping algebras. M.I.T. Preprint, December 1987 |

[14] | [Ma] MacLane, S.: Natural associativity and commutavity. Rice Univ. Studies49, 28–46 (1963) · Zbl 0244.18008 |

[15] | [MS] Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys.123, 177–254 (1989) · Zbl 0694.53074 · doi:10.1007/BF01238857 |

[16] | [Re1] Reshetikhin, N.Yu.: Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links. I. LOMI-preprint E-4-87 (1988), II. LOMI-preprint E-17-87 (1988) |

[17] | [Re2] Reshetikhin, N.Yu.: Quasitriangluar Hopfalgebras and invariants of Tangles. Algebra and Anal.1, (1988) (in Russian) |

[18] | [RS] Reshetikhin, N.Yu.: Semenov: Tian-Shansky, M.A.: QuantumR-matrices and factorization problem in quatum groups. J. Geom. Phys. v. V (1988) |

[19] | [RO] Rosso, M.: Représentations irréductibles de dimension finie duq-analogue de l’algébre enveloppante d’une algèbre de Lie simple. C.R. Acad. Sci. paris.305, Se’rie 1, 587–590 (1987) |

[20] | [Tu1] Turaev, V.G.: The Yang-Baxter equation and invariants of links. Invent. Math.92, 527–553 (1988) · Zbl 0648.57003 · doi:10.1007/BF01393746 |

[21] | [Tu2] Turaev, V.G.: The Conways and Kauffman modules of the solid torus with an appendix on the operator invariants of tangles. LOMI Preprint E-6-88. Leningrad (1988) |

[22] | [Ye] Yetter, D.N.: Category theoretic representations of Knotted graphs in S3, Preprint |

[23] | [Wi] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. (in press) · Zbl 0818.57014 |

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