zbMATH — the first resource for mathematics

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).

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)
Full Text:
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.