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Hyperplane arrangements and holonomy equations. (English) Zbl 0848.18004
In this beautifully written paper, the authors study the solutions of the Knizhnik-Zamolodchikov equation associated to some hyperplane arrangements and made an important contribution to the subject. Firstly, a convenient framework for hyperplane arrangements was set up to handle the combinatorics. Then, using this framework, they recovered some of the fundamental results of Drinfeld’s theory, for example the Drinfeld associatior in quantum group. In the case of the hyperplane arrangements associated to a root system, they gave a description of the monodromy representation which can be regarded as an extension of the result of {\it A. Kono} for the root system $A_n$. Finally, they explained how the universal Vassiliev invariants constructed by {\it M. Kontsevich} can be viewed as values of certain monodromy. Many of the above topics can be found in separate treatments by others and often in some abstract categorical framework. The advantage of the authors’ approach is that it presents these topics from a uniform and geometric viewpoint.

MSC:
18D10Monoidal, symmetric monoidal and braided categories
17B37Quantum groups and related deformations
57M25Knots and links in the $3$-sphere
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References:
[1] K. Aomoto. Functions hyperlogarithmiques et groupes de monodromie unipotents. J. Fac. Univ. Tokyo25 (1978), 149--156. · Zbl 0416.32020
[2] V.I. Arnold.The Vassiliev theory of discriminants and knots. in First European congress of Mathematics, Birkhäuser Basel, 1994. · Zbl 0869.57006
[3] D. Bar Natan.On the Vassiliev knot invariants. Topology,34 (1995), 423--472. · Zbl 0898.57001 · doi:10.1016/0040-9383(95)93237-2
[4] D. Bar Natan.Non-associative tangles. Harvard preprint (1993). · Zbl 0888.57008
[5] N. BourbakiGroupes et algèbres de Lie Ch 4-5-6. Hermann, Paris, 1981.
[6] E. Brieskorn.Sur les groupes de tresses (d’après V I Arnold). Séminaire Bourbaki 1971/72, S.L.N.317 (1973).
[7] E. Brieskorn. Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe. Inv. Math.,12 (1971) 57--61. · Zbl 0204.56502 · doi:10.1007/BF01389827
[8] P. Cartier. Construction combinatoire des invariants de Vassiliev-Kontsevich des noeuds. C.R. Acad. Sci. Paris, Sér. I Math.,316 (1993) 1205--1210. · Zbl 0791.57006
[9] K.T. Chen.Iterated integrals of differential forms and loop space cohomology. Ann. of Math., (1973) 217--246. · Zbl 0227.58003
[10] I.V. Cherednik.Generalized Braid Groups and local r-matrix systems. Doklady Akad. Nauk SSSR,307 (1989) 27--34. · Zbl 0747.17017
[11] I.V. CherednikMonodromy Representations for Generalized Knizhnik-Zamolodchikov Equations and Hecke Algebras. Publ. RIMS, Kyoto Univ.27 (1991) 711--726. · Zbl 0753.17035 · doi:10.2977/prims/1195169268
[12] C. De Concini and C. Procesi.Wonderful models of subspace arrangements. Selecta Math., (to appear) (1995). · Zbl 0842.14038
[13] P. Deligne. Les immeubles de groupes de tresses généralisés. Invent. Math.,17 (1972) 273--302. · Zbl 0238.20034 · doi:10.1007/BF01406236
[14] P. Deligne. Le groupe fondamental de la droite projective moins trois points. In ”Galois groups over $\mathbb{Q}$”, ed. Ihara, Ribet, Serre, Publ. M.S.R.I,16 (1987) 79--298.
[15] P. Deligne.Théorie de Hodge II. Publ. Math. I.H.E.S.,40 (1971) 5--58. · Zbl 0219.14007
[16] V.G. Drinfeld.Quasi Hopf algebras. Leningrad Math. J.,1 (1990) 1419--1457. · Zbl 0718.16033
[17] V. G. Drinfeld.On quasi triangular quasi-Hopf algebras and a group closely connected with Gal . Leningrad Math. J.,2 (1991), 829--860. · Zbl 0728.16021
[18] J. Humphreys.Reflection groups and Coxeter groups. Cambridge Studies in Adv. Math.,29 (1992). · Zbl 0768.20016
[19] M. Kontsevich.Vassiliev’s knot invariants. Advances in Soviet Math.,16 (1993) 137--150. · Zbl 0839.57006
[20] C. Kassel.Quantum Groups. Graduate texts in Math., Springer155 (1995)
[21] S. Keel.Intersection theory of moduli space of stable N-pointed curves of genus 0. T.A.M.S.,330 (1992), 545--574. · Zbl 0768.14002 · doi:10.2307/2153922
[22] T. Khono.On the holonomy Lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces. Nagoya Math. J.,93 (1983) 21--37.
[23] T. Khono.Monodromy representations of braid groups and Yang-Baxter equations. Ann. Inst. Fourier,37 (1987) 139--160. · Zbl 0634.58040
[24] M.M. KapranovThe permutoassociahedron, Mac Lane’s coherence theorem and asymptotic zones for the K-Z equation. J. Pure and Appl. Alg.85 (1993) 119--142. · Zbl 0812.18003 · doi:10.1016/0022-4049(93)90049-Y
[25] V.G. Knizhnik and A.B. Zamolodchikov.Current algebra and the Wess-Zumino model in two dimensions. Soviet J. of nuclear Physics,247 (1984), 83--103. · Zbl 0661.17020 · doi:10.1016/0550-3213(84)90374-2
[26] J.A. Lappo-Danilevsky.Mémoires sur la théorie des systèmes différentiels linéaires. Chelsea Publ. N.Y. (1953) · Zbl 0051.32301
[27] T.Q.T. Le and J. Murakani.Representations of the category of tangles by Kontsevich’s iterated integral. Max-Planck-Institut Bonn, preprint.
[28] S. Piunikhin.Combinatorial expression for universal Vassiliev’s link invariant. Harvard Univ. preprint (1993). · Zbl 0811.58011
[29] V.A. Vassiliev.Complements of discriminants of smooth maps. A. M. S. Transl.98 (1992). · Zbl 0779.58040
[30] E.T. Whittaker and G.N. Watson.A course in modern analysis. Cambridge Univ. Press, 4th ed. (1962). · Zbl 0105.26901