Geodesic
Algebraic Characterization of the Monodromy of Generalized Knizhnik--Zamolodchikov Equations of $B_n$ Type
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Monodromy in problems of algebraic geometry and differential equations, Tome 238 (2002), pp. 124-143.

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The Drinfeld–Kohno theorem describes the monodromy of the Knizhnik–Zamolodchikov equation in terms of quasi-bialgebras. The present paper contains a generalization of this theorem for the case of a Knizhnik–Zamolodchikov type equation associated with the root system $B_n$. The characterization is given to those representations of the fundamental group of the complement to the singular divisor of the equation that can be realized as representations of the monodromy of the equation. 
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V. A. Golubeva; V. P. Leksin. Algebraic Characterization of the Monodromy of Generalized Knizhnik--Zamolodchikov Equations of $B_n$ Type. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Monodromy in problems of algebraic geometry and differential equations, Tome 238 (2002), pp. 124-143. http://geodesic.mathdoc.fr/item/TM_2002_238_a7/

[1] Regge T., “Algebraic topology methods in the theory of Feynman relativistic amplitudes”, Battelle rencontres, 1967, eds. C. DeWitt, J. Wheeler, Benjamin, New York, 1968, 433–458

[2] Golubeva V. A., “Nekotorye voprosy analiticheskoi teorii feinmanovskikh integralov”, UMN, 31:2 (1976), 174–202 | MR

[3] Deligne P., Equations différentielles à points singuliers réguliers, Lect. Notes Math., 163, Springer, Berlin, 1970 | MR | Zbl

[4] Gérard R., “Théorie de Fuchs sur une varieté analytique complexe”, J. Math. Pures et Appl. Sér. 9, 47 (1968), 321–404 | MR | Zbl

[5] Bolibrukh A. A., “Sistemy Pfaffa tipa Fuksa na kompleksnom analiticheskom mnogoobrazii”, Mat. sb., 103:1 (1977), 112–123 | MR | Zbl

[6] Golubeva V. A., “O meromorfnom sechenii kompleksno-analiticheskogo vektornogo rassloennogo prostranstva nad kompleksnym proektivnym prostranstvom”, Mat. sb., 103:4 (1977), 505–518 | MR | Zbl

[7] Suzuki O., “The problems of Riemann and Hilbert and the relations of Fuchs in several complex variables”, Lect. Notes Math., 712, 1979, 325–364 | MR | Zbl

[8] Kita M., “The Riemann–Hilbert problem and its application to analytic functions of several complex variables”, Tokyo J. Math., 2:1 (1979), 1–27 | DOI | MR | Zbl

[9] Kita M., “The Riemann–Hilbert problem in several complex variables, II”, Tokyo J. Math., 2:2 (1979), 293–300 | DOI | MR | Zbl

[10] Katz N., “An overview of Deligne's works on Hilbert's twenty-first problem”, Proc. Symp. Pure Math., 28 (1976), 537–557 | MR | Zbl

[11] Hain R., “On a generalization of Hilbert's 21st problem”, Ann. Sci. Ecole Norm. Super. Sér. 4, 19:4 (1986), 609–627 | MR | Zbl

[12] Bolibrukh A. A., “O fundamentalnoi matritse sistemy Pfaffa tipa Fuksa”, Izv. AN SSSR. Ser. mat., 41:5 (1977), 1084–1109 | MR | Zbl

[13] Bolibrukh A. A., “Primer nerazreshimoi problemy Rimana–Gilberta na $\mathbb{CP}^2$”, Geometricheskie metody v zadachakh algebry i analiza, no. 2, Izd. Yarosl. gos. un-ta, Yaroslavl, 1980, 60–64 | MR

[14] Leksin V. P., “O fuksovykh predstavleniyakh fundamentalnoi gruppy kompleksnogo mnogoobraziya”, Kachestvennye i priblizhennye metody issledovaniya operatornykh uravnenii, no. 4, Izd. Yarosl. gos. un-ta, Yaroslavl, 1979, 109–114 | MR

[15] Golubeva V. A., “On the Riemann–Hilbert problem in a class of the Knizhnik–Zamolodchikov equations”, Funct. Diff. Equat., 8:3/4 (2001), 241–256 | MR | Zbl

[16] Drinfeld V. G., “Kvazikhopfovy algebry”, Algebra i analiz, 1:6 (1989), 114–148 | MR

[17] Drinfeld V. G., “O kvazitreugolnykh kvazikhopfovykh algebrakh i odnoi gruppe, tesno svyazannoi s $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$”, Algebra i analiz, 2:4 (1990), 149–181 | MR | Zbl

[18] Kohno T., “Linear representations of braid groups and classical Yang–Baxter equations”, Contemp. Math., 78 (1988), 339–363 | MR | Zbl

[19] Kohno T., Quantized universal envelopping algebras and monodromy of braid groups, Preprint Nagoya Univ., 1991

[20] Knizhnik V. G., Zamolodchikov A. B., “Current algebra and Wess–Zumino models in two dimensions”, Nucl. Phys. B., 247:1 (1984), 83–103 | DOI | MR | Zbl

[21] Kassel K., Kvantovye gruppy, Fazis, M., 1999

[22] Le T. Q. T., Murakami I., “Representations of the category of tangles by Kontsevich's iterated integral”, Commun. Math. Phys., 168:3 (1995), 535–562 | DOI | MR | Zbl

[23] Tanisaki T., “Killing forms, Harish-Chandra isomorphisms, and universal $R$-matrices for quantum algebras”, Intern. J. Mod. Phys. A., 7:Suppl. 1B (1992), 941–961 | DOI | MR | Zbl

[24] Piunikhin S., “Combinatorial expression for universal Vassiliev link invariant”, Commun. Math. Phys., 168:1 (1995), 1–22 | DOI | MR | Zbl

[25] Leibman A., “Some monodromy representations of generalized braid groups”, Commun. Math. Phys., 164:2 (1994), 293–304 | DOI | MR | Zbl

[26] Markushevich D. G., “The $D_n$ generalized pure braid groups”, Geom. Dedicata, 40:1 (1991), 73–96 | DOI | MR | Zbl

[27] Vershinin V., “Gruppy kos i prostranstva petel”, UMN, 54:2 (1999), 3–84 | MR | Zbl

[28] Mathieu O., “Équations de Knizhnik–Zamolodchikov et théorie des representations”, Séminaire Bourbaki, 1993/94, Astérisque, 227, no. 777, 1995, 47–67 | MR | Zbl

[29] Leibman A., Markushevich D., “The monodromy of the Brieskorn bundle”, Contemp. Math., 164 (1994), 91–117 | MR | Zbl

[30] Leibman A., “Fiber bundles with degenerations and their applications to computing fundamental groups”, Geom. Dedicata, 48:1 (1993), 93–126 | DOI | MR | Zbl

[31] tom Dieck T., “Knotentheorie und Wurzelsysteme, I, II”, Math. Göttingenis, 21 (1993), 44

[32] Lambropoulou S., “Solid torus links and Hecke algebras of $B$ type”, Proc. Conf. on Quantum Topology, World Sci., Singapore, 1994, 225–245 | MR | Zbl

[33] Häring-Oldenburg R., “Tensor categories of Coxeter type $B$ and QFT on the half plane”, J. Math. Phys., 38:10 (1997), 5371–5382 | DOI | MR | Zbl

[34] Golubeva V. A., Leksin V. P., “On two types of representations of the braid group associated with the Knizhnik–Zamolodchikov equation of the $B_n$ type”, J. Dyn. and Control Syst., 5:4 (1999), 565–596 | DOI | MR | Zbl

[35] Cherednik I. V., “Obobschennye gruppy kos i lokalnye $r$-matrichnye sistemy”, DAN SSSR, 307:1 (1989), 49–53 | Zbl

[36] Cherednik I. V., “Monodromy representations for generalized Knizhnik–Zamolodchikov equations and Hecke algebras”, Publ. RIMS. Kyoto Univ., 27:5 (1991), 711–726 | DOI | MR | Zbl

[37] de Concini C., Procesi C., “Hyperplane arrangements and holonomy equations”, Sel. Math. New ser., 1:3 (1995), 495–535 | DOI | MR | Zbl

[38] Goryunov V. V., “Kogomologii grupp kos serii $C$ i $D$”, Tr. Mosk. mat. o-va, 42, 1981, 234–242 | MR

[39] Lexin V. P., “Monodromy for KZ equations of the $B_n$-type and accompanying algebraic structures”, Funct. Diff. Equat., 8:3/4 (2001), 335–344 | MR | Zbl

[40] Loday J.-L., Cyclic homology, Grundl. Math. Wissensch., 301, Springer, Berlin, 1998 | MR | Zbl