Abstract
The paper studies Pfaffian systems of the Fuchs type on complex linear spaces defined by configurations of vectors in these spaces. It is shown that the Veselov ∨-condition imposed on a configuration of vectors is equivalent to the integrability of these system. For configuration of vectors that are root systems, a description of monodromy representations of the Pfaffian systems considered is given. These representations are deformations of the standard representations of reflection groups of the Burau type and were defined earlier by Squier and Givental’.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. Bourbaki, Lie Groups and Lie Algebras, Chapters IV–VI [Russian translation], Mir, Moscow (1972).
M. Broué, G. Malle, and R. Rouquier, “Complex reflection groups, braid groups, Hecke algebras,” J. Reine Angew. Math., 500, 127–190 (1998).
I. V. Cherednik, “Monodromy of R-systems and generalized braid groups,” Dokl. Akad. Nauk SSSR, 256, 15–18 (1989).
I. Cherednik, “Monodromy representations for generalized Knizhnik-Zamolodchikov equations,” Publ. RIMS. Kyoto Univ., 27, 711–726 (1991).
M. Freedman, A. Kitaev, M. Larsen, and Z. Wang, Topological quantum computation, e-print q-ph/0101025.
G. Giorgadze, “Monodromy approach to quantum computing,” Int. J. Mod. Phys. B, 16, No. 30, 4593–4605 (2002).
A. B. Givental’, “Twisted Picard-Lefschetz formulas,” Funkts. Anal. Pril., 22, No. 1, 12–22 (1988).
T. Kohno, “Linear representations of braid groups and classical Yang-Baxter equations,” Contemp. Math., 78, 339–363 (1988).
V. P. Leksin, “On the Riemann-Hilbert problem for analytic families of representations,” Mat. Zametki, 50, No. 2, 89–97 (1991).
V. P. Leksin, “WDVV equations and generalized Burau representations,” Proc. Int. Conf. “Mathematical Ideas of P. L. Chebyshev and Their Applications to Contemporary Problems,” Obninsk, May 14–18, 2002 [in Russian], Obninsk (2002), pp. 59–60.
V. P. Leksin, “Monodromy of rational KZ-equations and some related topics,” Acta Appl. Math., 75, Nos. 1–3, 105–115 (2003).
V. P. Leksin, “Monodromy of Veselov-Cherednik systems for real and complex configurations of vectors,” in: Proc. Int. Conf. “Differential Equations and Related Topics” Dedicated to the 103rd Anniv. of I. G. Petrovskii, Moscow, May 16–22, 2004 [in Russian], Moscow (2004), pp. 127–128.
V. P. Leksin, “Geometry vector configurations and monodromy KZ-type connections,” Proc. Int. Conf. on Geometry and Analysis Dedicated to the Memory of A. D. Aleksandrov, Novosibirsk, Sept. 9–20, 2002 [in Russian], Novosibirsk (2002), pp. 80–81.
E. M. Opdam, Lecture notes on Dunkl operators for real and complex reflection groups, MSJ Memoirs, 8 (2000).
C. C. Squier, “Matrix representations of Artin groups,” Proc. Amer. Math. Soc., 103, 49–53 (1988).
V. Toledano Laredo, “Flat connections and quantum groups,” Acta Appl. Math., 73, 155–173 (2002).
N. D. Vasilevich and N. N. Ladis, “Integrability of the Pfaffian equations on ℂP n,” Differents. Uravn., 17, No. 4, 732–733 (1979).
A. P. Veselov, “On geometry of a special class of solutions to generalized WDVV equations,” in: Integrability: The Seiberg-Witten and Whitham Equations: Edinburg, 1998, Gordon and Breach, Amsterdam (2000), pp. 125–136.
Author information
Authors and Affiliations
Additional information
__________
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 36, Suzdal Conference-2004, Part 2, 2005.
Rights and permissions
About this article
Cite this article
Leksin, V.P. Monodromy of Veselov-Kohno-Cherednik R-systems. J Math Sci 145, 5281–5294 (2007). https://doi.org/10.1007/s10958-007-0353-5
Issue Date:
DOI: https://doi.org/10.1007/s10958-007-0353-5