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Arrangements of hyperplanes and Lie algebra homology. (English) Zbl 0754.17024
The authors investigate the cohomology of one-dimensional local systems over complements of hyperplanes in complex affine subspaces. The main results of the paper consist of the study of “discriminantal” arrangements. The cohomology of certain local systems over them is closely connected with homology of nilpotent subalgebras of Kac-Moody type Lie bialgebras. Under certain conditions, the complete set of solutions of the Knizhnik-Zamolodchikov differential equations in terms of generalized hypergeometric integrals are given.

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B55 Homological methods in Lie (super)algebras
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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[1] [A] Aomoto, K.: Un théorème du typeM?M concernant l’integrale des fonctions multiformes. J. Math. Pures Appl.52, 1-11 (1973) · Zbl 0276.32003 · doi:10.1016/S0079-8169(08)62455-1
[2] [BB] Beilinson, A., Bernstein, J.: Localisation ofg-modules II. The Jantzen conjectures. Moscow (Preprint 1989)
[3] [Bj] Bjorner, A.: On the homology of geometric lattices. Algebra Univers.14, No 1, 107-128 (1982) · Zbl 0484.06014 · doi:10.1007/BF02483913
[4] [Br] Brieskorn, E.: Sur les group des tresses (d’après V.I. Arnold). Séminaire Bourbaki 24e année 1971/72. (Lect. Notes Math., vol. 317) Berlin Heidelberg New York: Springer 1973
[5] [C] Cartier, P.: Les arrangements d’hyperplanes: un chapitre de géométrie combinatoire. Séminaire Bourbaki 33e année 1980/81. (Lect. Notes Math., vol. 901) Berlin Heidelberg New York: Springer 1981
[6] [Ch] Cherednik, I.: Integral solutions of trigonometric Knizhnik-Zamolodchikov equations and Kac-Moody algebras. Moscow (Preprint 1990)
[7] [CF] Christe, P., Flume, R.: The four-point correlations of all primary operators of thed=2 conformally invariant SU(2)o-model with Wess-Zumino term. Nucl. Phys. B282, 466-494 (1987) · doi:10.1016/0550-3213(87)90693-6
[8] [DJMM] Date, E., Jimbo, M., Matsuo, A., Miwa, T.: Hypergeometric-type integrals and the sl(2,C) Knizhnik-Zamolodchikov equation. RIMS-667 (Preprint 1989) · Zbl 0722.33007
[9] [DF] Dotsenko, Vl.S., Fateev, V.A.: Conformal algebra and multipoint correlation functions in 2D statistical models. Nucl. Phys. N240, 312-348 (1984) · doi:10.1016/0550-3213(84)90269-4
[10] [D] Drinfeld, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematicians, Berkeley 1986, vol. 1, pp. 798-820. Providence, R.I.: Am. Math. Soc. 1987
[11] [GZ] Gelfand, I.M., Zelevinsky, A.V.: Algebraic and combinatorial aspects of the general theory of hypergeometric functions (in Russian). Funct. Anal. Appl.20, No. 3, 17-34 (1986) · Zbl 0629.58017 · doi:10.1007/BF01077310
[12] [J] Jantzen, J.C.: Moduln mit einem höchsten Gewicht. (Lect. Notes Math., vol. 75) Berlin Heidelberg New York: Springer 1980 · Zbl 0426.17001
[13] [K] Kac, V.G.: Infinite dimensional Lie algebras. Cambridge: Cambridge University Press 1985 · Zbl 0574.17010
[14] [KZ] Knizhnik, V.G., Zamolodchikov, A.B.: Current algebra and Wess-Zumino model in two dimensions. Nucl. Phys. B247, 83-103 (1984) · Zbl 0661.17020 · doi:10.1016/0550-3213(84)90374-2
[15] [Ko] Kohno, T.: Quantized universal enveloping algebras and monodromy of braid groups. (Preprint 1988)
[16] [Kos] Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. Math.74, No. 2, 329-387 (1961) · Zbl 0134.03501 · doi:10.2307/1970237
[17] [L] Lawrence, R.J.: Homology representations of braid groups. Dissertation. Oxford: 1989
[18] [M] Matsuo, A.: An application of Aomoto-Gelfand hypergeometric functions to the SU(n) Knizhnik-Zamolodchikov equation. RIMS-683 (Preprint 1990) · Zbl 0714.33012
[19] [N] Novikov, S.P.: Bloch homology. Critical points of functions and 1-forms (in Russian). Dokl. Akad. Nauk SSSR287, No. 6, 1321-1424 (1986)
[20] [OS] Orlik, P., Solomon, L.: Combinatorics and topology of complements of hyperplanes. Invent. Math.56, 167-189 (1980) · Zbl 0432.14016 · doi:10.1007/BF01392549
[21] [SV1] Schechtman, V.V., Varchenko, A.N.: Integral representations ofn-point conformal correlators in the WZW model. MPI/89-51, Bonn (Preprint 1989)
[22] [SV2] Schechtman, V.V., Varchenko, A.N.: Hypergeometric solutions of Knizhnik-Zamolodchikov equations. Lett. Math. Phys.20, 279-283 (1990) · Zbl 0719.35079 · doi:10.1007/BF00626523
[23] [SV3] Schechtman, V.V., Varchenko, A.N.: Quantum groups and homology of local systems. IAS (Preprint 1990)
[24] [Sh] Shapovalov, N.N.: On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra (in Russian). Funct. Anal. Appl.6, No. 4, 65-70 (1972) · Zbl 0262.47009 · doi:10.1007/BF01075514
[25] [V] Varchenko, A.N.: Euler Beta-function, Vander Monde determinant. Legendre equation and critical values of linear functions on an arrangement of hyperplanes. I. (in Russian). Izv. Akad. Nauk SSSR, Ser. Mat.53, No. 6, 1206-1235 (1989) · Zbl 0695.33004
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