Geodesic
On $q$-monodromy groups of singularities
Izvestiya. Mathematics , Tome 60 (1996) no. 1, pp. 119-136.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper some facts are proved concerning the $q$-analogue of symmetric and skew-symmetric monodromy groups of singularities. The image of the Burau representation of the group of braids on three strings is described. For $q$ a root of 1 the image of the group of $q$-coloured braids of a singularity in the skew-symmetric monodromy group is studied. The latter is distinguished among the $q$-monodromy groups for the roots of 1. The orbits of the braid monodromy group over $\mathbb Z$ are described. A connection is established between the spectra of the generalized Cartan $q$-matrix and the Coxeter $q$-operator for Coxeter–Dynkin diagrams without cycles of odd length. 
@article{IM2_1996_60_1_a4,
     author = {G. G. Ilyuta},
     title = {On $q$-monodromy groups of singularities},
     journal = {Izvestiya. Mathematics },
     pages = {119--136},
     publisher = {mathdoc},
     volume = {60},
     number = {1},
     year = {1996},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1996_60_1_a4/}
}
TY  - JOUR
AU  - G. G. Ilyuta
TI  - On $q$-monodromy groups of singularities
JO  - Izvestiya. Mathematics 
PY  - 1996
SP  - 119
EP  - 136
VL  - 60
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1996_60_1_a4/
LA  - en
ID  - IM2_1996_60_1_a4
ER  - 
%0 Journal Article
%A G. G. Ilyuta
%T On $q$-monodromy groups of singularities
%J Izvestiya. Mathematics 
%D 1996
%P 119-136
%V 60
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1996_60_1_a4/
%G en
%F IM2_1996_60_1_a4
G. G. Ilyuta. On $q$-monodromy groups of singularities. Izvestiya. Mathematics , Tome 60 (1996) no. 1, pp. 119-136. http://geodesic.mathdoc.fr/item/IM2_1996_60_1_a4/

[1] Arnold V. I., Varchenko A. N., Gusein-Zade S. M., Osobennosti differentsiruemykh otobrazhenii, T. II, Nauka, M., 1984 | MR

[2] Burbaki N., Algebra. Moduli, koltsa, formy, Nauka, M., 1965 | MR

[3] Burbaki N., Gruppy i algebra Li. Gruppy Kokstera i sistemy Titsa, Mir, M., 1972 | Zbl

[4] Varchenko A. N., “Asimptotiki integralov i struktury Khodzha”, Itogi nauki. Sovremennye problemy matematiki, 22, VINITI, M., 1983, 130–166 | MR | Zbl

[5] Ganning R. K., “Lektsii o modulyarnykh formakh”, Matematika, 8:6 (1964), 3–68

[6] Givental A. B., “Skruchennye formuly Pikara–Lefshetsa”, Funktsion. analiz i ego prilozh., 22:4 (1988), 12–22 | MR | Zbl

[7] Dolgachev I. V., “Avtomorfnye formy i kvaziodnorodnye osobennosti”, Funktsion. analiz i ego prilozh., 9:2 (1975), 67–68 | MR | Zbl

[8] Ilyuta G. G., “Otnosheniya mezhdu gruppami monodromii osobennosti”, Funktsion. analiz i ego prilozh., 26:1 (1992), 9–16 | MR | Zbl

[9] Ilyuta G. G., “Gruppy monodromii kraevykh osobennostei”, Matem. sb., 183:5 (1993), 29–42 | MR

[10] Ilyuta G. G., “Teorema A'Kampo na diskriminante”, Funktsion. analiz i ego prilozh., 28:2 (1994), 12–20 | MR | Zbl

[11] Savelev I. V., “Krugovye mnogochleny i osobennosti kompleksnykh giperpoverkhnostei”, UMN, 48:2 (1993), 197–198 | MR | Zbl

[12] Serr Zh.-P., “Problema kongruentspodgrupp dlya $\operatorname {SL}_2$”, Matematika, 15:6 (1971), 12–45 | MR | Zbl

[13] Feit V., “Isklyuchitelnye podgruppy v $\operatorname {GL}_2$”, Prilozhenie k knige: Leng S, Vvedenie v teoriyu modulyarnykh form, Mir, M., 1979

[14] Berman S., Lee Y. S., Moody R. V., “The Spectrum of a Coxeter transformation, affine Coxeter transformations and the defect map”, J. of Algebra, 121 (1989), 339–357 | DOI | MR | Zbl

[15] Birman J., “Braids, links and mapping class groups”, Ann. of Math. Stud., 82, Princeton Univ. Press, Princeton–N. J., 1974 | MR

[16] Brenner J. L., “The linear homogeneous group, III”, Ann. Math., 71 (1960), 210–223 | DOI | MR | Zbl

[17] Brieskorn E., “The Unfolding of exeptional singularities”, Nova acta Leopoldina (NF), 52:240 (1981), 65–93 | MR | Zbl

[18] Chmutov S. V., “Monodromy groups of critical point of functions”, Invent. Math., 67 (1982), 123–131 | DOI | MR | Zbl

[19] Chmutov S. V., “Monodromy groups of critical point of functions, II”, Invent. Math., 73 (1983), 491–510 | DOI | MR | Zbl

[20] Ebeling W., “Vanishing lattices and monodromy groups of isolated complete intersection singularities”, Invent. Math., 90 (1987), 653–668 | DOI | MR | Zbl

[21] Janssen W. A. M., “Skew-summetric vanishing lattices and their monodromy groups, II”, Math. Ann., 272 (1985), 17–22 | DOI | MR | Zbl

[22] Janssen W. A. M., Skew-symmetric vanishing lattices and their monodromy groups, Thesis, Univ. of Utrecht, Utrecht, 1984 | Zbl

[23] Klein F., Fricke R., Vozlesungen über die Theorie der elliptische Modulfunctionen, V. 1, Teubner, Leipzig, 1890 | Zbl

[24] Knopp M. I., “A note on subgroups of the modular group”, Proc. Amer. Math. Soc., 14 (1963), 95–97 | DOI | MR | Zbl

[25] Lindon R., “Two notes on Rankin's book on the modular group”, J. Austr. Math. Soc., 16 (1973), 454–457 | DOI | MR

[26] Lusztig G., “Quantum groups at root of $1$”, Geom. Dedicata, 35 (1990), 89–114 | DOI | MR | Zbl

[27] Magnus W., Peluzo A., “On a theorem of V. I. Arnol'd”, Comm. Pure Appl. Math., 22 (1969), 683–692 | DOI | MR | Zbl

[28] Mennicke J., “Finite factor groups of the unimodular group”, Ann. Math., 81 (1965), 31–37 | DOI | MR | Zbl

[29] Newman M., “A complete description of the normal subgroups of genus one of the modular group”, Amer. J. Math., 86 (1964), 17–24 | DOI | MR | Zbl

[30] Rankin R. A., The modular group and its subgroups, The Ramanujan Institute, University of Madras, 1969 | MR

[31] Reiner I., “Subgroups of the unimodular group”, Proc. Amer. Math. Soc., 12 (1961), 173–174 | DOI | MR | Zbl

[32] Wajnrib B., “On the monodromy of plane curve singularities”, Ann. Math., 246 (1980), 141–154 | DOI

[33] Wenzl H., “Hecke algebras of tupe $A$ and subfactors”, Invent. Math., 92 (1988), 349–383 | DOI | MR | Zbl

[34] Wohlfahrt K., “An extension of F. Klein's level concept”, Illinois J. Math., 8 (1964), 529–535 | MR | Zbl