no. 4
Monodromy representations of braid groups and Yang-Baxter equations
Annales de l'Institut Fourier, Tome 37 (1987) no. 4, pp. 139-160.

Motivé par la théorie des champs conforme en dimension deux avec la symétrie de jauge, nous allons étudier la monodromie des connexions intégrables associées aux algébres de Lie simples. Ceci donne une série de représentations linéaires du groupe de tresses dont la forme explicite est exprimée par solutions de l’équation de Yang-Baxter quantique.

Motivated by the two dimensional conformal field theory with gauge symmetry, we shall study the monodromy of the integrable connections associated with the simple Lie algebras. This gives a series of linear representations of the braid group whose explicit form is described by solutions of the quantum Yang-Baxter equation.

@article{AIF_1987__37_4_139_0,
     author = {Kohno, Toshitake},
     title = {Monodromy representations of braid groups and {Yang-Baxter} equations},
     journal = {Annales de l'Institut Fourier},
     pages = {139--160},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {37},
     number = {4},
     year = {1987},
     doi = {10.5802/aif.1114},
     mrnumber = {89h:17030},
     zbl = {0634.58040},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1114/}
}
TY  - JOUR
AU  - Kohno, Toshitake
TI  - Monodromy representations of braid groups and Yang-Baxter equations
JO  - Annales de l'Institut Fourier
PY  - 1987
SP  - 139
EP  - 160
VL  - 37
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - http://archive.numdam.org/articles/10.5802/aif.1114/
DO  - 10.5802/aif.1114
LA  - en
ID  - AIF_1987__37_4_139_0
ER  - 
%0 Journal Article
%A Kohno, Toshitake
%T Monodromy representations of braid groups and Yang-Baxter equations
%J Annales de l'Institut Fourier
%D 1987
%P 139-160
%V 37
%N 4
%I Institut Fourier
%C Grenoble
%U http://archive.numdam.org/articles/10.5802/aif.1114/
%R 10.5802/aif.1114
%G en
%F AIF_1987__37_4_139_0
Kohno, Toshitake. Monodromy representations of braid groups and Yang-Baxter equations. Annales de l'Institut Fourier, Tome 37 (1987) no. 4, pp. 139-160. doi : 10.5802/aif.1114. http://archive.numdam.org/articles/10.5802/aif.1114/

[1] K. Aomoto, Fonctions hyperlogarithmiques et groupes de monodromie unipotents, J. Fac. Sci. Tokyo, 25 (1978), 149-156. | MR | Zbl

[2] J. Birman, Braids, links, and mapping class groups, Ann. Math. Stud., 82 (1974).

[3] A. A. Belavin and V. G. Drinfel'D, Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl., 16 (1982), 1-29. | MR | Zbl

[4] N. Bourbaki, Groupes et algèbres de Lie, IV, V, VI, Masson, Paris (1982). | Zbl

[5] A. A. Belavin, A. N. Polyakov and A. B. Zamolodchikov, Infinite dimensional symmetries in two dimensional quantum field theory, Nucl. Phys., B241 (1984), 333-380. | MR | Zbl

[6] K. T. Chen, Iterated path integrals, Bull. Amer. Math. Soc., 83 (1977), 831-879. | MR | Zbl

[7] V. G. Drinfel'D, Quantum groups, preprint, ICM Berkeley (1986). | Zbl

[8] R. Hain, On a generalization of Hilbert 21st problem, Ann. ENS, 49 (1986), 609-627. | Numdam | MR | Zbl

[9] M. Jimbo, A q-difference analogue of U(g) and Yang-Baxter equation, Lett. in Math. Phys., 10 (1985), 63-69. | MR | Zbl

[10] M. Jimbo, Quantum R matrix for the generalized Toda system, Comm. Math. Phys., 102 (1986), 537-547. | MR | Zbl

[11] M. Jimbo, A q-analogue of U(gl(N + 1)), Hecke algebra, and the Yang-Baxter equation. Lett. in Math. Phys., 11 (1986), 247-252. | MR | Zbl

[12] M. Jimbo, Quantum R matrix related to the generalized Toda system : an algebraic approach, Lect. Note in Phys., 246 (1986), Springer. | MR | Zbl

[13] V. Jones, Index of subfactors, Invent. Math., 72 (1983), 1-25. | MR | Zbl

[14] V. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math., 126 (1987), 335-388. | MR | Zbl

[15] V. G. Kac, Infinite dimensional Lie algebras, Progress in Math., 44, Birkhäuser (1983). | MR | Zbl

[16] T. Kohno, Série de Poincaré-Koszul associée aux groupes de tresses pures, Invent. Math., 82 (1985), 57-75. | MR | Zbl

[17] T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations, to appear in Contemp. Math., "Artin's braid groups". | Zbl

[18] V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and Wess-Zumino models in two dimensions, Nucl. Phys., B247 (1984), 83-103. | MR | Zbl

[19] J. Morgan, The algebraic topology of smooth algebraic varieties, Publ. IHES, 48 (1978), 103-204. | Numdam | Zbl

[20] J. Murakami, On the Jones invariant of paralleled links and linear representations of braid groups, preprint (1986).

[21] D. Sullivan, Infinitesimal computations in topology, Publ. IHES, 47 (1977), 269-331. | Numdam | MR | Zbl

[22] A. Tsuchiya and Y. Kanie, Vertex operators in two dimensional conformal field theory on P1 and monodromy representations of braid groups, preprint (1987), to appear in Adv. Stud. In Pure Math. | Zbl

[23] J. L. Verdier, Groupes quantiques, Séminaire Bourbaki, 1987 juin. | Numdam

[24] H. Wenzl, Representations of Hecke algebras and subfactors, Thesis, Univ. of Pensylvenia (1985).

Cité par Sources :