Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups

Geometry of

q

-hypergeometric functions, quantum affine algebras and elliptic quantum groups

Tarasov, V. ; Varchenko, A.

Astérisque, no. 246 (1997) , 141 p.

MR 1646561 | Zbl 0938.17012 | 1 citation dans Numdam

@book{AST_1997__246__R1_0,
     author = {Tarasov, V. and Varchenko, A.},
     title = {Geometry of <span class="mathjax-formula">$q$</span>-hypergeometric functions, quantum affine algebras and elliptic quantum groups},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {246},
     year = {1997},
     zbl = {0938.17012},
     mrnumber = {1646561},
     language = {en},
     url = {http://www.numdam.org/item/AST_1997__246__R1_0/}
}

Tarasov, V.; Varchenko, A. Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups. Astérisque, no. 246 (1997), 141 p. http://numdam.org/item/AST_1997__246__R1_0/

Bibliographie

[A] K. Aomoto, $q$ -analogue of de Rham cohomology associated with Jackson integrals, I, Proceedings of Japan Acad. 66 Ser. A (1990), 161-164. | Article | MR 1078398 | Zbl 0718.33011

[A] K. Aomoto, $q$ -analogue of de Rham cohomology associated with Jackson integrals, II, Proceedings of Japan Acad. 66 Ser. A (1990), 240-244. | Article | MR 1077605 | Zbl 0718.33012

[AK] K. Aomoto and Y. Kato, Gauss decomposition of connection matrices for symmetric $A$ -type Jackson integrals, Selecta Math., New Series 1 (1995), 623-666. | Article | MR 1383582 | Zbl 0854.33015

[As] R. Askey, Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. Anal. 11 (1980), 938-951. | Article | MR 595822 | Zbl 0458.33002

[CP] V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press, 1994. | MR 1300632 | Zbl 0839.17009

[D1] V. G. Drinfeld, Quantum groups, in Proceedings of the ICM, Berkeley, 1986 (A.M. Gleason, ed.), AMS, Providence, 1987, pp. 798-820. | MR 934283

[D2] V. G. Drinfeld, Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419-1457. | MR 1047964 | Zbl 0718.16033

[DF] J. Ding and I. B. Frenkel, Isomorphism of two realizations of quantum affine algebra $U_{q} (\hat{𝔤𝔩} (n))$ , Comm. Math. Phys. 156 (1993), 277-300. | Article | MR 1233847 | Zbl 0786.17008

[E] R. J. Evans, Multidimensional beta and gamma integrals, Contemp. Math. 166 (1994), 341-357. | Article | MR 1284073 | Zbl 0820.33001

[F] G. Felder, Conformal field theory and integrable systems associated to elliptic curves, in Proceedings of the ICM, Zürich, 1994, Birkhäuser, 1994, pp. 1247-1255. | MR 1404026 | Zbl 0852.17014

G. Felder, Elliptic quantum groups, in Proceedings of the ICMP, Paris 1994 (D. Iagolnitzer, ed.), Intern. Press, Boston, 1995, pp. 211-218. | Zbl 0998.17015

[FR] I. B. Frenkel and N. Yu. Reshetikhin, Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146 (1992), 1-60. | Article | MR 1163666 | Zbl 0760.17006

[FTV1] G. Felder, V. Tarasov and A. Varchenko, Solutions of elliptic qKZB equations and Bethe ansatz I, Amer. Math. Society Transl., Ser. 2 180 (1997), 45-75. | MR 1767112 | Zbl 0884.65127

[FTV2] G. Felder, V. Tarasov and A. Varchenko, Monodromy of solutions of the elliptic quantum Knizhnik-Zamolodchikov-Bernard difference equations, Preprint (1997), 1-20. | MR 1739369 | Zbl 1038.32500

[FV] G. Felder and A. Varchenko, On representations of the elliptic quantum group $E_{τ, η} (s l_{2})$ , Comm. Math. Phys. 181 (1996), 741-761. | Article | MR 1414309 | Zbl 0870.17012

[GR] G. Gasper and M. Rahman, Basic hypergeometric series, Encycl. Math. Appl., Cambridge University Press, 1990. | MR 1052153 | Zbl 0695.33001

[IK] A. Izergin and V. E. Korepin, The quantum scattering method approach to correlation functions, Comm. Math. Phys. 94 (1984), 67-92. | Article | MR 763763 | Zbl 1093.82504

[J] M. Jimbo, Quantum R-matrix for the generalized Toda system, Comm. Math. Phys. 102 (1986), 537-547. | Article | MR 824090 | Zbl 0604.58013

[JM] M. Jimbo and T. Miwa, Algebraic analysis of solvable lattice models, CBMS Regional Conf. Series in Math. 85 (1995). | MR 1308712 | Zbl 0828.17018

T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Contemp. Math. 78 (1988), 339-363. | Article | MR 975088 | Zbl 0661.20026

[KL] D. Kazhdan and G. Lusztig, Affine Lie algebras and quantum groups, Intern. Math. Research Notices 2 (1991), 21-29. | Article | MR 1104840 | Zbl 0726.17015

D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras, I, J. Amer. Math. Society 6 (1993), 905-947. | Article | MR 1186962 | Zbl 0786.17017

D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras, II, J. Amer. Math. Society 6 (1993), 949-1011. | Article | MR 1186962 | Zbl 0786.17017

[KS] D. Kazhdan and Ya. S. Soibelman, Representation theory of quantum affine algebras, Selecta Math., New Series 3 (1995), 537-595. | Article | MR 1366624 | Zbl 0842.17021

[Lu] S. Lukyanov, Free field representation for massive integrable models, Comm. Math. Phys. 167 (1995), 183-226. | Article | MR 1316504 | Zbl 0818.46079

[RS] N. Yu. Reshetikhin and M. A. Semenov-Tian-Shansky, Central extensions of the quantum current groups, Lett. Math. Phys. 19 (1990), 133-142. | Article | MR 1039522 | Zbl 0692.22011

[S] F. A. Smirnov, Form factors in completely integrable models of quantum field theory, Advanced Series in Math. Phys., vol.14, World Scientific, Singapore, 1992. | Article | MR 1253319 | Zbl 0788.46077

[SV1] V. V. Schechtman and A. N. Varchenko, Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990), 279-283. | Article | MR 1077959 | Zbl 0719.35079

[SV2] V. V. Schechtman and A. N. Varchenko, Arrangements of hyperplanes and Lie algebras homology, Invent. Math. 106 (1991), 139-194. | Article | EuDML 143938 | MR 1123378 | Zbl 0754.17024

V. V. Schechtman and A. N. Varchenko, Quantum groups and homology of local systems, in Algebraic Geometry and Analytic Geometry, Proceedings of the ICM Satellite Conference, Tokyo, 1990, Springer-Verlag, Berlin, 1991, pp. 182-191. | MR 1260946 | Zbl 0760.17014

[T] V. O. Tarasov, Irreducible monodromy matrices for the $R$ -matrix of the $X X Z$ -model and lattice local quantum Hamiltonians, Theor. Math. Phys. 63 (1985), 440-454. | Article | MR 800062

[TV1] V. O. Tarasov and A. N. Varchenko, Jackson integral representations of solutions of the quantized Knizhnik-Zamolodchikov equation, St. Petersburg Math. J. 6 (1995), 275-313. | Zbl 0824.33012

[TV2] V. Tarasov and A. Varchenko, Asymptotic solution to the quantized Knizhnik-Zamolodchikov equation and Bethe vectors, Amer. Math. Society Transl., Ser. 2 174 (1996), 235-273. | MR 1386664 | Zbl 0883.17021

[TV3] V. Tarasov and A. Varchenko, Geometry of $q$ -hypergeometric functions as a bridge between Yangians and quantum affine algebras, Invent. Math. 128 (1997), 501-588. | Article | MR 1452432 | Zbl 0877.33013

[V1] A. N. Varchenko, The Euler beta-function, the Vandermonde determinant, Legendre's equation, and critical values of linear functions on a configuration of hyperplanes, I, Math. USSR, Izvestia 35 (1990), 543-571. | Article | MR 1039962 | Zbl 0709.33001

A. N. Varchenko, The Euler beta-function, the Vandermonde determinant, Legendre's equation, and critical values of linear functions on a configuration of hyperplanes, II, Math. USSR Izvestia 36 (1991), 155-168. | Article | MR 1044052 | Zbl 0709.33002

A. N. Varchenko, Determinant formula for Selberg type integrals, Funct. Anal. Appl. 4 (1991), 65-66. | MR 1167731 | Zbl 0778.33008

[V2] A. Varchenko, Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, Advanced Series in Math. Phys., vol. 21, World Scientific, Singapore, 1995. | MR 1384760 | Zbl 0951.33001

[V3] A. Varchenko, Quantized Knizhnik-Zamolodchikov equations, quantum Yang-Baxter equation, and difference equations for $q$ -hypergeometric functions, Comm. Math. Phys. 162 (1994), 499-528. | Article | MR 1277474 | Zbl 0807.17014

[V4] A. Varchenko, Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal base, Comm. Math. Phys. 171 (1995), 99-137. | Article | MR 1341696 | Zbl 0893.17009