Book Review
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2995002
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Book Information:
Author:
Pavel Etingof and Frédéric Latour
Title:
The dynamical Yang–Baxter equation, representation theory, and quantum integrable systems
Additional book information:
Oxford Lecture Series in Mathematics and Its Applications, Vol. 29,
Oxford University Press, Oxford,
2005,
xii+138 pp.,
ISBN 978-0-19-853068-8,
US $115.00
D. Arnaudon, E. Buffenoir, E. Ragoucy, and Ph. Roche, Universal solutions of quantum dynamical Yang-Baxter equations, Lett. Math. Phys. 44 (1998), no. 3, 201–214. MR 1627497, DOI 10.1023/A:1007498022373
Rodney J. Baxter, Exactly solved models in statistical mechanics, Integrable systems in statistical mechanics, Ser. Adv. Statist. Mech., vol. 1, World Sci. Publishing, Singapore, 1985, pp. 5–63. MR 826537
Denis Bernard, On the Wess-Zumino-Witten models on the torus, Nuclear Phys. B 303 (1988), no. 1, 77–93. MR 952765, DOI 10.1016/0550-3213(88)90217-9
Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 1300632
Vyjayanthi Chari and Andrew Pressley, Quantum affine algebras and their representations, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 59–78. MR 1357195
V. G. Drinfel′d, Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR 283 (1985), no. 5, 1060–1064 (Russian). MR 802128
V. G. Drinfel′d, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$, Algebra i Analiz 2 (1990), no. 4, 149–181 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 4, 829–860. MR 1080203
V. G. Drinfel′d, On the structure of quasitriangular quasi-Hopf algebras, Funktsional. Anal. i Prilozhen. 26 (1992), no. 1, 78–80 (Russian); English transl., Funct. Anal. Appl. 26 (1992), no. 1, 63–65. MR 1163026, DOI 10.1007/BF01077082
P. Etingof, O. Schiffmann, and A. Varchenko, Traces of intertwiners for quantum groups and difference equations, Lett. Math. Phys. 62 (2002), no. 2, 143–158. MR 1952123, DOI 10.1023/A:1021619920915
P. Etingof and A. Varchenko, Dynamical Weyl groups and applications, Adv. Math. 167 (2002), no. 1, 74–127. MR 1901247, DOI 10.1006/aima.2001.2034
P. Etingof and A. Varchenko, Orthogonality and the QKZB-heat equation for traces of $U_q(\mathfrak {g})$-intertwiners, Duke Math. J. 128 (2005), no. 1, 83–117. MR 2137950, DOI 10.1215/S0012-7094-04-12814-3
Pavel Etingof and Olivier Schiffmann, Lectures on quantum groups, 2nd ed., Lectures in Mathematical Physics, International Press, Somerville, MA, 2002. MR 2284964
Pavel Etingof and Alexander Varchenko, Traces of intertwiners for quantum groups and difference equations. I, Duke Math. J. 104 (2000), no. 3, 391–432. MR 1781477, DOI 10.1215/S0012-7094-00-10433-4
Pavel I. Etingof and Alexander A. Kirillov Jr., Representation-theoretic proof of the inner product and symmetry identities for Macdonald’s polynomials, Compositio Math. 102 (1996), no. 2, 179–202. MR 1394525
Giovanni Felder, Conformal field theory and integrable systems associated to elliptic curves, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 1247–1255. MR 1404026
Giovanni Felder, Elliptic quantum groups, XIth International Congress of Mathematical Physics (Paris, 1994) Int. Press, Cambridge, MA, 1995, pp. 211–218. MR 1370676
Giovanni Felder and Alexander Varchenko, Integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations, Internat. Math. Res. Notices 5 (1995), 221–233. MR 1333749, DOI 10.1155/S1073792895000171
Giovanni Felder and Alexander Varchenko, Algebraic Bethe ansatz for the elliptic quantum group $E_{\tau ,\eta }(\textrm {sl}_2)$, Nuclear Phys. B 480 (1996), no. 1-2, 485–503. MR 1421711, DOI 10.1016/S0550-3213(96)00461-0
Giovanni Felder and Alexander Varchenko, $q$-deformed KZB heat equation: completeness, modular properties and $\textrm {SL}(3,\Bbb Z)$, Adv. Math. 171 (2002), no. 2, 228–275. MR 1936497, DOI 10.1006/aima.2002.2080
Edward Frenkel and Nicolai Reshetikhin, The $q$-characters of representations of quantum affine algebras and deformations of $\scr W$-algebras, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998) Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp. 163–205. MR 1745260, DOI 10.1090/conm/248/03823
I. B. Frenkel and N. Yu. Reshetikhin, Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146 (1992), no. 1, 1–60. MR 1163666
J.-L. Gervais and A. Neveu, Novel triangle relation and absence of tachyons in Liouville string field theory, Nuclear Phys. B 238 (1984), no. 1, 125–141. MR 747093, DOI 10.1016/0550-3213(84)90469-3
M. Jimbo, H. Konno, S. Odake, and J. Shiraishi, Quasi-Hopf twistors for elliptic quantum groups, Transform. Groups 4 (1999), no. 4, 303–327. MR 1726695, DOI 10.1007/BF01238562
Michio Jimbo, A $q$-difference analogue of $U({\mathfrak {g}})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63–69. MR 797001, DOI 10.1007/BF00704588
V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and Wess-Zumino model in two dimensions, Nuclear Phys. B 247 (1984), no. 1, 83–103. MR 853258, DOI 10.1016/0550-3213(84)90374-2
L. A. Tahtadžjan and L. D. Faddeev, The quantum method for the inverse problem and the $XYZ$ Heisenberg model, Uspekhi Mat. Nauk 34 (1979), no. 5(209), 13–63, 256 (Russian). MR 562799
Kenji Ueno, Conformal field theory with gauge symmetry, Fields Institute Monographs, vol. 24, American Mathematical Society, Providence, RI; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2008. MR 2433154, DOI 10.1090/fim/024
References
- D. Arnaudon, E. Buffenoir, E. Ragoucy, and Ph. Roche, Universal solutions of quantum dynamical Yang-Baxter equations, Lett. Math. Phys. 44 (1998), no. 3, 201–214. MR 1627497 (2000b:81058)
- Rodney J. Baxter, Exactly solved models in statistical mechanics, Integrable systems in statistical mechanics, Ser. Adv. Statist. Mech., vol. 1, World Sci. Publishing, Singapore, 1985, pp. 5–63. MR 826537
- Denis Bernard, On the Wess-Zumino-Witten models on the torus, Nuclear Phys. B 303 (1988), no. 1, 77–93. MR 952765 (89k:81130)
- Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 1300632 (95j:17010)
- —, Quantum affine algebras and their representations, Representations of groups (Banff, AB, 1994), CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 59–78. MR 1357195 (96j:17009)
- V. G. Drinfel$’$d, Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR 283 (1985), no. 5, 1060–1064. MR 802128 (87h:58080)
- —, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$, Algebra i Analiz 2 (1990), no. 4, 149–181. MR 1080203 (92f:16047)
- —, On the structure of quasitriangular quasi-Hopf algebras, Funktsional. Anal. i Prilozhen. 26 (1992), no. 1, 78–80. MR 1163026 (93i:16052)
- P. Etingof, O. Schiffmann, and A. Varchenko, Traces of intertwiners for quantum groups and difference equations, Lett. Math. Phys. 62 (2002), no. 2, 143–158. MR 1952123 (2003k:17018)
- P. Etingof and A. Varchenko, Dynamical Weyl groups and applications, Adv. Math. 167 (2002), no. 1, 74–127. MR 1901247 (2003d:17004)
- —, Orthogonality and the QKZB-heat equation for traces of $U_q(\mathfrak {g})$-intertwiners, Duke Math. J. 128 (2005), no. 1, 83–117. MR 2137950 (2006a:17012)
- Pavel Etingof and Olivier Schiffmann, Lectures on quantum groups, Second Ed., Lectures in Mathematical Physics, International Press, Somerville, MA, 2002. MR 2284964 (2007h:17017)
- Pavel Etingof and Alexander Varchenko, Traces of intertwiners for quantum groups and difference equations. I, Duke Math. J. 104 (2000), no. 3, 391–432. MR 1781477 (2001k:17021)
- Pavel I. Etingof and Alexander A. Kirillov, Jr., Representation-theoretic proof of the inner product and symmetry identities for Macdonald’s polynomials, Compositio Math. 102 (1996), no. 2, 179–202. MR 1394525 (97j:17013)
- Giovanni Felder, Conformal field theory and integrable systems associated to elliptic curves, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, pp. 1247–1255. MR 1404026 (97m:81015)
- —, Elliptic quantum groups, XIth International Congress of Mathematical Physics (Paris, 1994), Int. Press, Cambridge, MA, 1995, pp. 211–218. MR 1370676 (96j:17012)
- Giovanni Felder and Alexander Varchenko, Integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations, Internat. Math. Res. Notices (1995), no. 5, 221–233. MR 1333749 (96g:32040)
- —, Algebraic Bethe ansatz for the elliptic quantum group $E_ {\tau ,\eta }(\textrm {sl}_ 2)$, Nuclear Phys. B 480 (1996), no. 1-2, 485–503. MR 1421711 (98b:82024)
- —, $q$-deformed KZB heat equation: completeness, modular properties and $\textrm {SL}(3,\mathbb Z)$, Adv. Math. 171 (2002), no. 2, 228–275. MR 1936497 (2003k:32026)
- Edward Frenkel and Nicolai Reshetikhin, The $q$-characters of representations of quantum affine algebras and deformations of $\scr W$-algebras, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp. 163–205. MR 1745260 (2002f:17022)
- I. B. Frenkel and N. Yu. Reshetikhin, Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146 (1992), no. 1, 1–60. MR 1163666 (94c:17024)
- J.-L. Gervais and A. Neveu, Novel triangle relation and absence of tachyons in Liouville string field theory, Nuclear Phys. B 238 (1984), no. 1, 125–141. MR 747093 (86f:81088)
- M. Jimbo, H. Konno, S. Odake, and J. Shiraishi, Quasi-Hopf twistors for elliptic quantum groups, Transform. Groups 4 (1999), no. 4, 303–327. MR 1726695 (2002c:17026)
- Michio Jimbo, A $q$-difference analogue of $U({\mathfrak {g}})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63–69. MR 797001 (86k:17008)
- V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and Wess-Zumino model in two dimensions, Nuclear Phys. B 247 (1984), no. 1, 83–103. MR 853258 (87h:81129)
- L. A. Tahtadžjan and L. D. Faddeev, The quantum method for the inverse problem and the $XYZ$ Heisenberg model, Uspekhi Mat. Nauk 34 (1979), no. 5(209), 13–63, 256. MR 562799 (81d:82066)
- Kenji Ueno, Conformal field theory with gauge symmetry, Fields Institute Monographs, vol. 24, American Mathematical Society, Providence, RI, 2008. MR 2433154 (2009k:81204)
Review Information:
Reviewer:
Giovanni Felder
Affiliation:
Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
Email:
felder@math.ethz.ch
Journal:
Bull. Amer. Math. Soc.
50 (2013), 181-186
DOI:
https://doi.org/10.1090/S0273-0979-2012-01367-8
Published electronically:
January 18, 2012
Additional Notes:
The reviewer was supported in part by the Swiss National Science Foundation
Review copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
