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Book Review
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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,
Oxford University Press, Oxford,
Vol. 29, 2005, xii+138 pp., US $115.00, ISBN 978-0-19-853068-8.
- 1.
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), http://dx.doi.org/10.1023/A:1007498022373
- 2.
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
- 3.
Denis
Bernard, On the Wess-Zumino-Witten models on the torus,
Nuclear Phys. B 303 (1988), no. 1, 77–93. MR 952765
(89k:81130), http://dx.doi.org/10.1016/0550-3213(88)90217-9
- 4.
Vyjayanthi
Chari and Andrew
Pressley, A guide to quantum groups, Cambridge University
Press, Cambridge, 1994. MR 1300632
(95j:17010)
- 5.
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 (96j:17009)
- 6.
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
(87h:58080)
- 7.
V.
G. Drinfel′d, On quasitriangular quasi-Hopf algebras and on a
group that is closely connected with
𝐺𝑎𝑙(\𝑜𝑣𝑒𝑟𝑙𝑖𝑛𝑒{𝑄}/𝑄),
Algebra i Analiz 2 (1990), no. 4, 149–181
(Russian); English transl., Leningrad Math. J. 2 (1991),
no. 4, 829–860. MR 1080203
(92f:16047)
- 8.
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
(93i:16052), http://dx.doi.org/10.1007/BF01077082
- 9.
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), http://dx.doi.org/10.1023/A:1021619920915
- 10.
P.
Etingof and A.
Varchenko, Dynamical Weyl groups and applications, Adv. Math.
167 (2002), no. 1, 74–127. MR 1901247
(2003d:17004), http://dx.doi.org/10.1006/aima.2001.2034
- 11.
P.
Etingof and A.
Varchenko, Orthogonality and the QKZB-heat equation for traces of
𝑈_{𝑞}(𝔤)-intertwiners, Duke Math. J.
128 (2005), no. 1, 83–117. MR 2137950
(2006a:17012), http://dx.doi.org/10.1215/S0012-7094-04-12814-3
- 12.
Pavel
Etingof and Olivier
Schiffmann, Lectures on quantum groups, 2nd ed., Lectures in
Mathematical Physics, International Press, Somerville, MA, 2002. MR 2284964
(2007h:17017)
- 13.
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), http://dx.doi.org/10.1215/S0012-7094-00-10433-4
- 14.
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)
- 15.
Giovanni
Felder, Conformal field theory and integrable systems associated to
elliptic curves, 2 (Zürich, 1994) Birkhäuser, Basel,
1995, pp. 1247–1255. MR 1404026
(97m:81015)
- 16.
Giovanni
Felder, Elliptic quantum groups, XIth International Congress
of Mathematical Physics (Paris, 1994) Int. Press, Cambridge, MA, 1995,
pp. 211–218. MR 1370676
(96j:17012)
- 17.
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
(96g:32040), http://dx.doi.org/10.1155/S1073792895000171
- 18.
Giovanni
Felder and Alexander
Varchenko, Algebraic Bethe ansatz for the elliptic quantum group
𝐸_{𝜏,𝜂}(𝑠𝑙₂), Nuclear
Phys. B 480 (1996), no. 1-2, 485–503. MR 1421711
(98b:82024), http://dx.doi.org/10.1016/S0550-3213(96)00461-0
- 19.
Giovanni
Felder and Alexander
Varchenko, 𝑞-deformed KZB heat equation: completeness,
modular properties and 𝑆𝐿(3,ℤ), Adv. Math.
171 (2002), no. 2, 228–275. MR 1936497
(2003k:32026), http://dx.doi.org/10.1006/aima.2002.2080
- 20.
Edward
Frenkel and Nicolai
Reshetikhin, The 𝑞-characters of representations of quantum
affine algebras and deformations of 𝒲-algebras, (Raleigh, NC,
1998) Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI,
1999, pp. 163–205. MR 1745260
(2002f:17022), http://dx.doi.org/10.1090/conm/248/03823
- 21.
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)
- 22.
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), http://dx.doi.org/10.1016/0550-3213(84)90469-3
- 23.
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), http://dx.doi.org/10.1007/BF01238562
- 24.
Michio
Jimbo, A 𝑞-difference analogue of 𝑈(𝔤) and
the Yang-Baxter equation, Lett. Math. Phys. 10
(1985), no. 1, 63–69. MR 797001
(86k:17008), http://dx.doi.org/10.1007/BF00704588
- 25.
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), http://dx.doi.org/10.1016/0550-3213(84)90374-2
- 26.
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
(81d:82066)
- 27.
Kenji
Ueno, Conformal field theory with gauge symmetry, Fields
Institute Monographs, vol. 24, American Mathematical Society,
Providence, RI, 2008. MR 2433154
(2009k:81204)
- 1.
- 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)
- 2.
- 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
- 3.
- Denis Bernard, On the Wess-Zumino-Witten models on the torus, Nuclear Phys. B 303 (1988), no. 1, 77-93. MR 952765 (89k:81130)
- 4.
- Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 1300632 (95j:17010)
- 5.
- -, 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)
- 6.
- 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)
- 7.
- -, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with
, Algebra i Analiz 2 (1990), no. 4, 149-181. MR 1080203 (92f:16047)
- 8.
- -, On the structure of quasitriangular quasi-Hopf algebras, Funktsional. Anal. i Prilozhen. 26 (1992), no. 1, 78-80. MR 1163026 (93i:16052)
- 9.
- 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)
- 10.
- P. Etingof and A. Varchenko, Dynamical Weyl groups and applications, Adv. Math. 167 (2002), no. 1, 74-127. MR 1901247 (2003d:17004)
- 11.
- -, Orthogonality and the QKZB-heat equation for traces of
-intertwiners, Duke Math. J. 128 (2005), no. 1, 83-117. MR 2137950 (2006a:17012)
- 12.
- Pavel Etingof and Olivier Schiffmann, Lectures on quantum groups, Second Ed., Lectures in Mathematical Physics, International Press, Somerville, MA, 2002. MR 2284964 (2007h:17017)
- 13.
- 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)
- 14.
- 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)
- 15.
- 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)
- 16.
- -, Elliptic quantum groups, XIth International Congress of Mathematical Physics (Paris, 1994), Int. Press, Cambridge, MA, 1995, pp. 211-218. MR 1370676 (96j:17012)
- 17.
- 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)
- 18.
- -, Algebraic Bethe ansatz for the elliptic quantum group
, Nuclear Phys. B 480 (1996), no. 1-2, 485-503. MR 1421711 (98b:82024)
- 19.
- -,
-deformed KZB heat equation: completeness, modular properties and , Adv. Math. 171 (2002), no. 2, 228-275. MR 1936497 (2003k:32026)
- 20.
- Edward Frenkel and Nicolai Reshetikhin, The
-characters of representations of quantum affine algebras and deformations of -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)
- 21.
- 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)
- 22.
- 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)
- 23.
- 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)
- 24.
- Michio Jimbo, A
-difference analogue of and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63-69. MR 797001 (86k:17008)
- 25.
- 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)
- 26.
- L. A. Tahtadžjan and L. D. Faddeev, The quantum method for the inverse problem and the
Heisenberg model, Uspekhi Mat. Nauk 34 (1979), no. 5(209), 13-63, 256. MR 562799 (81d:82066)
- 27.
- 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:
http://dx.doi.org/10.1090/S0273-0979-2012-01367-8
PII:
S 0273-0979(2012)01367-8
Posted:
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.
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