Book Review
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MathSciNet review:
2651087
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Book Information:
Author:
Alexander Molev
Title:
Yangians and classical Lie algebras
Additional book information:
Mathematical Surveys and Monographs, 143,
American Mathematical Society, Providence, RI,
2007,
xviii+400 pp.,
ISBN 13: 978-0-8218-4374-1,
US$99 hardcover
Tomoyuki Arakawa, Drinfeld functor and finite-dimensional representations of Yangian, Comm. Math. Phys. 205 (1999), no. 1, 1–18. MR 1706920, DOI 10.1007/s002200050664
Jonathan Brown, Twisted Yangians and finite $W$-algebras, Transform. Groups 14 (2009), no. 1, 87–114. MR 2480853, DOI 10.1007/s00031-008-9041-x
Jonathan Brundan and Alexander Kleshchev, Shifted Yangians and finite $W$-algebras, Adv. Math. 200 (2006), no. 1, 136–195. MR 2199632, DOI 10.1016/j.aim.2004.11.004
I. V. Cherednik, A new interpretation of Gel′fand-Tzetlin bases, Duke Math. J. 54 (1987), no. 2, 563–577. MR 899405, DOI 10.1215/S0012-7094-87-05423-8
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
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, Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 69–70 (Russian). MR 831053
V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
V. G. Drinfel′d, A new realization of Yangians and of quantum affine algebras, Dokl. Akad. Nauk SSSR 296 (1987), no. 1, 13–17 (Russian); English transl., Soviet Math. Dokl. 36 (1988), no. 2, 212–216. MR 914215
I. M. Gel′fand and M. L. Cetlin, Finite-dimensional representations of the group of unimodular matrices, Doklady Akad. Nauk SSSR (N.S.) 71 (1950), 825–828 (Russian). MR 0035774
A. I. Molev, Finite-dimensional irreducible representations of twisted Yangians, J. Math. Phys. 39 (1998), no. 10, 5559–5600. MR 1642338, DOI 10.1063/1.532551
A. Molev, M. Nazarov, and G. Ol′shanskiĭ, Yangians and classical Lie algebras, Uspekhi Mat. Nauk 51 (1996), no. 2(308), 27–104 (Russian); English transl., Russian Math. Surveys 51 (1996), no. 2, 205–282. MR 1401535, DOI 10.1070/RM1996v051n02ABEH002772
Maxim Nazarov and Vitaly Tarasov, Representations of Yangians with Gelfand-Zetlin bases, J. Reine Angew. Math. 496 (1998), 181–212. MR 1605817, DOI 10.1515/crll.1998.029
G. I. Ol′shanskiĭ, Twisted Yangians and infinite-dimensional classical Lie algebras, Quantum groups (Leningrad, 1990) Lecture Notes in Math., vol. 1510, Springer, Berlin, 1992, pp. 104–119. MR 1183482, DOI 10.1007/BFb0101183
N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev, Quantization of Lie groups and Lie algebras, Algebra i Analiz 1 (1989), no. 1, 178–206 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 1, 193–225. MR 1015339
Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158
References
- T. Arakawa, Drinfeld functor and finite-dimensional representations of Yangians, Comm. Math. Phys. 205 (1999), 1–18. MR 1706920 (2001c:17011)
- J. Brown, Twisted Yangians and finite $W$-algebras, Transform. Groups 14 (2009), 87–114. MR 2480853
- J. Brundan and A. Kleshchev, Shifted Yangians and finite $W$-algebras, Adv. Math. 200 (2006), 136–195. MR 2199632 (2006m:17010)
- I. Cherednik, A new interpretation of Gelfand-Zetlin bases, Duke Math. J. 54 (1987), 563–577. MR 899405 (88k:17005)
- J. Dixmier, Algébres Enveloppantes, Gauthier-Villars, Paris, 1974. MR 0498737 (58:16803a)
- V. G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 254–258. MR 802128 (87h:58080)
- V. G. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl. 20 (1986), 56–58. MR 831053 (87m:22044)
- V. G. Drinfeld, Quantum groups, in: “International Congress of Mathematicians (Berkeley, 1986)”, AMS, Providence, RI, 1987, pp. 798–820. MR 934283 (89f:17017)
- V. G. Drinfeld, A new realization of Yangians and quantized affine algebras, Soviet Math. Dokl. 36 (1988), 212–216. MR 914215 (88j:17020)
- I. M. Gelfand and M. L. Tsetlin, Finite-dimensional representations of the group of unimodular matrices, Dokl. Akad. Nauk. SSSR 71 (1950), 825–828; in: I. M. Gelfand, “Collected Papers”, vol. II, Springer-Verlag, Berlin, 1988, pp. 653–656. MR 0035774 (12:9j)
- A. Molev, Finite-dimensional irreducible representations of twisted Yangians, J. Math. Phys. 39 (1998), 5559–5600. MR 1642338 (99i:81106)
- A. Molev, M. Nazarov and G. Olshanski, Yangians and classical Lie algebras, Russian Math. Surveys 51 (1996), 205–282. MR 1401535 (97f:17019)
- M. Nazarov and V. Tarasov, Representations of Yangians with Gelfand-Zetlin bases, J. Reine Angew. Math. 496 (1998), 181–212. MR 1605817 (99c:17030)
- G. Olshanski, Twisted Yangians and infinite-dimensional classical Lie algebras, in: “Quantum Groups” (P. P. Kulish, Ed.), Lecture Notes in Math. 1510, Springer, Berlin-Heidelberg, 1992, pp. 103–120. MR 1183482 (93h:17039)
- N. Yu. Reshetikhin, L. A. Takhtajan and L. D. Faddeev, Quantization of Lie Groups and Lie Algebras, Leningrad Math. J. 1 (1990), 193–225. MR 1015339 (90j:17039)
- H. Weyl, Classical Groups, their Invariants and Representations, Princeton University Press, 1946. MR 1488158 (98k:01049)
Review Information:
Reviewer:
Jonathan Brundan
Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email:
brundan@uoregon.edu
Journal:
Bull. Amer. Math. Soc.
47 (2010), 561-566
DOI:
https://doi.org/10.1090/S0273-0979-10-01280-2
Published electronically:
February 9, 2010
Additional Notes:
The reviewer was supported in part by NSF Grant DMS-0635607.
Review copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
