Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

Book Review

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.

Full text of review: PDF   This review is available free of charge.
Book Information:

Author: Leonid L. Vaksman
Title: Quantum bounded symmetric domains
Additional book information: Translations of Mathematical Monographs, Vol. 238, American Mathematical Society, Providence, RI, 2010, xii+256 pp., ISBN 978-0-8218-4909-5, US $105.00, hardcover

References [Enhancements On Off] (What's this?)

  • [Bor98] A. Borel, Semisimple groups and Riemannian symmetric spaces, Texts and Readings in Mathematics, no. 16, Hindustan Book Agency, New Delhi, 1998. MR 1661166 (2000e:53063)
  • [BSV07] O. Bershtein, A. Stolin, and L. Vaksman, Spherical principal series of quantum Harish-Chandra modules, Mat. Fiz. Anal. Geom. 3 (2007), no. 2, 157-175. MR 2338185 (2009h:17011)
  • [Car35] E. Cartan, Sur les domaines bornés homogènes de $ n$ variables complexes, Abh. Math. Sem. Hamburg (1935), no. 11, 116-162.
  • [Dix77] J. Dixmier, $ {C}^*$-algebras. Translated from the French by Francis Jellett, North-Holland Mathematical Library, no. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. MR 0458185 (56:16388)
  • [Dri87] V. G. Drinfeld, Quantum groups, Proceedings ICM 1986, Amer. Math. Soc., 1987, pp. 798-820. MR 934283 (89f:17017)
  • [Hel78] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, no. 80, Academic Press, New York, 1978. MR 514561 (80k:53081)
  • [Hel84] -, Groups and geometric analysis, Academic Press, Orlando, 1984. MR 754767 (86c:22017)
  • [Jak96] H. P. Jakobsen, Quantized Hermitian symmetric spaces, Lie Theory and Its Applications in Physics (Clausthal, 1995) (River Edge, NJ), World Sci. Publ., 1996, pp. 105-116. MR 1634488 (99e:17022)
  • [Jim85] M. Jimbo, A $ q$-analogue of $ {U}(\mathfrak{g})$ and the Yang-Baxter equation, Lett. Math. Phys. 11 (1985), 63-69. MR 797001 (86k:17008)
  • [Kas91] M. Kashiwara, On crystal bases of the $ {Q}$-analog of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465-515. MR 1115118 (93b:17045)
  • [KK03] E. Koelink and J. Kustermans, A locally compact quantum group analogue of the normalizer of $ \operatorname {SU}(1,1)$ in $ \operatorname {SL}(2,{\mathbb{C}})$, Commun. Math. Phys. 233 (2003), no. 2, 231-296. MR 1962042 (2004c:46136)
  • [KL93] S. Klimek and A. Lesniewski, A two-parameter quantum deformation of the unit disc, J. Funct. Anal. 115 (1993), no. 1, 1-23. MR 1228139 (94e:46128)
  • [KMT98] A. Kamita, Y. Morita, and T. Tanisaki, Quantum deformations of certain prehomogeneous vector spaces. I., Hiroshima Math. J. 28 (1998), no. 3, 527-540. MR 1657547 (99i:17019)
  • [Kna86] A. Knapp, Representation theory of semisimple groups. An overview based on examples, Princeton Mathematical Series, no. 36, Princeton University Press, Princeton, 1986. MR 855239 (87j:22022)
  • [Koo89] T. Koornwinder, Representation of twisted $ \operatorname {SU}(2)$ quantum group and some $ q$-hypergeometric orthogonal polynomials, Indag. Math. (N.S.) 51 (1989), 97-117. MR 993682 (90h:17016)
  • [Koo93] -, Askey-Wilson polynomials as zonal spherical functions on the $ \operatorname {SU}(2)$ quantum group, SIAM J. Math. Anal. 24 (1993), no. 3, 795-813. MR 1215439 (94k:33042)
  • [Kor94] L. Korogodsky, Quantum group $ \operatorname {SU}(1,1)\rtimes {\mathbb{Z}}$ and ``super-tensor'' products, Commun. Math. Phys. 163 (1994), 433-460. MR 1284791 (95g:81081)
  • [Kor99] A. Korányi, Function spaces on bounded symmetric domains, Analysis and Geometry on Complex Homogeneous Domains (J. Faraut, S. Kaneyuki, L. Qikeng, and G. Roos, eds.), Progress in Math., vol. 185, Birkhäuser, Boston, 1999, pp. 183-281.
  • [KV00] J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. École Norm. Sup. 33 (2000), 837-934. MR 1832993 (2002f:46108)
  • [Let04] G. Letzter, Quantum zonal spherical functions and Macdonald polynomials, Adv. Math. 189 (2004), 88-147. MR 2093481 (2005i:33019)
  • [Lus90] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447-498. MR 1035415 (90m:17023)
  • [MMN+88] T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi, and K. Ueno, Representations of quantum groups and a $ q$-analogue of orthogonal polynomials, C. R. Math. Acad. Sci. Paris, Série I 307 (1988), 559-564. MR 967361 (90a:17013)
  • [MMN+91] -, Representations of the quantum group $ \operatorname {SU}_q(2)$ and the little $ q$-Jacobi polynomials, J. Funct. Anal. 99 (1991), 357-386. MR 1121618 (93c:17027)
  • [Nou96] M. Noumi, Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. Math. 123 (1996), 16-77. MR 1413836 (98a:33004)
  • [Rub92] H. Rubenthaler, Algèbres de Lie et espaces préhomogènes, Hermann, Paris, 1992.
  • [SSSV01] D. Shklyarov, S. Sinel'shchikov, A. Stolin, and L. L. Vaksman, Non-compact quantum groups and quantum Harish-Chandra modules, Nucl. Phys. B (Proc. Suppl.) 102/103 (2001), 334-337. MR 1922881 (2003g:17028)
  • [SSV01] S. Sinel'shchikov, A. Stolin, and L. L. Vaksman, Spherical principal non-degenerate series of representations for the quantum group $ \operatorname {SU}_{2,2}$, Czech. J. Phys. 51 (2001), no. 12, 1431-1440. MR 1917717 (2003e:17024)
  • [SV98] S. D. Sinel'shchikov and L. L. Vaksman, On $ q$-analogues of bounded symmetric domains and Dolbeault complexes, Math. Phys. Anal. Geom. 1 (1998), no. 1, 75-100. MR 1687517 (2000f:58016)
  • [Vak01] L. Vaksman (ed.), Lectures on q-analogues of Cartan domains and associated Harish-Chandra modules, preprint, arXiv:math/0109198, 2001.
  • [VK91] L. Vaksman and L. Korogodskiĭ, Spherical functions on the quantum group $ \operatorname {SU}(1,1)$ and a $ q$-analogue of the Mehler-Fock formula, Funct. Anal. Appl. 25 (1991), 48-49. MR 1113123
  • [VS88] L. Vaksman and Y. Soibelman, Algebra of functions on the quantum group $ \operatorname {SU}(2)$, Funct. Anal. Appl. 22 (1988), 170-181. MR 961757 (90f:17019)
  • [Wor87a] S. L. Woronowicz, Compact matrix pseudogroups, Commun. Math. Phys. 111 (1987), 613-665. MR 901157 (88m:46079)
  • [Wor87b] -, Twisted $ \operatorname {SU}(2)$ group. An example of a non-commutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), no. 1, 117-181. MR 890482 (88h:46130)
  • [Wor91] -, Unbounded elements affiliated with $ {C}^*$-algebras and noncompact quantum groups, Commun. Math. Phys. 136 (1991), 399-432. MR 1096123 (92b:46117)

Review Information:

Reviewer: Erik Koelink
Affiliation: IMAPP, Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
Reviewer: Stefan Kolb
Affiliation: School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom
Journal: Bull. Amer. Math. Soc. 50 (2013), 337-345
MSC (2010): Primary 32M15, 20G42; Secondary 17B37, 32A50, 33D80
Published electronically: January 9, 2012
Review copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
American Mathematical Society