Book Review
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MathSciNet review:
3020830
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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
Armand Borel, Semisimple groups and Riemannian symmetric spaces, Texts and Readings in Mathematics, vol. 16, Hindustan Book Agency, New Delhi, 1998. MR 1661166
O. Bershtein, A. Stolin, and L. Vaksman, Spherical principal series of quantum Harish-Chandra modules, Zh. Mat. Fiz. Anal. Geom. 3 (2007), no. 2, 157–175, 283 (English, with English and Ukrainian summaries). MR 2338185
E. Cartan, Sur les domaines bornés homogènes de $n$ variables complexes, Abh. Math. Sem. Hamburg (1935), no. 11, 116–162.
Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
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
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
Hans Plesner Jakobsen, Quantized Hermitian symmetric spaces, Lie theory and its applications in physics (Clausthal, 1995) World Sci. Publ., River Edge, NJ, 1996, pp. 105–116. MR 1634488
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
M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
Erik Koelink and Johan Kustermans, A locally compact quantum group analogue of the normalizer of $\rm SU(1,1)$ in $\textrm {SL}(2,\Bbb C)$, Comm. Math. Phys. 233 (2003), no. 2, 231–296. MR 1962042, DOI 10.1007/s00220-002-0736-x
Sławomir Klimek and Andrzej Lesniewski, A two-parameter quantum deformation of the unit disc, J. Funct. Anal. 115 (1993), no. 1, 1–23. MR 1228139, DOI 10.1006/jfan.1993.1078
Atsushi Kamita, Yoshiyuki Morita, and Toshiyuki Tanisaki, Quantum deformations of certain prehomogeneous vector spaces. I, Hiroshima Math. J. 28 (1998), no. 3, 527–540. MR 1657547
Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR 855239, DOI 10.1515/9781400883974
T. H. Koornwinder, Representations of the twisted $\textrm {SU}(2)$ quantum group and some $q$-hypergeometric orthogonal polynomials, Nederl. Akad. Wetensch. Indag. Math. 51 (1989), no. 1, 97–117. MR 993682
Tom H. Koornwinder, Askey-Wilson polynomials as zonal spherical functions on the $\textrm {SU}(2)$ quantum group, SIAM J. Math. Anal. 24 (1993), no. 3, 795–813. MR 1215439, DOI 10.1137/0524049
Leonid I. Korogodsky, Quantum group $\textrm {SU}(1,1)\rtimes Z_2$ and “super-tensor” products, Comm. Math. Phys. 163 (1994), no. 3, 433–460. MR 1284791
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.
Johan Kustermans and Stefaan Vaes, Locally compact quantum groups, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 6, 837–934 (English, with English and French summaries). MR 1832993, DOI 10.1016/S0012-9593(00)01055-7
Gail Letzter, Quantum zonal spherical functions and Macdonald polynomials, Adv. Math. 189 (2004), no. 1, 88–147. MR 2093481, DOI 10.1016/j.aim.2003.11.007
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
Tetsuya Masuda, Katsuhisa Mimachi, Yoshiomi Nakagami, Masatoshi Noumi, and Kimio Ueno, Representations of quantum groups and a $q$-analogue of orthogonal polynomials, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 11, 559–564 (English, with French summary). MR 967361
Tetsuya Masuda, Katsuhisa Mimachi, Yoshiomi Nakagami, Masatoshi Noumi, and Kimio Ueno, Representations of the quantum group $\textrm {SU}_q(2)$ and the little $q$-Jacobi polynomials, J. Funct. Anal. 99 (1991), no. 2, 357–386. MR 1121618, DOI 10.1016/0022-1236(91)90045-7
Masatoshi Noumi, Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. Math. 123 (1996), no. 1, 16–77. MR 1413836, DOI 10.1006/aima.1996.0066
H. Rubenthaler, Algèbres de Lie et espaces préhomogènes, Hermann, Paris, 1992.
D. Shklyarov, S. Sinel′shchikov, A. Stolin, and L. Vaksman, Non-compact quantum groups and quantum Harish-Chandra modules, Nuclear Phys. B Proc. Suppl. 102/103 (2001), 334–337. Supersymmetry and quantum field theory (Kharkov, 2000). MR 1922881, DOI 10.1016/S0920-5632(01)01576-6
S. Sinel′shchikov, L. Vaksman, and A. Stolin, Spherical principal non-degenerate series of representations for the quantum group $\textrm {SU}_{2,2}$, Czechoslovak J. Phys. 51 (2001), no. 12, 1431–1440. Quantum groups and integrable systems (Prague, 2001). MR 1917717, DOI 10.1023/A:1013355011139
S. Sinel′shchikov and L. Vaksman, On $q$-analogues of bounded symmetric domains and Dolbeault complexes, Math. Phys. Anal. Geom. 1 (1998), no. 1, 75–100. MR 1687517, DOI 10.1023/A:1009704002239
L. Vaksman (ed.), Lectures on q-analogues of Cartan domains and associated Harish-Chandra modules, preprint, arXiv:math/0109198, 2001.
L. L. Vaksman and L. I. Korogodskiĭ, Spherical functions on the quantum group $\textrm {SU}(1,1)$ and a $q$-analogue of the Mehler-Fock formula, Funktsional. Anal. i Prilozhen. 25 (1991), no. 1, 60–62 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 1, 48–49. MR 1113123, DOI 10.1007/BF01090677
L. L. Vaksman and Ya. S. Soĭbel′man, An algebra of functions on the quantum group $\textrm {SU}(2)$, Funktsional. Anal. i Prilozhen. 22 (1988), no. 3, 1–14, 96 (Russian); English transl., Funct. Anal. Appl. 22 (1988), no. 3, 170–181 (1989). MR 961757, DOI 10.1007/BF01077623
S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613–665. MR 901157
S. L. Woronowicz, Twisted $\textrm {SU}(2)$ group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), no. 1, 117–181. MR 890482, DOI 10.2977/prims/1195176848
S. L. Woronowicz, Unbounded elements affiliated with $C^*$-algebras and noncompact quantum groups, Comm. Math. Phys. 136 (1991), no. 2, 399–432. MR 1096123
References
- 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)
- 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)
- E. Cartan, Sur les domaines bornés homogènes de $n$ variables complexes, Abh. Math. Sem. Hamburg (1935), no. 11, 116–162.
- 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)
- V. G. Drinfeld, Quantum groups, Proceedings ICM 1986, Amer. Math. Soc., 1987, pp. 798–820. MR 934283 (89f:17017)
- S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, no. 80, Academic Press, New York, 1978. MR 514561 (80k:53081)
- —, Groups and geometric analysis, Academic Press, Orlando, 1984. MR 754767 (86c:22017)
- 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- —, 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)
- 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)
- 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.
- J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. École Norm. Sup. 33 (2000), 837–934. MR 1832993 (2002f:46108)
- G. Letzter, Quantum zonal spherical functions and Macdonald polynomials, Adv. Math. 189 (2004), 88–147. MR 2093481 (2005i:33019)
- G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415 (90m:17023)
- 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)
- —, 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)
- 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)
- H. Rubenthaler, Algèbres de Lie et espaces préhomogènes, Hermann, Paris, 1992.
- 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)
- 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)
- 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)
- L. Vaksman (ed.), Lectures on q-analogues of Cartan domains and associated Harish-Chandra modules, preprint, arXiv:math/0109198, 2001.
- 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
- 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)
- S. L. Woronowicz, Compact matrix pseudogroups, Commun. Math. Phys. 111 (1987), 613–665. MR 901157 (88m:46079)
- —, 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)
- —, 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
DOI:
https://doi.org/10.1090/S0273-0979-2012-01363-0
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.
