Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Symplectic leaves and deformation quantization
HTML articles powered by AMS MathViewer

by Albert J. L. Sheu PDF
Proc. Amer. Math. Soc. 124 (1996), 95-100 Request permission

Abstract:

In this paper, we show that for any classical simple compact Poisson Lie group $K$, there is no quantization of $K$ using the quantum group $K_q$, which is both group-preserving and symplectic leaf-preserving.
References
  • A. A. Belavin and V. G. Drinfel′d, Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funktsional. Anal. i Prilozhen. 16 (1982), no. 3, 1–29, 96 (Russian). MR 674005
  • A. Connes, A survey of foliations and operator algebras, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 521–628. MR 679730
  • 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
  • Branislav Jurčo and Pavel Šťovíček, Quantum dressing orbits of compact groups, Comm. Math. Phys. 152 (1993), no. 1, 97–126. MR 1207671, DOI 10.1007/BF02097059
  • Serge Levendorskiĭ and Yan Soibelman, Algebras of functions on compact quantum groups, Schubert cells and quantum tori, Comm. Math. Phys. 139 (1991), no. 1, 141–170. MR 1116413, DOI 10.1007/BF02102732
  • Jiang-Hua Lu and Alan Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), no. 2, 501–526. MR 1037412
  • L. D. Faddeev, N. Yu. Reshetikhin, and L. A. Takhtajan, Quantization of Lie groups and Lie algebras, Algebraic analysis, Vol. I, Academic Press, Boston, MA, 1988, pp. 129–139. MR 992450
  • Marc A. Rieffel, Deformation quantization and operator algebras, Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 411–423. MR 1077400
  • —, Quantization and C*-algebras, $C^*$-Algebras: 1943-1993, Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 66–97.
  • Marc A. Rieffel, Compact quantum groups associated with toral subgroups, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 465–491. MR 1216204, DOI 10.1090/conm/145/1216204
  • Albert Jeu-Liang Sheu, Quantization of the Poisson $\textrm {SU}(2)$ and its Poisson homogeneous space—the $2$-sphere, Comm. Math. Phys. 135 (1991), no. 2, 217–232. With an appendix by Jiang-Hua Lu and Alan Weinstein. MR 1087382, DOI 10.1007/BF02098041
  • —, Weyl quantization of Poisson $SU(2)$, Pacific J. Math. (to appear).
  • —, Leaf-preserving quantizations of Poisson $SU(2)$ are not coalgebra homomorphisms, Comm. Math. Phys. (to appear).
  • —, Compact quantum groups and groupoid C*-algebras, preprint.
  • Ya. S. Soĭbel′man, Algebra of functions on a compact quantum group and its representations, Algebra i Analiz 2 (1990), no. 1, 190–212 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 1, 161–178. MR 1049910
  • Ya. S. Soĭbel′man and L. L. Vaksman, On some problems in the theory of quantum groups, Representation theory and dynamical systems, Adv. Soviet Math., vol. 9, Amer. Math. Soc., Providence, RI, 1992, pp. 3–55. MR 1166194
  • 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
  • Alan Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), no. 3, 523–557. MR 723816
  • 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, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613–665. MR 901157, DOI 10.1007/BF01219077
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46L87, 81R50
  • Retrieve articles in all journals with MSC (1991): 46L87, 81R50
Additional Information
  • Albert J. L. Sheu
  • Email: sheu@kuhub.cc.ukans.edu
  • Received by editor(s): June 21, 1994
  • Additional Notes: Partially supported by NSF-Grant DMS-9303231
  • Communicated by: Palle E. T. Jorgensen
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 95-100
  • MSC (1991): Primary 46L87, 81R50
  • DOI: https://doi.org/10.1090/S0002-9939-96-03016-X
  • MathSciNet review: 1286007