Abstract
We define nontempered (exponential growth) function spaces on the Lie group ax+b which are stable under some left-invariant (convergent) star product. The techniques used to achieved the latter come from symmetric spaces geometry and star representation theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnal, D. and Cortet, J.-C.: *-products in the method of orbits for nilpotent groups, J.Geom.Phys. 2(2) (1985), 83–116.
Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A. and Sternheimer, D.: Deformation theory and quantization. II. Physical applications, Ann.Phys. 111(1) (1978), 111–151; Deformation theory and quantization. I. Deformations of symplectic structures, Ann.Phys. 111(1) (1978), 61–110.
Bertrand, J. and Bertrand, P.: Symbolic calculus on the time-frequency half-plane, J.Math.Phys. 39 (1998), 4071–4090; physics/9708027.
Bieliavsky, P.: Strict Quantization of solvable symmetric spaces. (math.QA/0010004), J.Symplectic Geom. 1(2) (2002), 269–320.
Bieliavsky, P. and Massar, M.: Oscillatory integral formulae for left-invariant star products on a class of Lie groups, Lett.Math.Phys. 58 (2001), 115–128.
Bieliavsky, P., Rooman, M. and Spindel, Ph.: Regular Poisson structures on massive nonrotating BTZ black holes, to appear in Nuclear Phys.B (hep-th/0206189).
Bieliavsky, P. and Pevzner, M.: Symmetric spaces and star representations. II. Causal symmetric spaces, J.Geom.Phys. 41(3) (2002), 224–234.
Douglas, M. R. and Nekrasov, N. A.: Noncommutative field theory, Rev.Modern Phys. 73 (2001), 977–1029 (hep-th/0106048).
Gulfand, I. M. and Chilov, G. E.: Les distributions.Tome II: Espaces fondamentaux. Traduit du russe par S. Vasilach (French), Collection univ. math. 15, Dunod, Paris (1964).
Giaquinto, A. and Zhang, J. J.: 339 Bialgebra actions, twists, and universal deformation formulas, J.Pure Appl.Algebra 128(2) (1998), 133–151.
Huynh, T. V.: Invariant ⋆-quantization associated with the affine group, J.Math.Phys. 23(6) (1984), 1082–1087.
Maillard, J-M.: Sur le produit de convolution gauche et la transformée de Weyl des distributions tempérées (On the twisted convolution product and the Weyl transform of the tempered distributions), C.R.Acad.Sci.Paris Ser.I 298 (1984), 35–38.
Manchon, D.: Weyl symbolic calculus on any Lie group, Acta Appl.Math. 30(2) (1993), 159–186.
Omori, H., Maeda, Y., Miyazaki, N. and Yoshioka, A.: Anomalous quadratic exponentials in the star-products, RIMS Kokyuroku 1150 (2000), 128–132.
Rieffel, M. A.: Deformation quantization for action of ℝd, Mem.Amer.Math.Soc. 506 (1993).
Schwartz, L.: Théorie des distributions (Distribution theory), Nouveau tirage, Hermann, Paris, 1998.
Troost, J.: Winding strings and AdS3 black holes, J.High Energy Phys. 0209 (2002), 041 (hep-th/0206118).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bieliavsky, P., Maeda, Y. Convergent Star Product Algebras on ‘ax+b’. Letters in Mathematical Physics 62, 233–243 (2002). https://doi.org/10.1023/A:1022211827438
Issue Date:
DOI: https://doi.org/10.1023/A:1022211827438