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Letters in Mathematical Physics

, Volume 81, Issue 1, pp 1–17 | Cite as

Coherent States for Hopf Algebras

  • Zoran ŠkodaEmail author
Article
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Abstract

Families of Perelomov coherent states are defined axiomatically in the context of unitary representations of Hopf algebras. A global geometric picture involving locally trivial noncommutative fibre bundles is involved in the construction. If, in addition, the Hopf algebra has a left Haar integral, then a formula for noncommutative resolution of identity in terms of the family of coherent states holds. Examples come from quantum groups.

Mathematics Subject Classification (2000)

14A22 16W30 14L30 58B32 

Keywords

coherent states Hopf algebra comodule algebra Ore localization localized coinvariants line bundle resolution of unity 
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References

  1. 1.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Enc. Math. Appl, vol. 71 Camb. Univ. Press xvi+664 pp. (Chap. 10: Introduction to q-series) (1999)Google Scholar
  2. 2.
    Askey R. (1980). Ramanujan’s extensions of the gamma and beta functions. Am. Math. Mon. 87(5): 346–359 zbMATHCrossRefGoogle Scholar
  3. 3.
    Bargmann V. (1961). On a Hilbert space of analytic functions and an associated integral transform I. Commun. Pure Appl. Math. 14: 187–214 zbMATHCrossRefGoogle Scholar
  4. 4.
    Berezin F.A. (1974). Quantization. Math. USSR-Izv. 38(5): 1109–1165 CrossRefGoogle Scholar
  5. 5.
    Brzeziński T., Majid S. (1993). Quantum group gauge theory on quantum spaces. Commun. Math. Phys. 157(3): 591–638 CrossRefADSGoogle Scholar
  6. 6.
    Connes A. (1994). Non-commutative Geometry. Academic, New York Google Scholar
  7. 7.
    Delbourgo, R., Zhang, R.B.: Minimal uncertainty states for quantum groups. J. Phys. A 30, L313–L316; arXiv:q-alg/9611021 (1997)Google Scholar
  8. 8.
    Drinfel’d, V.G.: Quantum Groups. Proc. ICM-86, pp. 798–820 (1986)Google Scholar
  9. 9.
    Furlan P., Stanev YA.S., Todorov I.T. (1991). Coherent state operators and n-point invariants for U q(sl(2)). Lett. Math. Physc. 22: 307–319 zbMATHCrossRefGoogle Scholar
  10. 10.
    Hadjiivanov L.K., Paunov R.R., Todorov I.T. (1992). U q–covariant oscillators and vertex operators. J. Math. Phys. 33(4): 1379–1394 CrossRefADSGoogle Scholar
  11. 11.
    Jantzen J.C. (1987). Representations of Algebraic Groups. Academic, zbMATHGoogle Scholar
  12. 12.
    Jurčo B. (1991). On coherent states for the simplest quantum groups. Lett. Math. Phys. 21: 51–58 CrossRefGoogle Scholar
  13. 13.
    Jurčo B., Štovíček P. (1993). Quantum dressing orbits on quantum groups. Commun. Math. Phys. 152: 97–126 CrossRefADSGoogle Scholar
  14. 14.
    Jurčo, B., Štovíček, P.: Coherent states for quantum compact groups. Commun. Math. Phys. 182, 221–251; arXiv:hep-th/9403114 (1996)Google Scholar
  15. 15.
    Klimyk A.U., Schmüdgen K. (1997). Quantum Groups and their Representations. Springer, Heidelberg zbMATHGoogle Scholar
  16. 16.
    Mack G., Schomerus V. (1992). Quasihopf quantum symmetry in quantum theory. Nucl. Phys. B 370: 185–230 CrossRefADSGoogle Scholar
  17. 17.
    Majid, S.: Foundations of Quantum Group Theory. Cambridge University Press, London (1995,2000)Google Scholar
  18. 18.
    Manin YU.I. (1988). Quantum Groups and Non-commutative Geometry. CRM, Montreal zbMATHGoogle Scholar
  19. 19.
    Montgomery, S.: Hopf algebras and their actions on rings. CBMS Regional Conference Series in Mathematics, vol. 82, AMS (1993)Google Scholar
  20. 20.
    Perelomov, A.M.: Coherent states for arbitrary Lie groups. Comm. Math. Phys. 26, 222–236; archived as arXiv:math-ph/0203002 (1972)Google Scholar
  21. 21.
    Perelomov, A.M.: Generalized Coherent States and their Applications. Texts and Monographs in Physics. Springer, Berlin xii+320 p. (1986)Google Scholar
  22. 22.
    Rawnsley J.H. (1977). Coherent states and Kähler manifolds. Quart. J. Math. Oxford 28(2): 403–415 zbMATHCrossRefGoogle Scholar
  23. 23.
    Rosenberg, A.L.: Non-commutative affine semischemes and schemes. Seminar on supermanifolds 26th edn. Dept. Math., U. Stockholm (1988)Google Scholar
  24. 24.
    Rosenberg A.L. (1998). Noncommutative schemes. Comp. Math. 112: 93–125 zbMATHCrossRefGoogle Scholar
  25. 25.
    Sazdjian H., Stanev Y.S., Todorov I.T. (1995). SU(3)-coherent state operators and invariant correlation functions and their quantum group counterparts. J. Math. Phys. 36: 2030–2052 zbMATHCrossRefADSGoogle Scholar
  26. 26.
    Schneider, H. J.: Lectures on Hopf algebras. Trabajos de Matemática, vol. 31. Universidad Nacional de Córdoba 58 p. (1995)Google Scholar
  27. 27.
    Schneider H.-J. (1990). Principal homogeneous spaces for arbitrary Hopf algebras. Israel J. Math. 72(1–2): 167–195 Google Scholar
  28. 28.
    Škoda, Z.: Coset spaces for quantum groups. Ph. D. Thesis, Univ. of Wisconsin-Madison, (available on request) (2002)Google Scholar
  29. 29.
    Škoda, Z.: Localizations for construction of quantum coset spaces. In: Hajac, P.M., Pusz, W. (eds.) Noncommutative geometry and quantum groups. vol.61, pp. 265–298, Banach Center Publications, Warszawa arXiv:math.QA/0301090 (2003)Google Scholar
  30. 30.
    Škoda, Z.: Globalizing Hopf–Galois extensions. preliminary versionGoogle Scholar
  31. 31.
    Škoda, Z.: Noncommutative localization in noncommutative geometry. In: London Math. Soc. Lec. Note Series, vol. 330, pp. 220–313, arXiv:math.QA/0403276Google Scholar
  32. 32.
    Škoda, Z.: Included-row exchange principle for quantum minors. math.QA/0510512Google Scholar
  33. 33.
    Škoda, Z.: Localized coinvariants I, II. (preprints)Google Scholar
  34. 34.
    Škoda, Z.: Every quantum minor generates an Ore set. math.QA/0604610Google Scholar
  35. 35.
    Soibelman Y. (1993). Orbit method for the algebras of functions on quantum groups and coherent states I. Int. Math. Res. Not. 6: 151–163 CrossRefGoogle Scholar
  36. 36.
    Spera M. (1993). On a generalized uncertainty principle, coherent states and the moment map. J. Geom. Phys. 12: 165–182 zbMATHCrossRefGoogle Scholar
  37. 37.
    Varilly, J.C.: Hopf algebras in noncommutative geometry. Lecture notes for the CIMPA Summer School in Villa de Leyva, arXiv:hep-th/0109077Google Scholar

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© Springer 2007

Authors and Affiliations

  1. 1.Theoretical Physics DivisionInstitute Rudjer BoškovićZagrebCroatia

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