Skip to main content
SpringerLink
Log in

Quantum deformation of lorentz group

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

A one parameter quantum deformationS μ L(2,ℂ) ofSL(2,ℂ) is introduced and investigated. An analog of the Iwasawa decomposition is proved. The compact part of this decomposition coincides withS μ U(2), whereas the solvable part is identified as a Pontryagin dual ofS μ U(2). It shows thatS μ L(2,ℂ) is the result of the dual version of Drinfeld's double group construction applied toS μ U(2). The same construction applied to any compact quantum groupG c is discussed in detail. In particular the explicit formulae for the Haar measures on the Pontryagin dualG d ofG c and on the double groupG are given. We show that there exists remarkable 1-1 correspondence between representations ofG and bicovariant bimodules (“tensor bundles”) overG c . The theory of smooth representations ofS μ L(2,ℂ) is the same as that ofSL(2,ℂ) (Clebsh-Gordon coefficients are however modified). The corresponding “tame” bicovariant bimodules onS μ U(2) are classified. An application to 4D + differential calculus is presented. The nonsmooth case is also discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abe, E.: Hopf algebras, Cambridge Tracts in Mathematics, vol.74. Cambridge: Cambridge Univ. Press 1977

    Google Scholar 

  2. Connes, A.: Non-commutative differential gometry. Institut des Hautes Etudes Scientifiques. Extrait des Publications Mathématiques No. 62 (1986).

  3. Drinfeld, V.G.: Quantum groups. Proceedings of the International Congress of Mathematicians. Berkeley, CA, USA (1986), pp. 793–820

  4. Kruszyński, P., Woronowicz, S.L.: A noncommutative Gelfand Naimark theorem. J. Operator Theory8, 361–389 (1982)

    MathSciNet  MATH  Google Scholar 

  5. Manin, Yu.I.: Quantum groups and noncommutative geometry. Centre de Recherche Mathemqtique, Montreal 1988

    Google Scholar 

  6. Phillips, N.C.: Inverse limits ofC *-algebras and applications. Operator algebras and application, Vol. 1. London Mathematical Society, Lecture Note Series, vol.135, pp. 127–186. Cambridge: Cambridge University Press 1988

    Google Scholar 

  7. Podleś, P.: Quantum spheres. Lett. Math. Phys.14, 193–202 (1987)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  8. Woronowicz, S.L.: Pseudospaces, pseudogroups and Pontryagin duality. Proceedings of the International Conference on Mathematics and Physics, Lausanne 1979. Lecture Notes in Physics, vol.116, pp. 407–412. Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  9. Woronowicz, S.L.: TwistedSU(2) group. An example of a noncommutative differential calculus. Publ. RIMS, Kyoto University23, 117–181 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  10. Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys.111, 613–665 (1987)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Woronowicz, S.L.: Tannaka-Krein duality for compact matrix pseudogroups. TwistedSU(N) groups. Invent. Math.93, 35–76 (1988)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys.122, 125–170 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Woronowicz, S.L.: Elements affiliated with aC *-algebra and non-compact quantum groups (to appear)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoza 74, PL-00-682, Warszawa, Poland

    P. Podleś & S. L. Woronowicz

Authors
  1. P. Podleś
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. L. Woronowicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Connes

Rights and permissions

Reprints and permissions

About this article

Cite this article

Podleś, P., Woronowicz, S.L. Quantum deformation of lorentz group. Commun.Math. Phys. 130, 381–431 (1990). https://doi.org/10.1007/BF02473358

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02473358

Keywords

Search

Navigation