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Communications in Mathematical Physics

, Volume 111, Issue 4, pp 613–665 | Cite as

Compact matrix pseudogroups

  • S. L. Woronowicz
Article

Abstract

The compact matrix pseudogroup is a non-commutative compact space endowed with a group structure. The precise definition is given and a number of examples is presented. Among them we have compact group of matrices, duals of discrete groups and twisted (deformed)SU(N) groups. The representation theory is developed. It turns out that the tensor product of representations depends essentially on their order. The existence and the uniqueness of the Haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived. The form of these relations differs from that in the group case. This is due to the fact that the Haar measure on pseudogroups is not central in general. The corresponding modular properties are discussed. The Haar measures on the twistedSU(2) group and on the finite matrix pseudogroup are found.

Keywords

Neural Network Matrix Element Nonlinear Dynamics Tensor Product Irreducible Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Barut, A.O., Raczka, R.: Theory of group representations and applications. Warszawa: PWN — Polish Scientific Publishers 1977Google Scholar
  2. 2.
    Bragiel, K.: TwistedSU(3) group (in preparation)Google Scholar
  3. 3.
    Dixmier, J.: LesC*-algèbres et leurs representations. Paris: Gauthier, Villars 1964Google Scholar
  4. 4.
    Drinfeld, V.S.: Quantum groups, will appear in Proceedings ICM — 1986Google Scholar
  5. 5.
    Enock, M., Schwartz, J.M.: Une dualité dans les algèbres de von Neumann. Bull. Soc. Math. France, Suplément mémoire44, 1–144 (1975)Google Scholar
  6. 5a.
    Schwartz, J.M.: Sur la structure des algèbres des Kac I. J. Funct. Anal.34, 370–406 (1979)Google Scholar
  7. 6.
    Kac, G.I.: Ring-groups and the principle of duality I and II. Trudy Moskov. Mat. Obsc.12, 259–301 (1963);13, 84–113 (1965)Google Scholar
  8. 7.
    Kruszynski, P., Woronowicz, S.L.: A noncommutative Gelfand-Naimark theorem. J. Oper. Theory8, 361–389 (1982)Google Scholar
  9. 8.
    Lang, S.: Algebra. Reading, MA: Addison-Wesley 1965Google Scholar
  10. 9.
    Maurin, K.: Analysis I. Warsaw-Dordrecht: PWN — Polish Scientific Publishers, Dordrecht: Reidel 1976Google Scholar
  11. 10.
    Ocneanu, A.: A Galois theory for operator algebras. PreprintGoogle Scholar
  12. 11.
    Takesaki, M.: Duality and von Neumann algebras. Lecture notes, Fall 1970, Tulane University, New Orleans, LouisianaGoogle Scholar
  13. 12.
    Tatsuuma, N.: An extension of AKHT theory of locally compact groups. Kokyuroku RIMS, 314 (1977)Google Scholar
  14. 13.
    Vallin, J.M.:C*-algèbres de Hopf etC*-algèbres de Kac. Proc. Lond. Math. Soc. (3),50, 131–174 (1985)Google Scholar
  15. 14.
    Vaksman, L.L., Soibelman, J.S.: The algebra of functions on quantum groupSU(2) (to appear)Google Scholar
  16. 15.
    Weyl, H.: The classical groups, their invariants and representations. Princeton, NS: Princeton University Press 1946Google Scholar
  17. 16.
    Woronowicz, S.L.: On the purification of factor states. Commun. Math. Phys.28, 221–235 (1972)Google Scholar
  18. 17.
    Woronowicz, S.L.: Pseudospaces, pseudogroups, and Pontryagin duality. Proceedings of the International Conference on Mathematics and Physics, Lausanne1979. Lecture Notes in Physics, Vol. 116. Berlin, Heidelberg, New York: Springer 1980Google Scholar
  19. 18.
    Woronowicz, S.L.: TwistedSU(2) group. An example of a non-commutative differential calculus, will appear in RIMS — Publ. University of Kyoto (1987)Google Scholar
  20. 19.
    Woronowicz, S.L.: Tannaka-Krein duality for compact matrix pseudogroups. TwistedSU(N) groups (in preparation)Google Scholar
  21. 20.
    Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (in preparation)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • S. L. Woronowicz
    • 1
  1. 1.Department of Mathematical Methods of Physics, Faculty of PhysicsUniversity of WarsawWarszawaPoland

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