Theoretical and Mathematical Physics

, Volume 106, Issue 3, pp 291–306 | Cite as

Ideal structure for superconformal semigroups

  • S. A. Duplii (Duplij)


The role of univertible transformations in superstring theories is discussed. A new parametrization of superconformal groups is presented that permits their ideal extensions—superconformal semigroups—to be constructed adequately. These semigroups consist of a group part containing the standard superconformal transformations and an ideal part whose abstract structure is analyzed in detail. A classification of all elements in terms of different indices of nilpotency is performed. An ideal series is constructed, the generalized Green's “vector” and “tensor” relations are defined, and some types of quasi-characters separating the elements of superconformal semigroups are introduced. The need for similar constructions in other supersymmetric and superquantum models is pointed out.


Ideal Structure Abstract Structure Similar Construction Ideal Part Group Part 
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  1. 1.
    D. V. Volkov and V. P. Akulov,Pis'ma Zh. Eksp. Teor. Fiz.,16, 621–624 (1972).Google Scholar
  2. 2.
    M. B. Green, J. H. Schwarz, and E. Witten,Superstring Theory, Vols. 1, 2, Cambridge Univ. Press, Cambridge (1987).Google Scholar
  3. 3.
    S. J. Gates, M. T. Grisaru, M. Rocek, et al.,Superspace, Benjamin, Reading (1983).Google Scholar
  4. 4.
    F. A. Berezin,Introduction to Superanalysis, Reidel, Dordrecht (1987).Google Scholar
  5. 5.
    A. H. Clifford and G. B. Preston,The Algebraic Theory of Semigroups, Vol. 1, Amer. Math. Soc., Providence (1961).Google Scholar
  6. 6.
    E. S. Ljapin,Semigroups, Amer. Math. Soc., Providence (1968).Google Scholar
  7. 7.
    K. Kawakubo,The Theory of Transformation Groups, Clarendon Press, New York (1991).Google Scholar
  8. 8.
    J. M. Howie,Math. Chronicle,16, 1–14 (1987).Google Scholar
  9. 9.
    S. Duplii,J. Math. Phys.,32, 2959–2965 (1991);Sov. J. Nucl. Phys.,52, No. 10, 742–745 (1990).Google Scholar
  10. 10.
    S. Duplii,Theor. Math. Phys.,86, 138–143 (1991);J. Phys.,A24, 3167–3179 (1991);Acta Phys. Pol.,B21, 783–811 (1990).Google Scholar
  11. 11.
    M. Petrich, in:Cayley Theorems for Semigroups. Semigroups and Their Applications. (S. M. Coberstein and P. M. Higgins, eds.), D. Reidel, Dordrecht (1987), pp. 133–138.Google Scholar
  12. 12.
    K. D. Magill,Semigroup Forum,11, 1–189 (1975).Google Scholar
  13. 13.
    A. J. Aizenstat,Izv. Vyssh. Uchebn. Zaved., Mat.,1, 3–11 (1965).Google Scholar
  14. 14.
    K. H. Hofmann and P. S. Mostert,Elements of Compact Semigroups, Merill, Columbus (1966).Google Scholar
  15. 15.
    D. A. Leites,Russ. Math. Surv.,35, 1–64 (1980).Google Scholar
  16. 16.
    A. Rogers,J. Math. Phys.,21, 1352–1365 (1980).Google Scholar
  17. 17.
    A. Y. Khrennikov,Russ. Math. Surv.,43, 87–114 (1988);Theor. Math. Phys.,72, 1313 (1987).Google Scholar
  18. 18.
    A. S. Schwarz,Theor. Math. Phys.,60, 37–42 (1984); A. A. Voronov,Theor. Math. Phys.,60, 43–48 (1984); V. S. Vladimirov and I. V. Volovich,Theor. Math. Phys.,59, 3–27 (1984).Google Scholar
  19. 19.
    C. P. Boyer and S. Gitler,Trans. Amer. Math. Soc.,285, 241–267 (1984).Google Scholar
  20. 20.
    R. Catenacci, C. Reina, and P. Teofilatto,J. Math. Phys.,26, 671–674 (1985).Google Scholar
  21. 21.
    A. I. Shtern,Funct. Anal. Appl.,25, 140–143 (1991).Google Scholar
  22. 22.
    F. A. Berezin and G. I. Kac,Mat. Sb.,11, 311–326 (1970).Google Scholar
  23. 23.
    A. S. Schwarz, “Superanalogs of symplectic and contact geometry and their applications to quantum field theory,” UC Davis preprint-94-06-01, HEP-TH-9406120 (1994).Google Scholar
  24. 24.
    B. S. D. Witt,Supermanifolds, Cambridge Univ. Press, Cambrige (1984).Google Scholar
  25. 25.
    B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov,Modern Geometry. Methods and Applications (Part 1. The Geometry of Surfaces, Transformation Groups and Fields), Springer, Berlin (1984); Springer, Berlin (1985).Google Scholar
  26. 26.
    V. N. Shander,Funct. Anal. Appl.,22, No. 1, 91–92 (1988).Google Scholar
  27. 27.
    M. A. Baranov, I. V. Frolov, and A. S. Schwarz,Theor. Math. Phys.,79, 509 (1989).Google Scholar
  28. 28.
    R. Gilmer,Multiplicative Ideal Theory, Dekker, New York (1972); M. D. Larsen and P. J. McCarthy,Multiplicative Theory of Ideals, Academic Press, New York (1971); A. Császár and E. Thümmel,Acta Math. Hung.,56, No. 3–4, 189–204 (1990).Google Scholar
  29. 29.
    A. M. Baranov and A. S. Schwarz,Int. J. Mod. Phys.,A2, 1773–1781 (1987).Google Scholar
  30. 30.
    L. Crane and J. M. Rabin,Commun. Math. Phys.,113, 601–623 (1988).Google Scholar
  31. 31.
    S. B. Giddings and P. Nelson,Commun. Math. Phys.,116, 607–634 (1988);Phys. Rev. Lett.,59, 2619–2622 (1987).Google Scholar
  32. 32.
    D. Friedan, “Notes on string theory and two-dimensional conformal field theory,” in:Unified String Theories (M. Green and D. Gross, ed.), World Sci., Singapore (1986), pp. 118–149.Google Scholar
  33. 33.
    M. A. Baranov, I. V. Frolov, and A. S. Schwarz,Theor. Math. Phys.,70, 64 (1987); A. A. Rosly, A. S. Schwarz, and A. A. Voronov,Commun. Math. Phys.,119, 129–152 (1988).Google Scholar
  34. 34.
    J. Distler and P. Nelson,Phys. Rev. Lett.,66, 1955–1959 (1991); S. Covindarajan, P. Nelson, and S.-J. Rey,Nucl. Phys.,B365, 633–652 (1991); S. Covindarajan, P. Nelson, and E. Wong,Commun. Math. Phys.,147, No. 2, 253–275 (1992).Google Scholar
  35. 35.
    S. B. Giddings and P. Nelson,Commun. Math. Phys.,118, 289–302 (1988); L. Hodgkin,J. Geom. Phys.,6, 333–338 (1989); A. Rogers, in:Super Riemann surfaces. The Interface of Mathematics and Particle Physics (D. S. Quillen, G. B. Segal, and S. T. Tson, eds.), Clarendon Press, New York (1990), pp. 87–96.Google Scholar
  36. 36.
    A. Rogers and M. Langer,Class. Q. Grav.,11, 2619–2626 (1994).Google Scholar
  37. 37.
    P. A. Grillet,Semigroup Forum,43, 187–201 (1991); R. P. Sullivan,J. Algebra,110, 324–343 (1987).Google Scholar
  38. 38.
    P. A. Grillet,Semigroup Forum,50, No. 1, 25–36 (1995); R. P. Sullivan,J. Austr. Math. Soc.,A43, 127–136 (1987).Google Scholar
  39. 39.
    P. A. Grillet and M. Petrich,Pacific J. Math.,26, 493–508 (1968); S. Bogdanović and M. Ćirić,Proc. Japan Acad.,A68, No. 6, 115–117 (1992); 126–130; L. M. Wang,Semigroup Forum,47, 353–358 (1993).Google Scholar
  40. 40.
    O. Steinfeld and T. T. Thang,Beitrage Alg. Geom.,26, 127–135 (1988); A. H. Clifford,Semigroup Forum,16, No. 2, 183–196 (1978); O. Steinfeld,Quasi-ideals in Rings and Semigroups, Akadémiai Kiado, Budapest (1978).Google Scholar
  41. 41.
    M. M. Miccoli,Note Mat.,7, No. 1, 83–89 (1987); F. Catino,Rev. Mat. Pure Appl.,4, 89–92 (1989); S. Lajos,Acta Sci. Math. Seged.,22, No. 1, 217–222 (1961).Google Scholar
  42. 42.
    I. L. Hmelnitsky,Semigroup Forum,32, 135–144 (1985); L. N. Shevrin and A. S. Prosvirov,Trans. Moscow Math. Soc.,29, 235–246 (1973); D. Y. Long,Chinese Ann. Math.,A13, No. 3, 360–363 (1992).Google Scholar
  43. 43.
    J. A. Huckaba,Commutative Rings with Zero Divisors, Dekker, New York (1988); S. Visweswaran,Bull. Austr. Math. Soc.,43, 233–240 (1991).Google Scholar
  44. 44.
    G. M. S. Gomes and J. M. Howie,Proc. Edinburgh Math. Soc.,30, 383–395 (1987); G. U. Garba,Semigroup Forum,48, No. 1, 37–49 (1994); M. Yamada,Proc. Japan Acad.,40, 94–98 (1964).Google Scholar
  45. 45.
    I. Levi and W. Williams,Semigroup Forum,43, 344–356 (1991); I. Levi,Glasgow Math. J.,29, 149–157 (1987); I. Levi and S. Seif,Semigroup Forum,43, 93–113 (1991); I. Levi, “Green's relations onG-normal semigroups,” Louisville preprint (1992).Google Scholar
  46. 46.
    B. M. Schein, in:Cosets in groups and semigroups. Semigroups With Applications (J. M. Howie, W. Munn, and H. J. Weinert, eds.), World Sci., River Edge (1992), pp. 205–221.Google Scholar
  47. 47.
    J. Meakin,J. London Math. Soc.,21, 244–256 (1980).Google Scholar
  48. 48.
    L. Anderson, R. Hunter, and R. Koch,Trans. Amer. Math. Soc.,117, 521–529 (1965).Google Scholar
  49. 49.
    Yu. Manin,Commun. Math. Phys.,123, 123–135 (1989); in:Topics in Noncommutative Differential Geometry, Princeton University Press, Princeton (1991); S. M. Khoroshkin and N. V. Tolstoy,Commun. Math. Phys.,141, 599 (1991).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • S. A. Duplii (Duplij)
    • 1
    • 2
  1. 1.Department of PhysicsUniversity of KaiserslauternKaiserslauternGermany
  2. 2.Khar'kov State UniversityGermany

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