Skip to main content
SpringerLink
Log in

Classification of irreducible super-unitary representations ofsu(p,q/n)

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

Abstract

In this paper we classify all the irreducible super-unitary representations ofsu(p,q/n), which can be integrated up to a unitary representation ofS(U(p,q)×U(n)), a Lie group corresponding to the even part ofsu(p,q/n). Note that a real form of the Lie superalgebrasl(m/n;ℂ) which has non-trivial superunitary representations is of the formsu(p,q/n)(p+q=m) orsu(m/r,s)(r+s=n). Moreover, we give an explicit realization for each irreducible super-unitary representation, using the oscillator representation of an orthosymplectic Lie superalgebra.

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. Berezin, F. A.: Introduction to superanalysis. Dordrecht: Reidel 1987

    Google Scholar 

  2. Enright, T., Howe, R., Wallach, N.: A classification of unitary highest weight modules. In: Representation theory of reductive groups, pp. 97–143. Basel: Birkhäuser 1983

    Google Scholar 

  3. Flato, M., Fronsdal, C.: Representations of super conformal supersymmetry. Lett. Math. Phys.8, 159–162 (1984)

    Article  Google Scholar 

  4. Fronsdal, C., Hirai, T.: Unitary representations of supergroups. In: Essays on Supersymmetry, pp. 15–66. Dordrecht: Reidel 1986

    Google Scholar 

  5. Furutsu, H.: Representations of Lie superalgebras, II. Unitary representations of Lie superalgebras of typeA(n, 0). J. Math. Kyoto Univ.29, 671–687 (1989)

    Google Scholar 

  6. Furutsu, H., Hirai, T.: Representations of Lie superalgebras, I. Extensions of representations of the even part. J. Math. Kyoto Univ.28, 695–749 (1988)

    Google Scholar 

  7. Gould, M. D.: Atypical representations for type-I Lie superalgebras. J. Phys. A., Math. Gen.22, 1209–1221 (1989)

    Google Scholar 

  8. Gould, M. D., Zhang, R. B.: Classification of all star and grade star irreps ofgl(n/1). J. Math. Phys.31, 1524–1534 (1990)

    Article  Google Scholar 

  9. Günaydin, M.: Unitary highest weight representations of non-compact supergroups. J. Math. Phys.29, 1275–1282 (1988)

    Article  Google Scholar 

  10. Günaydin, M., Hyun, S. J.: Unitary lowest weight representations of the non-compact supergroupsOSp(2n/2m,ℝ). J. Math. Phys.29, 2367–2376 (1988)

    Article  Google Scholar 

  11. Harish-Chandra: Representations of a semisimple Lie group on a Banach space I. Trans. Am. Math. Soc.75, 185–243 (1953)

    Google Scholar 

  12. Heidenreich, W.: All linear unitary irreducible representations of de Sitter supersymmetry with positive energy. Phys. Lett.110B, 461–464 (1982)

    Google Scholar 

  13. Jakobsen, H. P.: On the range of unitarity for highest weight representations of classical Lie superalgebras. In: Topological and geometrical method in field theory. Hietarinta, J., Westerholm, J. (eds.), pp. 103–109. Singapore: World Scientific 1986

    Google Scholar 

  14. Jakobsen, H. P.: The full set of unitarizable highest weight modules of basic classical Lie superalgebras. preprint, 1989

  15. Van der Jeugt, J.: Representations ofN=2 extended supergravity and unitary conditions inOsp(N, 4). J. Math. Phys.28, 758–764 (1987)

    Article  Google Scholar 

  16. Kac, V. G.: Characters of typical representations of classical Lie superalgebras. Commun. Alg.5, 889–897 (1977)

    Google Scholar 

  17. Kac, V. G.: Lie superalgebras. Adv. Math.26, 8–96 (1977)

    Article  Google Scholar 

  18. Kac, V. G.: Representations of classical Lie superalgebras. Lecture Notes in Mathematics, vol.676, 597–626. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  19. Kostelecký, V. A., Campbell, D. K.: Introduction and overview. Physica15D, 3–21 (1985)

    Google Scholar 

  20. Kostelecký, V. A., Campbell, D. K. (eds.): Supersymmetry in physics. Physica15D (1985)

  21. Nishiyama, K.: Characters and super characters of discrete series representations for orthosymplectic Lie superalgebras, preprint

  22. Nishiyama, K.: Oscillator representations for orthosymplectic algebras. J. Alg.129, 231–262 (1990)

    Article  Google Scholar 

  23. Nishiyama, K.: Super dual pairs and unitary highest weight modules of orthosymplectic algebras, preprint

  24. Scheunert, M., Nahm, W., Rittenberg, V.: Graded Lie algebras: Generalization of Hermitian representations. J. Math. Phys.18, 146–154 (1977)

    Article  Google Scholar 

  25. Tilgner, H.: Graded generalizations of Weyl- and Clifford algebras. J. Pure Appl. Alg.10, 163–168 (1977)

    Article  Google Scholar 

  26. Vogan, D. A.: Representations of real reductive Lie groups. Basel: Birkhäuser 1981

    Google Scholar 

  27. Wakimoto, M.: A certain series of unitarizable representations of Lie superalgebrasgl(p/q), preprint

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Science and Technology, Nihon University, Kanda-Surugadai 1-8, 101, Chiyoda, Tokyo, Japan

    Hirotoshi Furutsu

  2. Department of Mathematics, Yoshida College, Kyoto University, 606, Sakyo, Kyoto, Japan

    Kyo Nishiyama

Authors
  1. Hirotoshi Furutsu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kyo Nishiyama
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by H. Araki

Rights and permissions

Reprints and permissions

About this article

Cite this article

Furutsu, H., Nishiyama, K. Classification of irreducible super-unitary representations ofsu(p,q/n). Commun.Math. Phys. 141, 475–502 (1991). https://doi.org/10.1007/BF02102811

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02102811

Keywords

Search

Navigation