Skip to main content
SpringerLink
Log in

Supertrace and Superquadratic Lie Structure on the Weyl Algebra, and Applications to Formal Inverse Weyl Transform

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

Abstract

Using the Moyal *-product and orthosymplectic supersymmetry, we construct a natural nontrivial supertrace and an associated nondegenerate invariant supersymmetric bilinear form for the Lie superalgebra structure of the Weyl algebra W. We decompose adjoint and twisted adjoint actions. We define a renormalized supertrace and a formal inverse Weyl transform in a deformation quantization framework and develop some examples

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. Arnal, D., Benamor, H., Benayadi, S., Pinczon, G.: Une algèbre de Lie non semi-simple rigide et sympathique: \(H^1 (\mathfrak{g})=H^2 (\mathfrak{g}) =H^0 (\mathfrak{g}, \mathfrak{g})=H\sp 1(\mathfrak{g}, \mathfrak{g})=H\sp 2(\mathfrak{g}, \mathfrak{g})= \{0\}\). C. R. Acad. Sci. Paris, Sér. I 315(3), 261–263 (1992)

    MathSciNet  MATH  Google Scholar 

  2. Arnaudon D., Bauer M., Frappat L. (1997). On Casimir’s ghost. Commun. Math. Phys. 187(2):429–439

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Bayen F., Flato M., Frønsdal C., Lichnerowicz A., Sternheimer D. Deformation theory and quantization I,II. Deformations of symplectic structures, Physical applications. Ann. Phys. 111, 61–110, 111–151 (1978)

    Google Scholar 

  4. Bonneau P., Flato M., Gerstenhaber M., Pinczon G. (1994). The hidden group structure of quantum groups: Strong duality, rigidity and preferred deformations. Commun. Math. Phys. 61(1):125–156

    Article  ADS  MathSciNet  Google Scholar 

  5. Connes A., Flato M., Sternheimer D. (1992). Closed star products and cyclic cohomology. Lett. Math. Phys. 24(1):1–12

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Dixmier, J.: Sur les algèbres de Weyl. Bull. Soc. Math. Fr. 96, 209–242 (1968). Sur les algèbres de Weyl II. Bull. Sci. Math. Sér94, 289–301 (1970)

    Google Scholar 

  7. Fronsdal C., Flato M., Hirai T (eds). (1986). Essays on supersymmetry. Mathematical physics studies, vol 8. D Reidel Publishing, Dordrecht

  8. Gié, P.-A., Pinczon G., Ushirobira R. (2003). Back to the Amitsur–Levitzki theorem: a super version for the orthosymplectic Lie superalgebra \(\mathfrak{osp}(1, 2n)\). Lett. Math. Phys. 66(1–2):141–155

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Gorelik M. (2000). On the ghost centre of Lie superalgebras. Ann. Inst. Fourier 50(6):1745–1764

    MathSciNet  MATH  Google Scholar 

  10. Kac V.G. (1990). Infinite dimensional Lie algebras. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  11. Lesimple M., Pinczon G. (2001). Deformations of the metaplectic representations. J. Math. Phys. 42(4):1887–1899

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Montgomery S. (1997). Constructing simple Lie superalgebras from associative graded algebras. J. Algebra 195(2):558–579

    Article  MathSciNet  MATH  Google Scholar 

  13. Musson I.M. (1999). Some Lie superalgebras associated to the Weyl algebras. Proc. Am. Math. Soc. 127(10):2821–2827

    Article  MathSciNet  MATH  Google Scholar 

  14. Pinczon G. (1997). On the equivalence between continuous and differential deformation theories. Lett. Math. Phys. 39(2):143–156

    Article  MathSciNet  MATH  Google Scholar 

  15. Pinczon G. (1990). The enveloping algebra of the Lie superalgebra \(\mathfrak{osp} (1,2)\). J. Algebra 132(1): 219–242

    Article  MathSciNet  MATH  Google Scholar 

  16. Pinczon G., Simon J. (1978). Non-linear representations of inhomogeneous groups. Lett. Math. Phys. 2:499–504

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Sternheimer D. Deformation quantization: Twenty years after. In: Rembielinski J (eds). Particles, fields, and gravitation, Papers from the conference dedicated to the memory of Ryszard Raczka, Lódz Poland, April 15–19, 1998. Woodbury, NY: American Institute of Physics. AIP Conf. Proc. 453, 107–145 (1998)

  18. Dito G., Sternheimer D. (2002). Deformation quantization: genesis, developments and metamorphoses. Deformation quantization (Strasbourg, 2001), IRMA Lect. Math. Theor. Phys. 1:9–54

    MathSciNet  Google Scholar 

  19. Szegö G. (1939). Orthogonal polynomials. American Mathematical Society Colloquium Publications, vol 23. American Mathematical Society, New York

    Google Scholar 

  20. Très F. (1967). Topological vector spaces, distributions and kernels. Academic Press, New York, London

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Bourgogne, Université de Bourgogne, B.P. 47870, F-21078, Dijon Cedex, France

    G. Pinczon & R Ushirobira

Authors
  1. G. Pinczon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R Ushirobira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Pinczon.

Additional information

Mathematics Subject Classification: 53D55, 17B05, 17B10, 17B20, 17B60, 17B65

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pinczon, G., Ushirobira, R. Supertrace and Superquadratic Lie Structure on the Weyl Algebra, and Applications to Formal Inverse Weyl Transform. Lett Math Phys 74, 263–291 (2005). https://doi.org/10.1007/s11005-005-0029-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-005-0029-3

Keywords

Search

Navigation