Deformations of Quadratic Algebras, the Joseph Ideal for Classical Lie Algebras, and Special Tensors

  • Petr Somberg
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 144)


The Joseph ideal is a unique ideal in the universal enveloping algebra of a simple Lie algebra attached to the minimal coadjoint orbit. Its construction using deformation theory was described by Braverman and Joseph. The same ideal appeared recently in connection with algebra of symmetries of differential operators. The paper presents a review of the corresponding circle of ideas and it discusses a role of special tensors in these questions.


Deformation Theory Coadjoint Orbit Hochschild Cohomology Quadratic Algebra Conformal Killing Vector 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    B. Binegar and R. Zierau, Unitarizatiou of a singular representation ofSO(p, q), Comm. Math. Phys. (1991), 138(2).Google Scholar
  2. [2]
    A. Braverman and D. Gaitsgory, Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type, J. of Algebra (1996), 181: 315–328.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    A. Braverman and A. Joseph, The minimal realization from deformation theory, J. of Algebra (1998), 205: 13–36.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    I.M. Benn and J.M. Kress, First-order Dirac symmetry operators, Class, quantum gravity, 2004, 21(2): 427–431.MATHMathSciNetGoogle Scholar
  5. [5]
    M. Eastwood, Higher symmetries of the Laplacian, Ann. Math. (2005), 161: 1645–1665.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    M. Eastwood, P. Somberg, and V. Soucek, The Uniqueness of the Joseph Ideal for the Classical Groups, math.RT/0512296.Google Scholar
  7. [7]
    D. Garfinkle, A new construction of the Joseph ideal, PhD thesis, MIT.Google Scholar
  8. [8]
    A. Joseph, The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. Sci. Ecole Norm. Sup. (1976), 9: 1–30.MATHMathSciNetGoogle Scholar
  9. [9]
    B. Kostant, Verma modules and the existence of quasi-invariant differential operators, Lecture Notes in Mathematics, 466, Springer-Verlag, 1974, pp. 101–129.CrossRefMathSciNetGoogle Scholar
  10. [10]
    R. Penrose and W. Rindler, Spinors and Space-time, Vol. 1, Cambridge University Press, 1984.Google Scholar
  11. [11]
    M.A. Vasiliev, Higher spin superalgebras in any dimension and their representations, preprint, hep-th 0404124.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Petr Somberg
    • 1
  1. 1.Karlova Universita (Charles University)Czech Republic

Personalised recommendations