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The Dixmier Map for Nilpotent Super Lie Algebras

Abstract

In this article we prove that there exists a Dixmier map for nilpotent super Lie algebras. In other words, if we denote by \({\mathrm{Prim}(\mathcal{U}(\mathfrak{g}))}\) the set of (graded) primitive ideals of the enveloping algebra \({\mathcal{U}(\mathfrak{g})}\) of a nilpotent Lie superalgebra \({\mathfrak{g}}\) and \({\mathcal{A}d_{0}}\) the adjoint group of \({\mathfrak{g}_{0}}\), we prove that the usual Dixmier map for nilpotent Lie algebras can be naturally extended to the context of nilpotent super Lie algebras, i.e. there exists a bijective map

$$I : \mathfrak{g}_{0}^{*}/\mathcal{A}d_{0} \rightarrow \mathrm{Prim}(\mathcal{U}(\mathfrak{g}))$$

defined by sending the equivalence class [λ] of a functional λ to a primitive ideal I(λ) of \({\mathcal{U}(\mathfrak{g})}\), and which coincides with the Dixmier map in the case of nilpotent Lie algebras. Moreover, the construction of the previous map is explicit, and more or less parallel to the one for Lie algebras, a major difference with a previous approach (cf. [18]). One key fact in the construction is the existence of polarizations for super Lie algebras, generalizing the concept defined for Lie algebras. As a corollary of the previous description, we obtain the isomorphism \({\mathcal{U}(\mathfrak{g})/I(\lambda) \simeq {\rm Cliff}_{q}(k) \otimes A_{p}(k)}\), where \({(p,q) = ({\rm dim}(\mathfrak{g}_{0}/\mathfrak{g}_{0}^{\lambda})/2,{\rm dim}(\mathfrak{g}_{1}/\mathfrak{g}_{1}^{\lambda}))}\), we get a direct construction of the maximal ideals of the underlying algebra of \({\mathcal{U}(\mathfrak{g})}\) and also some properties of the stabilizers of the primitive ideals of \({\mathcal{U}(\mathfrak{g})}\).

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References

  1. 1

    Anderson, F.W., Fuller, K.R.: Rings and categories of modules, 2nd ed., Graduate Texts in Mathematics, Vol. 13, New York: Springer-Verlag, 1992

  2. 2

    Behr E.J.: Enveloping algebras of Lie superalgebras. Pacific J. Math. 130(1), 9–25 (1987)

    MathSciNet  MATH  Google Scholar 

  3. 3

    Bell A.D., Musson I.M.: Primitive factors of enveloping algebras of nilpotent Lie superalgebras. J. London Math. Soc. (2) 42(3), 401–408 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4

    Cohen M., Montgomery S.: Group-graded rings, smash products, and group actions. Trans. Amer. Math. Soc. 282(1), 237–258 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5

    Connes A., Dubois-Violette M.: Yang-Mills algebra. Lett. Math. Phys. 61(2), 149–158 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6

    Deligne, P., Morgan, J.W.: Notes on supersymmetry (following Joseph Bernstein), (Princeton, NJ, 1996/1997), Providence, RI: Amer. Math. Soc., 1999, pp. 41–97

  7. 7

    Dixmier J.: Représentations irréductibles des algèbres de Lie nilpotentes. An. Acad. Brasil. Ci. 35, 491–519 (1963) (French)

    MathSciNet  Google Scholar 

  8. 8

    Dixmier, J.: Enveloping algebras. Graduate Studies in Mathematics, Vol. 11, Providence, RI: Amer. Math. Soc., 1996, revised reprint of the 1977 translation

  9. 9

    Elduque A., Laliena J., Sacristán S.: Maximal subalgebras of associative superalgebras. J. Algebra 275(1), 40–58 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10

    Farb, B., Dennis, R.K.: Noncommutative algebra. Graduate Texts in Mathematics, Vol. 144, New York: Springer-Verlag, 1993

  11. 11

    Goodearl K.R.: Prime ideals in skew polynomial rings and quantized Weyl algebras. J. Algebra 150(2), 324–377 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12

    Grothendieck, A.: Revêtements étales et groupe fondamental. Fasc. I: Exposés 1 à 5. Séminaire de Géométrie Algébrique, Vol. 1960/61, Paris: Institut des Hautes Études Scientifiques, 1963

  13. 13

    Herscovich E., Solotar A.: Representation theory of Yang-Mills algebras. Ann. of Math. (2) 173(2), 1043–1080 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14

    Kac V.G.: Lie superalgebras. Adv. in Math. 26(1), 8–96 (1977)

    MATH  Article  Google Scholar 

  15. 15

    Karoubi, M.: K-theory. Grundlehren der Mathematischen Wissenshcaften, Vol. 226, Berlin, Heidel-berg-New York: Springer-Verlag, 1978

  16. 16

    Lam, T.Y.: The algebraic theory of quadratic forms. Reading, MA: Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, 1980, revised second printing; Mathematics Lecture Note Series

  17. 17

    Letzter E.: Primitive ideals in finite extensions of Noetherian rings. J. London Math. Soc. (2) 39(3), 427–435 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18

    Letzter, E.S.: Prime and primitive ideals in enveloping algebras of solvable Lie superalgebras. In: Abelian groups and noncommutative rings, Contemp. Math., Vol. 130, Providence, RI: Amer. Math. Soc., 1992, pp. 237–255

  19. 19

    Năstăsescu, C., Van Oystaeyen, F.: Methods of graded rings. Lecture Notes in Mathematics, Vol. 1836, Berlin: Springer-Verlag, 2004

  20. 20

    Quillen D.: On the endomorphism ring of a simple module over an enveloping algebra. Proc. Amer. Math. Soc. 21, 171–172 (1969)

    MathSciNet  MATH  Google Scholar 

  21. 21

    Racine M.L.: Primitive superalgebras with superinvolution. J. Algebra 206(2), 588–614 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22

    Ross L.E.: Representations of graded Lie algebras. Trans. Amer. Math. Soc. 120, 17–23 (1965)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23

    Scheunert, M.: The theory of Lie superalgebras: An introduction. Lecture Notes in Mathematics, Vol. 716, Berlin: Springer, 1979

  24. 24

    Sergeev A.: Irreducible representations of solvable Lie superalgebras. Represent. Theory 3, 435–443 (1999) (electronic)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25

    Tauvel, P., Yu, R.W.T.: Lie algebras and algebraic groups. Springer Monographs in Mathematics, Berlin: Springer-Verlag, 2005

  26. 26

    Varadarajan, V.S.: Supersymmetry for mathematicians: an introduction. Courant Lecture Notes in Mathematics, Vol. 11, New York: New York University Courant Institute of Mathematical Sciences, 2004

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Correspondence to Estanislao Herscovich.

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The author is an Alexander von Humboldt fellow.

Communicated by A. Connes

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Herscovich, E. The Dixmier Map for Nilpotent Super Lie Algebras. Commun. Math. Phys. 313, 295–328 (2012). https://doi.org/10.1007/s00220-012-1505-0

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Keywords

  • Maximal Ideal
  • Symmetric Bilinear Form
  • Homogeneous Element
  • Isotropic Subspace
  • Primitive Ideal