Advertisement

The Poisson Lie Algebra, Rumin’s Complex and Base Change

  • Alessandro D’AndreaEmail author
Chapter
First Online:
  • 162 Downloads
Part of the Springer INdAM Series book series (SINDAMS, volume 37)

Abstract

Results from the forthcoming papers [4] and [8] are announced. We introduce a singular current construction, or base change, for pseudoalgebras which may be used to obtain a primitive Lie pseudoalgebra of type H from a suitable one of type K. When applied to representations, it derives the pseudo de Rham complex of type H from that of type K—which is related to Rumin’s construction from [15]—both with standard coefficients and with nontrivial Galois coefficients. In the latter case, the construction yields exact complexes of modules for the Poisson linearly compact Lie algebra \(P_{2N}\) exhibiting a nontrivial central action.

Keywords

Representation theory Lie algebras and pseudoalgebras Conformally symplectic geometry Hopf–Galois extensions 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Bakalov, B., D’Andrea, A., Kac, V.G.: Theory of finite pseudoalgebras. Adv. Math. 162, 1–140 (2001)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bakalov, B., D’Andrea, A., Kac, V.G.: Irreducible modules over finite simple Lie pseudoalgebras I. Primitive pseudoalgebras of type \(W\) and \(S\). Adv. Math. 204, 278–346 (2006)Google Scholar
  3. 3.
    Bakalov, B., D’Andrea, A., Kac, V.G.: Irreducible modules over finite simple Lie pseudoalgebras II. Primitive pseudoalgebras of type \(K\). Adv. Math. 232, 188–237 (2013)Google Scholar
  4. 4.
    Bakalov, B., D’Andrea, A., Kac, V.G.: Irreducible modules over finite simple Lie pseudoalgebras III. Primitive pseudoalgebras of type \(H\), work in progressGoogle Scholar
  5. 5.
    Cartan, E.: Les groupes de transformation continus, infinis, simples. Ann. Sci. ÉNS 26, 93–161 (1909)MathSciNetzbMATHGoogle Scholar
  6. 6.
    D’Andrea, A.: Finite vertex algebras and nilpotence. J. Pure Appl. Algebr. 212(4), 669–688 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D’Andrea, A.: A remark on simplicity of vertex and Lie conformal algebras. J. Algebr. 319(5), 2106–2112 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    D’Andrea, A.: Projective modules over pseudoalgebras and base change, work in progressGoogle Scholar
  9. 9.
    De Sole, A., Kac, V.G.: Finite vs. infinite \(W\)-algebras. Jap. J. of Math. 1(1), 137–261 (2006)Google Scholar
  10. 10.
    D’Andrea, A., Kac, V.G.: Structure theory of finite conformal algebras. Sel. Math. 4, 377–418 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    D’Andrea, A., Marchei, G.: Representations of Lie pseudoalgebras with coefficients. J. Algebr. (2010)Google Scholar
  12. 12.
    D’Andrea, A., Marchei, G.: A root space decomposition for finite vertex algebras. Doc. Math. 17, 783–805 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    De Commer, K.: On projective representations for compact quantum groups. J. Funct. Anal. 260, 3596–3644 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kac, V.G.: Vertex Algebras for Beginners. Mathematical Lecture Series, vol. 10, 2nd Edn, 1996. AMS (1998)Google Scholar
  15. 15.
    Rumin, M.: Formes différentielles sur les variétés de contact. J. Diff. Geom. 39, 281–330 (1994)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Sweedler, M.E.: Cohomology of algebras over Hopf algebras. Trans. AMS 133, 205–239 (1968)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di Roma “La Sapienza”RomeItaly

Personalised recommendations

Cite chapter

Buy options