Structure of classical (finite and affine) -algebras

  • Alberto De Sole

    Università di Roma La Sapienza, Italy
  • Victor G. Kac

    Massachusetts Institute of Technology, Cambridge, USA
  • Daniele Valeri

    SISSA, Trieste, Italy

Abstract

First, we derive an explicit formula for the Poisson bracket of the classical finite -algebra , the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra and its nilpotent element . On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine -algebra . As an immediate consequence, we obtain a Poisson algebra isomorphism between and the Zhu algebra of . We also study the generalized Miura map for classical -algebras.

Cite this article

Alberto De Sole, Victor G. Kac, Daniele Valeri, Structure of classical (finite and affine) -algebras. J. Eur. Math. Soc. 18 (2016), no. 9, pp. 1873–1908

DOI 10.4171/JEMS/632