Hopf-Galois algebras and their Poisson structures

Abstract

Hopf-Galois objects are natural generalizations of Hopf algebras with a Galois-theoretic flavor. In this paper, we prove a criterion for an Ore extension of a Hopf-Galois algebra to be a Hopf-Galois algebra, introduce a concept of Poisson Hopf-Galois algebra and establish a relationship between Poisson Hopf-Galois algebras and Poisson Hopf algebras. More importantly, we study Poisson Hopf-Galois structures on Poisson polynomial algebras, and give a necessary and sufficient condition for the Poisson enveloping algebra of a Poisson Hopf-Galois algebra to be a Hopf-Galois algebra.