Hopf-Galois algebras and their Poisson structures

TL;DR

This paper explores the structure of Hopf-Galois algebras, introduces Poisson Hopf-Galois algebras, and establishes conditions under which certain algebraic extensions retain Hopf-Galois properties, linking them to Poisson structures.

Contribution

It provides a criterion for Ore extensions of Hopf-Galois algebras to be Hopf-Galois and introduces Poisson Hopf-Galois algebras, connecting them with Poisson Hopf algebras.

Findings

Criterion for Ore extension to be Hopf-Galois algebra

Introduction of Poisson Hopf-Galois algebras

Necessary and sufficient condition for Poisson enveloping algebra to be Hopf-Galois

Abstract

As is known to all, Hopf-Galois objects have a significant research value for analyzing tensor categories of comodules and classification questions of pointed Hopf algebras, and are natural generalizations of Hopf algebras with a Galois-theoretic flavour. In this paper, we mainly prove a criterion for an Ore extension of a Hopf-Galois algebra to be a Hopf-Galois algebra, and introduce the conception of Poisson Hopf-Galois algebras, and establish the relationship between Poisson Hopf-Galois algebras and Poisson Hopf algebras. Moreover, we study Poisson Hopf-Galois structures on Poisson polynomial algebras, and mainly give a necessary and sufficient condition for the Poisson enveloping algebra of a Poisson Hopf-Galois algebra to be a Hopf-Galois algebra.