Some interactions between Hopf Galois extensions and noncommutative rings

Abstract

In this memoir, we consider Hopf Galois extensions and families of noncommutative rings known as skew polynomial rings, PBW extensions and skew PBW extensions. We address remarkable examples of Hopf Galois extensions (e.g., Hopf algebras, field extensions, strongly graded algebras, crossed products, principal bundles), some properties and recent advances in the theory such as quantum torsors and Hopf Galois systems. Next, we study the families of noncommutative rings and some of its basic properties and recall remarkable examples appearing in the theory. Finally, some interactions between all families of rings are presented.