Commutative-by-finite Hopf algebras

A commutative-by-finite Hopf algebra is an extension of a commutative Hopf algebra by a finite dimensional Hopf algebra. Examples include many large classes of algebras - commutative Hopf algebras and finite Hopf algebras of course,  group algebras of abelian-by-finite groups, enveloping algebras of Lie algebras in positive characteristic, quantum groups at a root of unity,.... I'll review aspects of their structure and their representation theory, and mention a number of open questions. I'll aim to make the talk accessible without previous knowledge of Hopf algebras. Joint work with Miguel Couto.