On Takeuchi's correspondence

TL;DR

This paper reviews Takeuchi's correspondence in Hopf algebras, focusing on cases without the faithfulness assumption, and explores open questions related to flatness and coflatness conditions.

Contribution

It extends the understanding of Takeuchi's correspondence by analyzing the non-faithful case and addressing open questions in the theory of Hopf algebras.

Findings

Describes the correspondence without faithfulness assumption

Identifies open questions in flatness and coflatness

Provides insights into the structure of Hopf algebra subobjects

Abstract

In this paper we review the Takeuchi correspondence between right coideal subalgebras and left $H$-module factor coalgebras of a Hopf algebra with bijective antipode. We are especially interested to describe the situation when the faithfulness assumption in the conditions of flatness and coflatness is dropped. This study was motivated by several questions which remain open.