Hopf algebras---Variant notions and reconstruction theorems

TL;DR

This paper surveys various generalizations of Hopf algebras through the lens of the Tannaka reconstruction theorem, highlighting their connections to monoidal categories and representation theory.

Contribution

It provides a comprehensive overview of different Hopf algebra generalizations and their relation to Tannaka reconstruction, emphasizing categorical perspectives.

Findings

Multiple generalizations of Hopf algebras are characterized via Tannaka reconstruction.

The categorical framework links Hopf algebra variants to monoidal and autonomous categories.

The survey clarifies the role of the forgetful functor in these generalizations.

Abstract

Hopf algebras are closely related to monoidal categories. More precise, $k$-Hopf algebras can be characterized as those algebras whose category of finite dimensional representations is an autonomous monoidal category such that the forgetful functor to $k$-vectorspaces is a strict monoidal functor. This result is known as the Tannaka reconstruction theorem (for Hopf algebras). Because of the importance of both Hopf algebras in various fields, over the last last few decades, many generalizations have been defined. We will survey these different generalizations from the point of view of the Tannaka reconstruction theorem.