The Tannaka-Krein formalism and (re)presentations of universal quantum groups

TL;DR

This paper explores the Tannaka-Krein formalism in the context of universal quantum groups, focusing on their (re)presentations and the associated universal Hopf algebras, as part of a broader survey in quantum algebra.

Contribution

It provides a detailed survey of the authors' work on Manin's universal Hopf algebras, emphasizing the Tannaka-Krein approach to quantum groups.

Findings

Clarifies the role of Tannaka-Krein formalism in quantum group representations

Highlights the structure of universal Hopf algebras in quantum algebra

Connects the formalism to Manin's universal quantum groups

Abstract

This is a draft version for an extra chapter in the second edition of the book Quantum Groups and Noncommutative Geometry by Yu. I. Manin. We survey our work on Manin's universal Hopf algebras, placing particular emphasis on the Tannaka-Krein formalism.