Hopf images and inner faithful representations

TL;DR

This paper develops a comprehensive theory of Hopf images and inner faithful representations for Hopf algebras, providing new insights and examples, and establishing a Tannaka duality framework.

Contribution

It introduces a general theory of Hopf images and inner faithful representations, extending classical notions and applying them to various Hopf algebra examples with a Tannaka duality perspective.

Findings

Established a general framework for Hopf images and inner faithful representations.

Analyzed multiple examples including group algebras and enveloping algebras.

Presented a Tannaka duality formulation for Hopf images.

Abstract

We develop a general theory of Hopf image of a Hopf algebra representation, with the associated concept of inner faithful representation, modelled on the notion of faithful representation of a discrete group. We study several examples, including group algebras, enveloping algebras of Lie algebras, pointed Hopf algebras, function algebras, twistings and cotwistings, and we present a Tannaka duality formulation of the notion of Hopf image.