A comonadicity theorem for partial comodules

TL;DR

This paper proves that the category of partial comodules over a Hopf algebra is comonadic over vector spaces, providing explicit constructions and exploring partial representations of algebraic groups, with special focus on finite-dimensional cases.

Contribution

It establishes the comonadicity of partial comodules over Hopf algebras and constructs the associated comonad explicitly, especially for finite-dimensional cases.

Findings

Category of partial comodules is comonadic over vector spaces.

Explicit construction of the comonad using topological vector spaces.

Connected linear algebraic groups do not admit partiality.

Abstract

We show that the category of partial comodules over a Hopf algebra $H$ is comonadic over ${\sf Vect}_k$ and provide an explicit construction of this comonad using topological vector spaces. The case when $H$ is finite dimensional is treated in detail. A study of partial representations of linear algebraic groups is initiated; we show that a connected linear algebraic group does not admit partiality.