# Affine commutative-by-finite Hopf algebras

## TL;DR

This paper studies affine commutative-by-finite Hopf algebras, exploring their structure, homological properties, and module dimensions, especially under semisimplicity and cosemisimplicity conditions.

## Contribution

It provides new structural insights and bounds on simple module dimensions for these Hopf algebras, highlighting constraints under certain conditions.

## Key findings

- Bounds on dimensions of simple modules
- Structural constraints when extension is semisimple and cosemisimple
- Examples of affine commutative-by-finite Hopf algebras

## Abstract

The objects of study in this paper are Hopf algebras $H$ which are finitely generated algebras over an algebraically closed field and are extensions of a commutative Hopf algebra by a finite dimensional Hopf algebra. Basic structural and homological properties are recalled and classes of examples are listed. Bounds are obtained on the dimensions of simple $H$-modules, and the structure of $H$ is shown to be severely constrained when the finite dimensional extension is semisimple and cosemisimple.