# Hopf algebra

## Contents

## History

### Motivation

The idea behind Hopf algebras was to generalize certain properties of the inverse map in the group algebra (which is the map that sends the basis elements to their inverses in the group and is extended linearly). It is/was also hoped that many properties that we study for groups and their group algebras, could also be studied for Hopf algebras.

## Definition

A **Hopf algebra** over a field is defined as a bialgebra over that field with an additional operation, called the **antipode map**, satisfying certain compatibility conditions.

### Notation

- Let be the set and the base field
- Let denote the multiplication and the unit
- Let denote the comultiplication and the counit
- Let be a linear map which we call the
**antipode map**

### Compatibility conditions

We require that equipped with the given multiplication, unit, comultiplication and counit forms a bialgebra, and that the following compatibility conditions hold:

## Properties

Check out Category:Properties of Hopf algebras