# Bialgebra

From Groupprops

*This article or section of article is sourced from*:Wikipedia

## Contents

## Definition

A **bialgebra** over a field is defined as a set equipped with the structure of a unital associative algebra over as well as a coalgebra over satisfying certain compatibility conditions.

### Notation

- Let be the set and be the underlying field
- Let denote the multiplication and the unit of (for its algebra structure)
- Let denote the comultiplication and the counit of .
- Let be the unique linear map from to itself that sends each pure tensor to .

### Compatibility conditions

The compatibility conditions are as follows:

- Compatibility between multiplication and comultiplication:

- Compatibility between multiplication and counit:

under the canonical identification of with .

- Compatibility between comultiplication and unit:

- Compatibility between unit and counit: