# Bialgebra

## Definition

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

### Notation

• Let $B$ be the set and $K$ be the underlying field
• Let $\nabla$ denote the multiplication and $\eta$ the unit of $B$ (for its algebra structure)
• Let $\Delta$ denote the comultiplication and $\epsilon$ the counit of $B$.
• Let $\tau$ be the unique linear map from $B \otimes B$ to itself that sends each pure tensor $x \otimes y$ to $y \otimes x$.

### Compatibility conditions

The compatibility conditions are as follows:

• Compatibility between multiplication and comultiplication:

$\Delta \circ \nabla = (\nabla \otimes \nabla) \circ (id \otimes \tau \otimes id) \circ (\Delta \otimes \Delta)$

• Compatibility between multiplication and counit:

$\epsilon \otimes \epsilon = \epsilon \circ \nabla$ under the canonical identification of $K \otimes K$ with $K$.

• Compatibility between comultiplication and unit:

$\Delta \circ \eta = \eta \otimes \eta$

• Compatibility between unit and counit:

$\epsilon \circ \eta = id$