# Hopf algebra

A Hopf algebra^{} is a bialgebra^{} $A$ over a field $\mathbb{K}$ with a $\mathbb{K}$-linear map $S:A\to A$,
called the antipode, such that

$$m\circ (S\otimes \mathrm{id})\circ \mathrm{\Delta}=\eta \circ \epsilon =m\circ (\mathrm{id}\otimes S)\circ \mathrm{\Delta},$$ | (1) |

where $m:A\otimes A\to A$ is the multiplication map $m(a\otimes b)=ab$ and $\eta :\mathbb{K}\to A$ is the unit map $\eta (k)=k1\mathrm{I}$.

In of a commutative diagram^{}: