Hopf algebra

A Hopf algebra is a bialgebra $A$ over a field $𝕂$ with a $𝕂$-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 $\eta :𝕂\to A$ is the unit map $\eta (k)=k1\mathrm{I}$.

In of a commutative diagram: