Hopf algebra

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

 $m\circ(S\otimes\mathrm{id})\circ\Delta=\eta\circ\varepsilon=m\circ(\mathrm{id}% \otimes S)\circ\Delta,$ (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 $\eta(k)=k\mathord{\mathrm{1\!\!\!\>I}}$.