Hopf algebra

Remarks

Example: the group algebra $KG$ of a group $G$, with $\Delta(g) = g \otimes g$, $\varepsilon(g) = 1$ and $S(g) = g^{-1}$. The check on a basis element $g$: $m(S \otimes \id)\Delta(g) = g^{-1} g = 1 = \varepsilon(g) 1$, and likewise $g g^{-1} = 1$. Example: the universal enveloping algebra of a Lie algebra $\mathfrak{g}$, with $\Delta(x) = x \otimes 1 + 1 \otimes x$, $\varepsilon(x) = 0$ and $S(x) = -x$ for $x \in \mathfrak{g}$. Non-example: the monoid algebra $K[x]$ of $(\N, +)$, with $\Delta(x) = x \otimes x$ and $\varepsilon(x) = 1$, is a bialgebra but not a Hopf algebra. An antipode would need $S(x) x = \varepsilon(x) 1 = 1$, but $x$ has no inverse in $K[x]$. The antipode of a Hopf algebra is unique, because it is the inverse of $\id_H$ in the convolution algebra $\End_K(H)$.