[[Hopf theory MOC]]
# Sweedler's large Hopf algebra

**Sweedler's large Hopf algebra** $H$ is an infinite-dimensional commutative noncocommutative [[Hopf algebra]] generated as a $\mathbb{K}$-monoid by $\{ x,g,g^{-1} \}$, #m/def/falg/hopf with comultiplication
$$
\begin{align*}
\Delta g &= g \otimes g, & \Delta x &= 1 \otimes x + x \otimes g;
\end{align*}
$$
coünit
$$
\begin{align*}
\epsilon(x) &= 0, & \epsilon(g) &= 1;
\end{align*}
$$
and thus antipous
$$
\begin{align*}
S(x) &= -x\,g^{-1}, & S(g) &= g^{-1}.
\end{align*}
$$
A quotient is [[Sweedler's small Hopf algebra]] $H_{4}$.

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#state/tidy | #lang/en | #SemBr