Larson–Sweedler theorem

Remarks

The proof applies the fundamental theorem of Hopf modules to the dual space $H^*$, which is a Hopf module over $H$. This gives an isomorphism $H^* \cong (H^*)^{\mathrm{co}H} \otimes H$, and the coinvariants of $H^*$ are the left integrals on $H$. Comparing dimensions shows that this space is one-dimensional, and the isomorphism $H^* \cong H$ of $H$-modules is the Frobenius property; applying this to $H^*$ gives the statement about integrals in $H$. Example: for a finite group $G$, the left integrals in $KG$ are the multiples of $\sum_{g \in G} g$. Indeed, if $\Lambda = \sum_g a_g g$ satisfies $h \Lambda = \Lambda$ for all $h \in G$, then all $a_g$ are equal. Without finite dimension the conclusion fails: in $K[x]$ with $x$ primitive the only integral is $0$. The same work shows that the antipode of a finite-dimensional Hopf algebra is bijective.

Going deeper

For a student

The theorem is the starting point for the structure theory of finite-dimensional Hopf algebras. Together with it, Larson and Sweedler proved the Hopf algebra form of Maschke's theorem: $H$ is semisimple if and only if $\varepsilon(\Lambda) \ne 0$ for a nonzero left integral $\Lambda$. For $H = KG$ this gives $\varepsilon(\sum_g g) = \abs{G}$, and so recovers the classical condition that the characteristic of $K$ does not divide $\abs{G}$. Integrals also define the distinguished group-like elements of $H$ and $H^*$, which appear in Radford's formula for $S^4$.