Integral of a Hopf algebra

Summary

A left integral in a Hopf algebra $H$ is an element $\Lambda$ that absorbs left multiplication through the counit: $h \Lambda = \varepsilon(h) \Lambda$ for all $h \in H$. For a finite group algebra $KG$, the sum of all group elements is an integral. Integrals generalize averaging over a finite group and the Haar measure of a compact group. By the Larson–Sweedler theorem, the left integrals of a finite-dimensional Hopf algebra form a one-dimensional space. Such an $H$ is semisimple exactly when $\varepsilon(\Lambda) \ne 0$ for a nonzero left integral $\Lambda$.