Integral of a Hopf algebra

Remarks

The left integrals form a subspace of $H$, often written $\int_H^l$. Example: for a finite group $G$, the element $\Lambda = \sum_{g \in G} g$ of the group algebra $KG$ is a left and right integral. The check: for $h \in G$, $h \Lambda = \sum_{g} hg = \Lambda = \varepsilon(h) \Lambda$, since $g \mapsto hg$ is a bijection of $G$; linearity gives all $h \in KG$. Non-example: in $KG$ with $G \ne 1$, the element $1$ is not an integral, because $g \cdot 1 = g \ne 1 = \varepsilon(g) 1$ for $g \ne 1$. In $K[x]$ with $\Delta(x) = x \otimes 1 + 1 \otimes x$ and $\varepsilon(x) = 0$, a left integral must satisfy $x \Lambda = 0$, so $0$ is the only integral. Some authors call these elements integrals in $H$, and call integrals on $H$ the linear forms $\lambda \colon H \to K$ with $\sum h_{(1)} \lambda(h_{(2)}) = \lambda(h) 1$.