---
tag: 1BK2B286W3
kind: definition
name: Integral of a Hopf algebra
msc: [16T05]
defines: RJA5026RGX
uses: [7XFYP0N84H, MNP2HBWB5Q]
sources:
  - url: https://en.wikipedia.org/wiki/Hopf_algebra
---

## Definition

Let $K$ be a field and let $H$ be a [[Hopf algebra|7XFYP0N84H]] over $K$ with counit $\varepsilon$.
A *left integral* in $H$ is an element $\Lambda \in H$ such that $h \Lambda = \varepsilon(h) \Lambda$ for all $h \in H$.
A *right integral* in $H$ is an element $\Lambda \in H$ such that $\Lambda h = \varepsilon(h) \Lambda$ for all $h \in H$.

## 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|MNP2HBWB5Q]] $KG$ 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 $\sum h_{(1)} \lambda(h_{(2)}) = \lambda(h) 1$.