# Integrals of a Hopf algebra: Why that name?

1. Context: The notion of an integral
Let $$H$$ be a Hopf algebra over a field $$\mathbb k$$. We call its $$\mathbb k$$-linear subspace $$I_l(H)= \{x \in H; h \cdot x=\epsilon(h)x \quad for \>all\>h\in H\}$$ the space of left integrals. In other words, $$I_l(H)$$ is the space of left invariants for $$H$$ acting on itself by multiplication. In a similar manner one can define (the space of) right (co)integrals.

Integrals seem to have a wide range of applications. For instance, they appear in a strong "(Hopf algebra) version" of Maschke's theorem, i.e. they are related to the semisimplicity of a Hopf algebra.

2. Question

• Why are integrals called integrals?
• Specifically, I think I overheard someone saying that they can be related to the notion of an integral in calculus. How so?

Let $$G$$ be a compact topological group. A Haar integral on $$G$$ is a linear functional $$\lambda$$ defined on the space of continuous functions $$\mathbb{R}^G = \text{Map}(G,\mathbb{R})$$, which is translation invariant, so $$\forall f \in \mathbb{R}^G, \forall x \in G, \lambda(xf)=\lambda(f)$$. We can restrict this to the Hopf algebra $$H$$ contained in $$\mathbb{R}^G$$ and the map
$$H \longrightarrow \mathbb{R} \\f \mapsto \int_G f(x) d\mu$$ is an integral in $$H^*$$ an precisely is the Haar integral. Indeed $$\int_G f(sx) d\mu = \int_G f(x) d\mu, \forall s \in G$$