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?
Best Answer
The terminology is related to topological groups and Haar integrals.
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 $