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?
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 $