1. Definitions
- We call a Hopf algebra $H$ unimodular if the space of left integrals $I_l(H)$ is equal to the space of right integrals $I_r(H)$.
- We call a square integer matrix $M$ unimodular if $det(M)=\pm 1$.
- Apparently, there exists a notion of unimodular group: "a locally compact group whose left Haar measure equals its right Haar measure.”
2. Questions
- (How) are these three notions related?
- I haven't heard of unimodular groups, let alone locally compact groups or the Haar measure before. However, looking at this answer here, the unimodularity of a Hopf algebra seems to be somehow related to unimodular groups. How so?
Yes, they are related.
A Lie group $G$ is unimodular if and only if, for all $g\in G$, $\operatorname{Ad}g\colon\mathfrak{g}\to\mathfrak{g}$ is a unimodular matrix when you choose a basis of $\mathfrak{g}$.
For a locally compact group $G$, we know there exists, up to multiplicative constants, a unique left-invariant Haar measure and similarly right-invariant. We can also build a Hopf algebra of continuous functions $H\subseteq\mathbb{R}^G$. Now you can choose to integrate a function with respect to the left-invariant Haar measure or right-invariant Haar measure and these give you the two integrals for the Hopf algebra. If $G$ is unimodular, any left-invariant Haar measure is also right-invariant, so the two integrals agree, giving you the notion of unimodular Hopf algebra.