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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?
Max Demirdilek
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1 Answers1

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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.

user10354138
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