# Unimodular group

A topological group whose left-invariant Haar measure is right invariant (equivalently, is invariant under the transformation $\alpha \mapsto \alpha^{-1}$). A Lie group $G$ is unimodular if and only if $$| \det \mathrm{Ad}\, g | = 1\ ,\ \ \ (g \in G),$$ where $\mathrm{Ad}$ is the adjoint representation (cf. Adjoint representation of a Lie group). For a connected Lie group $G$ this is equivalent to requiring that $\mathrm{tr}\,\mathrm{ad}\, g = 0$ ($g \in \mathfrak{g}$), where $\mathrm{ad}$ is the adjoint representation of the Lie algebra $\mathfrak{g}$ of $G$. Any compact, discrete or Abelian locally compact group, as well as any connected reductive or nilpotent Lie group, is unimodular.

Sometimes, more rarely, the phrase unimodular group means the group of unimodular matrices (of a given size) over a ring, i.e. the group of matrices of determinant $1$, that is more usually called the "special linear group" , cf. e.g. [a3].