Unimodular group

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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].


[a1] H. Reiter, "Classical harmonic analysis and locally compact groups" , Clarendon Press (1968) MR0306811 Zbl 0165.15601
[a2] N. Bourbaki, "Intégration" , Eléments de mathématiques , Hermann (1963) pp. Chapt. 7 MR0179291 Zbl 0156.03204
[a3] H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) pp. 45 MR0000255 Zbl 1024.20502
How to Cite This Entry:
Unimodular group. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article