# 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.

#### Comments

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].

#### References

[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 |

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Unimodular group.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Unimodular_group&oldid=33579