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Difference between revisions of "Unimodular group"

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A [[Topological group|topological group]] whose left-invariant [[Haar measure|Haar measure]] is right invariant (equivalently, is invariant under the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095350/u0953501.png" />). A Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095350/u0953502.png" /> is unimodular if and only if
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A [[Topological group|topological group]] whose left-invariant [[Haar measure|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
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095350/u0953503.png" /></td> </tr></table>
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| \det \mathrm{Ad}\, g | = 1\ ,\ \ \ (g \in G),
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095350/u0953504.png" /> is the adjoint representation (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]). For a connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095350/u0953505.png" /> this is equivalent to requiring that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095350/u0953506.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095350/u0953507.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095350/u0953508.png" /> is the adjoint representation of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095350/u0953509.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095350/u09535010.png" />. Any compact, discrete or Abelian locally compact group, as well as any connected reductive or nilpotent Lie group, is unimodular.
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where $\mathrm{Ad}$ is the adjoint representation (cf. [[Adjoint representation of a Lie group|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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095350/u09535011.png" />, that is more usually called the "special linear groupspecial linear group" , cf. e.g. [[#References|[a3]]].
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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. [[#References|[a3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Reiter, "Classical harmonic analysis and locally compact groups" , Clarendon Press (1968) {{MR|0306811}} {{ZBL|0165.15601}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Intégration" , ''Eléments de mathématiques'' , Hermann (1963) pp. Chapt. 7 {{MR|0179291}} {{ZBL|0156.03204}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) pp. 45 {{MR|0000255}} {{ZBL|1024.20502}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Reiter, "Classical harmonic analysis and locally compact groups" , Clarendon Press (1968) {{MR|0306811}} {{ZBL|0165.15601}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Intégration" , ''Eléments de mathématiques'' , Hermann (1963) pp. Chapt. 7 {{MR|0179291}} {{ZBL|0156.03204}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) pp. 45 {{MR|0000255}} {{ZBL|1024.20502}} </TD></TR></table>

Latest revision as of 18:19, 12 October 2014

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
How to Cite This Entry:
Unimodular group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unimodular_group&oldid=24171
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article