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Difference between revisions of "Lattice in a Lie group"

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A [[Discrete subgroup|discrete subgroup]] $\Gamma$ of a [[Lie group|Lie group]] $G$ such that $G/\Gamma$ has a finite volume relative to the (induced) $G$-invariant measure.
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A [[discrete subgroup]] $\Gamma$ of a [[Lie group]] $G$ such that $G/\Gamma$ has a finite volume relative to the (induced) $G$-invariant measure.
  
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A lattice of dimension $n$ (or rank) $n$ in a vector space $V$ over $\mathbf R$ or $\mathbf C$ is a free Abelian subgroup in $V$ generated by $n$ linearly independent vectors over $\mathbf R$. A subgroup of the additive group of a finite-dimensional vector space $V$ over $\mathbf R$ is discrete if and only if it is a lattice [[#References|[1]]].
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A lattice of dimension $n$ (or rank $n$) in a vector space $V$ over $\mathbf R$ or $\mathbf C$ is a free Abelian subgroup in $V$ generated by $n$ linearly independent vectors over $\mathbf R$. A subgroup of the additive group of a finite-dimensional vector space $V$ over $\mathbf R$ is discrete if and only if it is a lattice [[#References|[1]]].
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.A. Morris,  "Pontryagin duality and the structure of locally compact Abelian groups" , ''London Math. Soc. Lecture Notes'' , '''29''' , Cambridge Univ. Press  (1977)</TD></TR></table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  S.A. Morris,  "Pontryagin duality and the structure of locally compact Abelian groups" , ''London Math. Soc. Lecture Notes'' , '''29''' , Cambridge Univ. Press  (1977)</TD></TR>
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Latest revision as of 18:43, 11 April 2023

A discrete subgroup $\Gamma$ of a Lie group $G$ such that $G/\Gamma$ has a finite volume relative to the (induced) $G$-invariant measure.

A lattice of dimension $n$ (or rank $n$) in a vector space $V$ over $\mathbf R$ or $\mathbf C$ is a free Abelian subgroup in $V$ generated by $n$ linearly independent vectors over $\mathbf R$. A subgroup of the additive group of a finite-dimensional vector space $V$ over $\mathbf R$ is discrete if and only if it is a lattice [1].

References

[1] S.A. Morris, "Pontryagin duality and the structure of locally compact Abelian groups" , London Math. Soc. Lecture Notes , 29 , Cambridge Univ. Press (1977)


Comments

See also Discrete group of transformations.

How to Cite This Entry:
Lattice in a Lie group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lattice_in_a_Lie_group&oldid=32811
This article was adapted from an original article by r group','../u/u095350.htm','Unitary transformation','../u/u095590.htm','Vector bundle, analytic','../v/v096400.htm')" style="background-color:yellow;">A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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