View source for Lattice in a Lie group

A [[Discrete subgroup|discrete subgroup]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057650/l0576501.png" /> of a [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057650/l0576502.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057650/l0576503.png" /> has a finite volume relative to the (induced) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057650/l0576504.png" />-invariant measure.

A lattice of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057650/l0576506.png" /> (or rank) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057650/l0576508.png" /> in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057650/l0576509.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057650/l05765010.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057650/l05765011.png" /> is a free Abelian subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057650/l05765012.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057650/l05765013.png" /> linearly independent vectors over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057650/l05765014.png" />. A subgroup of the additive group of a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057650/l05765015.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057650/l05765016.png" /> is discrete if and only if it is a lattice [[#References|[1]]].

====References====
<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>



====Comments====
See also [[Discrete group of transformations|Discrete group of transformations]].