Namespaces
Variants
Actions

Difference between revisions of "Diagonal subgroup"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
The subgroup of a Cartesian power of a given group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315401.png" /> consisting of all elements with identical components. For instance, the diagonal group of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315402.png" /> is the group of pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315404.png" />.
+
<!--
 +
d0315401.png
 +
$#A+1 = 15 n = 0
 +
$#C+1 = 15 : ~/encyclopedia/old_files/data/D031/D.0301540 Diagonal subgroup
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
 +
{{TEX|auto}}
 +
{{TEX|done}}
  
 +
The subgroup of a Cartesian power of a given group  $  G $
 +
consisting of all elements with identical components. For instance, the diagonal group of the product  $  G \times G $
 +
is the group of pairs  $  ( g , g ) $,
 +
$  g \in G $.
  
 
====Comments====
 
====Comments====
The phrase diagonal group (or diagonal subgroup) is also used for the subgroup scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315405.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315406.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315407.png" /> whose points with values in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315408.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315409.png" /> are the diagonal invertible matrices with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d03154010.png" />.
+
The phrase diagonal group (or diagonal subgroup) is also used for the subgroup scheme $  D ( n, k) $
 +
of $  \mathop{\rm GL} ( n, k) $
 +
over a field $  k $
 +
whose points with values in a $  k $-
 +
algebra $  R $
 +
are the diagonal invertible matrices with coefficients in $  R $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d03154011.png" /> be a commutative group. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d03154012.png" /> from commutative rings with unit element to groups then defines a [[Group scheme|group scheme]]. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d03154013.png" /> is the group of invertible elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d03154014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d03154015.png" /> is the category of groups. Group schemes isomorphic to such group schemes are called diagonizable group schemes.
+
Let $  \Gamma $
 +
be a commutative group. The functor $  D ( \Gamma ) : R \rightarrow {\mathcal G} {\mathcal r} ( \Gamma , R  ^ {*} ) $
 +
from commutative rings with unit element to groups then defines a [[Group scheme|group scheme]]. Here $  R  ^ {*} $
 +
is the group of invertible elements of $  R $
 +
and $  {\mathcal G} {\mathcal r} $
 +
is the category of groups. Group schemes isomorphic to such group schemes are called diagonizable group schemes.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Demazure,  P. Gabriel,  "Groupes algébriques" , '''1''' , North-Holland  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Demazure,  P. Gabriel,  "Groupes algébriques" , '''1''' , North-Holland  (1970)</TD></TR></table>

Latest revision as of 17:33, 5 June 2020


The subgroup of a Cartesian power of a given group $ G $ consisting of all elements with identical components. For instance, the diagonal group of the product $ G \times G $ is the group of pairs $ ( g , g ) $, $ g \in G $.

Comments

The phrase diagonal group (or diagonal subgroup) is also used for the subgroup scheme $ D ( n, k) $ of $ \mathop{\rm GL} ( n, k) $ over a field $ k $ whose points with values in a $ k $- algebra $ R $ are the diagonal invertible matrices with coefficients in $ R $.

Let $ \Gamma $ be a commutative group. The functor $ D ( \Gamma ) : R \rightarrow {\mathcal G} {\mathcal r} ( \Gamma , R ^ {*} ) $ from commutative rings with unit element to groups then defines a group scheme. Here $ R ^ {*} $ is the group of invertible elements of $ R $ and $ {\mathcal G} {\mathcal r} $ is the category of groups. Group schemes isomorphic to such group schemes are called diagonizable group schemes.

References

[a1] M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , North-Holland (1970)
How to Cite This Entry:
Diagonal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagonal_subgroup&oldid=46643
This article was adapted from an original article by Yu.I. Merzlyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article