Difference between revisions of "Diagonal subgroup"
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| + | 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 | + | 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 | + | 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
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