Namespaces
Variants
Actions

Difference between revisions of "Linear classical group"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A group of non-singular linear transformations of a finite-dimensional [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l0591201.png" /> over a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l0591202.png" /> that is a [[Classical group|classical group]] (see also [[Linear group|Linear group]]). The most important types of linear classical groups are the following: the [[General linear group|general linear group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l0591203.png" />, the [[Special linear group|special linear group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l0591204.png" /> and the [[Unitary group|unitary group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l0591205.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l0591206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l0591207.png" /> is a Hermitian or skew-Hermitian form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l0591208.png" />, relative to an involution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l0591209.png" />). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l05912010.png" /> is also commutative, special important cases are: the [[Symplectic group|symplectic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l05912011.png" /> and the [[Orthogonal group|orthogonal group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l05912012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l05912013.png" /> a quadratic form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l05912014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059120/l05912015.png" /> of characteristic not 2).
+
<!--
 +
l0591201.png
 +
$#A+1 = 15 n = 0
 +
$#C+1 = 15 : ~/encyclopedia/old_files/data/L059/L.0509120 Linear classical group
 +
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}}
 +
 
 +
A group of non-singular linear transformations of a finite-dimensional [[Vector space|vector space]] $  E $
 +
over a skew-field $  K $
 +
that is a [[Classical group|classical group]] (see also [[Linear group|Linear group]]). The most important types of linear classical groups are the following: the [[General linear group|general linear group]] $  \mathop{\rm GL} _ {n} ( K) $,  
 +
the [[Special linear group|special linear group]] $  \mathop{\rm SL} _ {n} ( K) $
 +
and the [[Unitary group|unitary group]] $  U _ {n} ( K , f  ) $(
 +
where $  n = \mathop{\rm dim}  E $
 +
and $  f $
 +
is a Hermitian or skew-Hermitian form on $  E $,  
 +
relative to an involution of $  K $).  
 +
When $  K $
 +
is also commutative, special important cases are: the [[Symplectic group|symplectic group]] $  \mathop{\rm Sp} _ {n} ( K) $
 +
and the [[Orthogonal group|orthogonal group]] $  O _ {n} ( K , f  ) $(
 +
$  f $
 +
a quadratic form on $  E $
 +
and $  K $
 +
of characteristic not 2).

Latest revision as of 22:17, 5 June 2020


A group of non-singular linear transformations of a finite-dimensional vector space $ E $ over a skew-field $ K $ that is a classical group (see also Linear group). The most important types of linear classical groups are the following: the general linear group $ \mathop{\rm GL} _ {n} ( K) $, the special linear group $ \mathop{\rm SL} _ {n} ( K) $ and the unitary group $ U _ {n} ( K , f ) $( where $ n = \mathop{\rm dim} E $ and $ f $ is a Hermitian or skew-Hermitian form on $ E $, relative to an involution of $ K $). When $ K $ is also commutative, special important cases are: the symplectic group $ \mathop{\rm Sp} _ {n} ( K) $ and the orthogonal group $ O _ {n} ( K , f ) $( $ f $ a quadratic form on $ E $ and $ K $ of characteristic not 2).

How to Cite This Entry:
Linear classical group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_classical_group&oldid=47649
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article