Difference between revisions of "Orthogonal IAPS"

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The '''orthogonal IAPS''' associated to a commutative unital ring <math>R</math> (usually a field) is the [[IAPS of groups]] defined as follows:
 
The '''orthogonal IAPS''' associated to a commutative unital ring <math>R</math> (usually a field) is the [[IAPS of groups]] defined as follows:
  
* Its <math>n^{th}<math> member is the orthogonal group <math>O(n,R)</math>: the group of <math>n \times n</math> matrices <math>A</math> such that <math>AA^t</math> is the identity matrix
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* Its <math>n^{th}</math> member is the orthogonal group <math>O(n,R)</math>: the group of <math>n \times n</math> matrices <math>A</math> such that <math>AA^t</math> is the identity matrix
 
* Its block concatenation map <math>\Phi_{m,n}</math> is described as follows:
 
* Its block concatenation map <math>\Phi_{m,n}</math> is described as follows:
  

Latest revision as of 14:21, 13 September 2017

This article describes a particular IAPS of groups, or family of such IAPSes parametrized by some structure

Definition

The orthogonal IAPS associated to a commutative unital ring R (usually a field) is the IAPS of groups defined as follows:

  • Its n^{th} member is the orthogonal group O(n,R): the group of n \times n matrices A such that AA^t is the identity matrix
  • Its block concatenation map \Phi_{m,n} is described as follows:

\Phi_{m,n}(A,B) = \begin{pmatrix}
A & 0 \\
0 & B \end{pmatrix}

Inside the GL IAPS

Further information: Orthogonal IAPS in GL IAPS

The orthogonal IAPS is a sub-IAPS of the GL IAPS, which comprises the general linear groups. It is in fact a saturated sub-IAPS, and the quotient space can be identified with diagonalizable bilinear forms.