# 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 | + | * 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 (usually a field) is the IAPS of groups defined as follows:

- Its member is the orthogonal group : the group of matrices such that is the identity matrix
- Its block concatenation map is described as follows:

## 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.