Orthogonal IAPS

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.