Orthogonal IAPS
From Apstheory
Contents |
[edit] Definition
Given a unital ring, the orthogonal IAPS over that unital ring is defined as an IAPS of groups where a square matrix Failed to parse (Can't write to or create math temp directory): A
lies inside the IAPS if and only if Failed to parse (Can't write to or create math temp directory): AA^T = I
.
Equivalently, the Failed to parse (Can't write to or create math temp directory): n^{th}
member of the orthogonal IAPS is defined as the group of matrices Failed to parse (Can't write to or create math temp directory): A that preserve the standard Euclidean inner product.
[edit] As a functor
The orthogonal IAPS is an APS sub-functor of the IAPS of groups.
[edit] Properties
[edit] Saturated
The orthogonal IAPS is a saturated sub-IAPS of the GL IAPS. The quotient of the GL IAPS by the orthogonal IAPS is the set-theoretic IAPS which, for any Failed to parse (Can't write to or create math temp directory): n , describes the space of bilinear forms equivalent to the identity form (in the case of reals, this is the space of positive definite symmetric bilinear forms; for complexi numbers, this is the space of all symmetric bilinear forms).
