GL IAPS
From Apstheory
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[edit] Definition
[edit] Symbol-free definition
Given a ring, the GL IAPS over that ring is an IAPS of groups whose member at a given index is the general linear group of that index, with the block concatenation maps being the usual block concatenation of matrices.
[edit] Definition with symbols
Given a ring Failed to parse (Can't write to or create math temp directory): R , the GL IAPS over Failed to parse (Can't write to or create math temp directory): R , denoted Failed to parse (Can't write to or create math temp directory): GL(R) , associates to each Failed to parse (Can't write to or create math temp directory): n
the group Failed to parse (Can't write to or create math temp directory): GL_n(R)
, viz the group of invertible Failed to parse (Can't write to or create math temp directory): n
→Failed to parse (Can't write to or create math temp directory): n matrices over Failed to parse (Can't write to or create math temp directory): R
. The block concatenation map Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}
takes a pair Failed to parse (Can't write to or create math temp directory): (a,b) of invertible matrices to a block matrix with Failed to parse (Can't write to or create math temp directory): a as the top left block, Failed to parse (Can't write to or create math temp directory): b as the bottom right block, and the other two blocks being zero.
[edit] As a functor
The map sending a ring to the corresponding GL IAPS is an IAPS functor from the category of rings to the category of groups. Set-theoretically (and monoid-theoretically), it can be viewed as a sub-functor of the matrix functor. Ring-theoretically, it is the unit group functor associated with the matrix functor.
[edit] Property theory
[edit] Sub-functors
Important sub-IAPSes of the general linear IAPS include:
- Special linear IAPS is the IAPS of matrices of determinant 1. It is a constant-quotient normal sub-IAPS.
- Orthogonal IAPS is the IAPS of matrices Failed to parse (Can't write to or create math temp directory): m
satisfying Failed to parse (Can't write to or create math temp directory): mm^T = 1
. It is a saturated sub-IAPS.
- Symplectic IAPS
- Special orthogonal IAPS is the intersection of the special linear IAPS and the orthogonal IAPS.
- Borel IAPS is the IAPS comprising upper triangular matrices.
Each of these can also be viewed as sub-functors of the GL functor.
