GL IAPS

From Apstheory

Revision as of 13:10, 30 December 2006 (edit) Vipul (Talk | contribs) (Started the page) ← Previous diff		Current revision as of 14:34, 30 December 2006 (edit) (undo) Vipul (Talk | contribs) (→Parabolic structure)
Line 28:		Line 28:
	===Parabolic structure===		===Parabolic structure===
		+
		+	[[Category: IAPSes of groups]]

---

Current revision as of 14:34, 30 December 2006

Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 1.3 As a functor 2 Property theory 2.1 Sub-functors 2.2 Parabolic structure

[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.

[edit] Parabolic structure