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special linear group

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Idea

Given a field kk and a natural number nn \in \mathbb{N}, the special linear group SL(n,k)SL(n,k) (or SL n(k)SL_n(k)) is the subgroup of the general linear group SL(n,k)GL(n,k)SL(n,k) \subset GL(n,k) consisting of those linear transformations that preserve the volume form on the vector space k nk^n. It can be canonically identified with the group of n×nn\times n matrices with entries in kk having determinant 11.

This group can be considered as a subvariety of the affine space M n×n(k)M_{n\times n}(k) of square matrices of size nn carved out by the equations saying that the determinant of a matrix is 1. This variety is an algebraic group over kk, and if kk is the field of real or complex numbers then it is a Lie group over kk.

Revised on April 10, 2014 02:06:49 by Urs Schreiber (145.116.131.80)