special linear group



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.

Last revised on January 11, 2016 at 17:53:47. See the history of this page for a list of all contributions to it.