group theory

# Contents

## Idea

Given a field $k$ and a natural number $n \in \mathbb{N}$, the special linear group $SL(n,k)$ (or $SL_n(k)$) is the subgroup of the general linear group $SL(n,k) \subset GL(n,k)$ consisting of those linear transformations that preserve the volume form on the vector space $k^n$. It can be canonically identified with the group of $n\times n$ matrices with entries in $k$ having determinant $1$.

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

Revised on January 11, 2016 17:53:47 by Urs Schreiber (78.35.58.176)