Monthly Archives: January 2017

SL(2,IR) is the commutator subgroup of GL(2,IR)

Here is a proof of the above fact.

Let N be the commutator subgroup of the general linear group GL(2,\mathbb R); i.e.,

N=\langle ABA^{-1}B^{-1}:A,B\in GL(2,\mathbb R)\rangle.

First, it is clear that N is contained in the special linear group SL(2,\mathbb R), since \det(ABA^{-1}B^{-1})=1 for any A,B\in GL(2,\mathbb R). Next, we claim that N contains all matrices

\begin{pmatrix} 1 & b\\ 0 & 1\end{pmatrix}.

This follows from noting that

\begin{pmatrix} 1 & b\\ 0 & 1\end{pmatrix}=\begin{pmatrix} 1 & b\\ 0 & b\end{pmatrix}\begin{pmatrix} 1 & 1\\ 0 & 1\end{pmatrix}\begin{pmatrix} 1 & b\\ 0 & b\end{pmatrix}^{-1}\begin{pmatrix} 1 & 1\\ 0 & 1\end{pmatrix}^{-1}.

By taking transposes, it also follows that N contains all matrices

\begin{pmatrix} 1 & 0\\ c & 1\end{pmatrix}.

Further, N contains all matrices

\begin{pmatrix} a & 0\\ 0 & 1/a\end{pmatrix}

since

\begin{pmatrix} a & 0\\ 0 & 1/a\end{pmatrix}=\begin{pmatrix} a & 0\\ 0 & 1\end{pmatrix}\begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}\begin{pmatrix} a & 0\\ 0 & 1\end{pmatrix}^{-1}\begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}^{-1}

for any a\neq 0.

Now let

\begin{pmatrix} a & b\\ c & d\end{pmatrix}\in SL(2,\mathbb R).

Then ad-bc=1. Using the above results,

\begin{pmatrix} a & b\\ c & d\end{pmatrix}=\begin{pmatrix} 1 & 0\\ c/a & 1\end{pmatrix}\begin{pmatrix} 1 & ab\\ 0 & 1\end{pmatrix}\begin{pmatrix} a & 0\\ 0 & 1/a\end{pmatrix}\in N

if a\neq 0, and

\begin{pmatrix} a & b\\ c & d\end{pmatrix}=\begin{pmatrix}0&1\\-1&0\end{pmatrix}\begin{pmatrix}1&-\frac{d}{b}\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ ab&1\end{pmatrix}\begin{pmatrix}1/b&0\\ 0&b\end{pmatrix}\in N

if b\neq 0, and the latter since

\begin{aligned}\begin{pmatrix} 0 & -1\\ 1 & 0\end{pmatrix}=&\begin{pmatrix}x&y\\0&-x-y\end{pmatrix}\begin{pmatrix}-x-y&0\\ x&y\end{pmatrix}\begin{pmatrix}x&y\\0&-x-y\end{pmatrix}^{-1}\begin{pmatrix}-x-y&0\\ x&y\end{pmatrix}^{-1}\\ \in &N\end{aligned}

for any x,y,x+y\neq 0. Thus SL(2,\mathbb R)\subseteq N, i.e., N=SL(2,\mathbb R).

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