References for affine groups of $\mathbb R^n$

Question

I should study affine group $\operatorname{Aff} (\mathbb R^n)$ in a Lie group and Lie algebras course but I have almost no information about it. So please if some one can suggest any books or links for more information about this subject, I will be very grateful !

Answer 1

The affine group ${\rm Aff}(V)$ for a vector space $V$ is by definition given by $GL(V)\ltimes V$. It is easy to see that it is isomorphic to the group of affine transformations $L_{A,v}\colon V\rightarrow V, \; x\mapsto Ax+v$. What is its Lie algebra?

Recall that the semidirect product $\mathfrak{g} \ltimes V$ with abelian Lie algebra $V$ becomes a Lie algebra by \begin{align*} [(x,v),(y,w)] & = ([x,y], D(x)(w)-D(y)(v)), \end{align*} for $x,y\in \mathfrak{g}$ and $v,w\in V$ and a representation $D\colon \mathfrak{g} \rightarrow \mathfrak{gl}(V)$.

For $\mathfrak{g}=\mathfrak{gl}(V)$ and $D={\rm id}$ we obtain the Lie algebra $$ \mathfrak{aff} (V):=\mathfrak{gl}(V)\ltimes V $$ with Lie bracket $[(A,v),(B,w)]=([A,B], Aw-Bv)$. Identifying $V$ with $\Bbb R^n$, we obtain that $\mathfrak{aff}(V)$ is isomorphic to the following subalgebra of $\mathfrak{gl}_{n+1}(\Bbb R)$: $$ \mathfrak{aff} (V) \cong \left \{ \begin{pmatrix} A & v \\ 0 & 0 \end{pmatrix} \mid A \in M_n(\Bbb R),\, v\in \Bbb R^n \right \}. $$ The Lie bracket here is given by the commutator of matrices,

$$ \Bigl[ \begin{pmatrix} A & v \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} B & w \\ 0 & 0 \end{pmatrix} \Bigr] = \begin{pmatrix} [A,B] & Aw-Bv \\ 0 & 0 \end{pmatrix} $$