Such automorphisms have the form f(z)=az+b,\quad a,b\in \Bbb C, a\neq 0.
(To be proved later) If f(z)=az+b,g(z)=a'z+b', then f\circ g(z)=aa'z+(ab'+b)\\f^{-1}(z)=a^{-1}z-a^{-1}b.
Expressed in matrix form, we have \begin{pmatrix}a&b\\0&1\end{pmatrix}\begin{pmatrix}a'&b'\\0&1\end{pmatrix}=\begin{pmatrix}aa'&ab'+b\\0&1\end{pmatrix}\\ \begin{pmatrix}a&b\\0&1\end{pmatrix}^{-1}=\begin{pmatrix}a^{-1}&-a^{-1}b\\0&1\end{pmatrix},
which encode the formulas for composing and inverting automophisms of the plane.
The correspondence f(z)=az+b \longleftrightarrow \begin{pmatrix}a&b\\0&1\end{pmatrix}
is a natural group isomorphism. We now define the parabolic group as P=\bigg\{\begin{pmatrix}a&b\\0&1\end{pmatrix}\bigg|\; a,b\in \Bbb C, a\neq 0\bigg\}.
Thus \text{Aut}(C)\cong P.
Two subgroups of P are its Levi component M=\bigg\{\begin{pmatrix}a&0\\0&1\end{pmatrix}\bigg|\; a\in \Bbb C, a\neq 0\bigg\},
describing the dilations f(z)=az, and its unipotent radical N=\bigg\{\begin{pmatrix}1&b\\0&1\end{pmatrix}\bigg|\; b\in \Bbb C\bigg\},
describing the translations f(z)=z+b. The parabolic group takes the form P=MN=NM
since \begin{pmatrix}a&b\\0&1\end{pmatrix}=\begin{pmatrix}a&0\\0&1\end{pmatrix}\begin{pmatrix}1&\color{blue}{a^{-1}b}\\0&1\end{pmatrix}=\begin{pmatrix}1&b\\0&1\end{pmatrix}\begin{pmatrix}a&0\\0&1\end{pmatrix}.
Geometrically, any affine map is the composition of a translation and a dilation, in which the order of translation and dilation does not matter. Also, M normalizes N, meaning that m^{-1}nm\in N\quad \forall m\in M, n\in N
since \begin{pmatrix}a^{-1}&0\\0&1\end{pmatrix}\begin{pmatrix}1&b\\0&1\end{pmatrix}\begin{pmatrix}a&0\\0&1\end{pmatrix}=\begin{pmatrix}1&a^{-1}b\\0&1\end{pmatrix}.
Geometrically, it says that a dilation followed by a translation followed by the reciprocal dilation is again a translation. On the other hand, N does not normalize M.
No comments:
Post a Comment