What is the generalization of Lie group of transformation?

Question

What is the generalization of Lie group of transformation?

I found $a_1x+a_2$ and $(a_1x+a_2)/(a_3x+a_4)$ are also called Lie group of transformation!! It contradicts with what we learn about the Type A, Type B, Type C ...

any book or web link that lists all Lie groups of transformation or family of Lie group of transformation for reference would be nice.

Any one using the following Maple code to demonstrate the steps from matrix to these equations?

Maple Code

restart;
with(LinearAlgebra):
with(linalg):
TypeB := proc(n)

m := Matrix(n);
for i from 1 to n do
    for j from 1 to n do
        if i = j  then
            if i+1 < n  then    
                m[i+1,j] := -1;
            end if;
            if i+1 = n  then    
                m[i+1,j] := -2;
            end if;
            m[i,j] := 2;
            if j+1 <= n  then   
                m[i,j+1] := -1;
            end if;
        end if;
    od;
od;
return m;
end proc;

TypeC := proc(n)

m := Matrix(n);
for i from 1 to n do
    for j from 1 to n do
        if i = j  then
            if i+1 <= n  then   
                m[i+1,j] := -1;
            end if;
            m[i,j] := 2;
            if j+1 <= n  then   
                m[i,j+1] := -1;
            end if;
            if j+1 = n  then    
                m[i,j+1] := -2;
            end if;
        end if;
    od;
od;
return m;
end proc;

TypeD := proc(n)

m := Matrix(n);
for i from 1 to n do
    for j from 1 to n do
        if i = j  then
            if i+1 <= n  then   
                m[i+1,j] := -1;
            end if;
            if i+2 = n then 
                m[i+2,j] := -1;
            end if;
            if i+1 = n  then    
                m[i+1,j] := 0;
            end if;
            m[i,j] := 2;
            if j+1 <= n  then   
                m[i,j+1] := -1;
            end if;
            if j+2 = n  then    
                m[i,j+2] := -1;
            end if;
            if j+1 = n  then    
                m[i,j+1] := 0;
            end if;
        end if;
    od;
od;
return m;
end proc;

TypeA := proc(n)

m := Matrix(n);
for i from 1 to n do
    for j from 1 to n do
        if i = j  then
            if i+1 <= n  then   
                m[i+1,j] := -1;
            end if;
            if i-2 = 0 then 
                m[i+2,j] := -1;
            end if;
            m[i,j] := 2;
            if j+1 <= n  then   
                m[i,j+1] := -1;
            end if;
        end if;
    od;
od;
return m;
end proc;

TypeF4 := proc(n)

m := Matrix(4);
for i from 1 to 4 do
    for j from 1 to 4 do
        if i = j  then
            if i+1 <= 4  then   
                m[i+1,j] := -1;
            end if;
            m[i,j] := 2;
            if j+1 <= 4  then   
                m[i,j+1] := -1;
            end if;
        end if;
    od;
od;
m[3,2] := -2;
return m;
end proc;

TypeG2 := Matrix([[2,-1],[-3,2]]);

Answer 1

Type A, B, etc., is the classification of non-abelian simple Lie algebras over the complex numbers. The classification of simple Lie algebras over the real numbers is more involved, and of simple Lie groups more involved still, since different real Lie algebras can give the same complex Lie algebra, and different simple Lie groups can give the same (real) Lie algebra.

Then, not all Lie groups are simple.

So the group of matrices $\begin{pmatrix} a_1 & a_2 \\ 0 & 1 \end{pmatrix}$ ($a_1 \neq 0$), which is your first example, is a solvable Lie algebra, which is an extension of one abelian Lie algebra (the multiplicative group of invertible real numbers, represented here by the variable $a_1$) by another (the additive group of all real matrices, represented here by the variable $a_2$).

Your second example is the matrix group $SL_2(\mathbb R)$ acting via linear fractional transformations. It is a simple Lie group of type $A_1$.

Answer 2

Concerning your second example, Wiki can tell you the following:

The general form of a Möbius transformation is given by $$ f(z) = \frac{a z + b}{c z + d} $$ where $a, b, c, d$ are any complex numbers satisfying $ad − bc ≠ 0$. (If $ad = bc$ the rational function defined above is a constant and is not considered a Möbius transformation.) In case $c≠0$ this definition is extended to the whole Riemann sphere by defining $$ f(-d/c) = \infty \mbox{ and } f(\infty) = a/c; $$ if c=0 we define $$ f(\infty) = \infty. $$ This turns $f(z)$ into a bijective holomorphic function from the Riemann sphere to the Riemann sphere.

The set of all Möbius transformations forms a group under composition. This group can be given the structure of a complex manifold in such a way that composition and inversion are holomorphic maps. The Möbius group is then a complex Lie group. The Möbius group is usually denoted $\mbox{Aut}(\widehat{\mathbf{C}})$ as it is the automorphism group of the Riemann sphere.