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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]]);

2 Answers 2

3

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 $A_1$.

  • 0
    i know now, but how a1, a2 etc plugged into these type? it seems not a matrix multiply.2012-07-02
  • 0
    @M-Askman: Dear M-Askman, I don't understand your question, and it makes me wonder if you understand what is meant by the classification of Lie algebras into Type A, B, and so on. Each Lie algebra in the classification can be represented (in many different ways) as a Lie algebra of matrices, but there is no simple formula relating the type of the Lie algebra to the kind of matrix groups that give rise to it. Perhaps you should look at some basic literature on classification of Lie groups and Lie algebras (say beginning with the wikipedia article). Regards,2012-07-02
  • 0
    updated Maple code for you to demonstrate the steps, what i learn from lie group do not teach how to do from group to these equations2012-07-11
1

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.

  • 0
    would you mind demonstrate the process of how it use one of type and form this transformation, i would like to practice on other type2012-07-02
  • 0
    @M-Askman, sorry I don't get it. What do you want or expect as example?2012-07-11
  • 0
    i want the steps from a group (matrix) to Mobius, updated Maple code for you to demonstrate from group(matrix) to these formula2012-07-11