Inverse of an action

In this section, we illustrate the functionality of the kernel functions on the Lie Group SL_2R. We quote from S. Lie 1930:
"Eine bekannte Gruppe dieser Art ist die folgende:

f[a_, x_] := (x + a_1)/(a_2x + a_3) ;

f[a, x]

(x + a_1)/(x a_2 + a_3)

welche drei Parameter a_1,a_2,a_3 enthält. Führt man die beiden Transformationen

{f[a, x], f[b, x]}

{(x + a_1)/(x a_2 + a_3), (x + b_1)/(x b_2 + b_3)}

nacheinander aus, so erhält man:

f[c, x]

f[b, f[a, x]]//S

(x + c_1)/(x c_2 + c_3)

(x + a_1 + (x a_2 + a_3) b_1)/((x + a_1) b_2 + (x a_2 + a_3) b_3)

wo c_1,c_2,c_3 als Functionen von den a und b durch die Relationen..."
So far from S. Lie. We tranform the above expression to match (x + c_1)/(c_2x + c_3) for proper coefficients c_i.

True

The coefficients c_i are easily read of the last fraction. However, the command GInv solves these kind of inverse problems. However, the coordinates have to be named {x_1,...,x_n} and {y_1,...,y_n} and input to GInv is the expression for {x_1,...,x_n}◦{y_1,...,y_n}.

{(a_1 b_1 + a_2 b_3)/(1 + a_3 b_2), (a_2 + a_1 b_2)/(1 + a_3 b_2), (a_3 b_1 + b_3)/(1 + a_3 b_2)}/.a→x/.b→y

{(x_1 y_1 + x_2 y_3)/(1 + x_3 y_2), (x_2 + x_1 y_2)/(1 + x_3 y_2), (x_3 y_1 + y_3)/(1 + x_3 y_2)}

The output of GInv is a function, which maps {x_1,...,x_n} to the inverse element {x_1, ..., x_n}^(-1)coordinates. If debug mode is on, GInv also states the necessary conditions, on which subset the formula is valid.

debug = True ;

inv = Inv[{(x_1 y_1 + x_2 y_3)/(1 + x_3 y_2), (x_2 + x_1 y_2)/(1 + x_3 y_2), (x_3 y_1 + y_3)/(1 + x_3 y_2)}, {1, 0, 0}]

debug = False ;

eqs: x_1≠0&&y_1 == 1/x_1&&y_2 == -x_2 y_1&&y_3 == -x_3 y_1&& -x_1 + x_2 x_3≠0   {{1/x_1, -x_2/x_1, -x_3/x_1}}

{1/x_1, -x_2/x_1, -x_3/x_1}

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