Pensieve header: Formulas for the Cartan involution (failed).

Code borrowed from Dequantizator.nb.

Monoblog on Oct 31, 2017:

Representing (𝔤^ϵ).

ME=MatrixExp; MF=MatrixForm;
Simp[sol_] := sol /. {ss_List} :> ss /. C[_]->0/. (var_->val_) :> (var -> Simplify[PowerExpand@val]);

ρt=(({{1, 0}, {0, 1}}));  ρy=(({{0, 0}, {ϵ, 0}}));  ρa=(({{(1+1/ϵ)/2, 0}, {0, -(1-1/ϵ)/2}})); ρx=(({{0, 1}, {0, 0}}));
Simplify@{ρa.ρx-ρx.ρa==ρx, ρa.ρy-ρy.ρa==-ρy, ρx.ρy-ρy.ρx==2ϵ ρa-ρt}

Finding θ

At ℏ=0, we have θ(t,y,a,x)=(-t,-x,-a,-y).

eqn=Simplify[ME[-τ1 ρt].ME[-η1 ρx].ME[-α1 ρa].ME[-ξ1 ρy]==ME[τ0 ρt].ME[η0 ρy].ME[α0 ρa].ME[ξ0 ρx]];
MF /@ eqn

sol=Solve[Thread[Flatten/@eqn], {τ0,η0,α0,ξ0}]

eqn=Simplify[ME[-η1 ρx].ME[0ρa].ME[-ξ1 ρy]==T0*ME[η0 ρy].ME[α0 ρa].ME[ξ0 ρx]];
MF /@ eqn
sol=Simp@Solve[eqn1=Thread[Flatten/@eqn], {T0,η0,α0,ξ0}]

FullSimplify[eqn1/.sol]

eqn=Simplify[ME[-η1 ρx].ME[0ρa].ME[-ξ1 ρy]==ME[η0 ρy].ME[α0 ρa].ME[ξ0 ρx]];
MF /@ eqn
sol=Simp@Solve[eqn1=Thread[Flatten/@eqn], {η0,α0,ξ0}]

FullSimplify[eqn1/.sol]