Pensieve header: Reproducing commutation relations from a representation (proves the math wrong!).

Some Shortcuts

ME[x_] := MatrixExp[x]; MF[x_] := MatrixForm[x];

Representing ϵyax

(Borrowed from pensieve://2017-08/Multi-beta-yax.nb and pensieve://2017-09/QuantizedLogos.nb)

q=(E^(ℏ γ ϵ)); ρI=(({{1, 0}, {0, 1}})); ρy=(({{0, 0}, {-1, 0}})); ρa=(({{0, 0}, {0, -γ}}));ρx=(({{0, ((-1+(E^(-γ ϵ ℏ)))/ℏ)}, {0, 0}})); ρb=(({{ϵ, 0}, {0, 0}}));
ρt=Simplify[ϵ ρa-γ ρb]; ρA=ME[-ℏ ϵ ρa];  ρB=ME[-ℏ γ ρb];  ρT=ME[ℏ ρt];
(#->MF@Simplify@ToExpression@#)& /@ {"{{q}}", "ρy","ρa","ρx","ρb", "ρt", "ρA", "ρB", "ρT", "ρI"}

{ρa.ρx-ρx.ρa==γ ρx, ρx.ρA==q ρA.ρx, ρa.ρy-ρy.ρa==-γ ρy, ρb.ρy-ρy.ρb==-ϵ ρy, ρx.ρy-q ρy.ρx==(ρI-ρT.ρA.ρA)/ℏ} // Simplify

The 2-Stitch of Exponentials

eqn=(ME[τ1 ρt].ME[η1 ρy].ME[α1 ρa].ME[ξ1 ρx].ME[τ2 ρt].ME[η2 ρy].ME[α2 ρa].ME[ξ2 ρx])==(T0 ME[η0 ρy].ME[α0 ρa].ME[ξ0 ρx]);
sol=Solve[Thread[Flatten/@Expand[eqn]], {T0,η0,α0,ξ0}][[1]];
ϕ=sol /. (T0->ℰ_) :> (τ0 -> ((Log[ℰ])/(-ϵ γ))) /. (v_->ℰ_) :> (v->FullSimplify@PowerExpand[ℰ])

Φ=Simplify[t0 τ0 + y0 η0 + a0 α0 + x0 ξ0 /. ϕ]

Collect[Limit[Φ, ϵ->0], {t0, y0, a0,xξ0}]

Reproducing the Commutation Relations

Simplify@PowerExpand[((∂_(α1,ξ2))(E^Φ)-(∂_(ξ1,α2))(E^Φ)) /. {τ1|η1|α1|ξ1|τ2|η2|α2|ξ2 -> 0}]

Simplify@PowerExpand[((∂_(α1,η2))(E^Φ)-(∂_(η1,α2))(E^Φ)) /. {τ1|η1|α1|ξ1|τ2|η2|α2|ξ2 -> 0}]

qbxy=Simplify@PowerExpand[((∂_(ξ1,η2))(E^Φ)-q(∂_(η1,ξ2))(E^Φ)) /. {τ1|η1|α1|ξ1|τ2|η2|α2|ξ2 -> 0}]

Limit[qbxy, ϵ->0]