Dror Bar-Natan: Talks: Les Diablerets-1708:

Efficient PBW-like Computations

Goal.

In (𝔤^ϵ)=⟨t,y,a,x⟩/([a,x]=x, [a,y]=-y, [x,y]=t-2ϵ a, [t,*]=0), compute

Representing (𝔤^ϵ).

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==ρt-2ϵ ρa}

ME=MatrixExp;
eqn=(ME[(Subscript[η, 1])ρy].ME[(Subscript[α, 1])ρa].ME[(Subscript[ξ, 1])ρx].ME[(Subscript[η, 2])ρy].ME[(Subscript[α, 2])ρa].ME[(Subscript[ξ, 2])ρx]==ME[τ ρt].ME[η ρy].ME[α ρa].ME[ξ ρx])

sol=Solve[Thread[Flatten/@eqn], {τ,η,α,ξ}][[1]]

ME=MatrixExp;
eqn=Simplify[ME[η1 ρy].ME[α1 ρa].ME[ξ1 ρx].ME[η2 ρy].ME[α2 ρa].ME[ξ2 ρx]==τ0*ME[η0 ρy].ME[α0 ρa].ME[ξ0 ρx]];
MatrixForm /@ eqn

sol=Solve[Thread[Flatten/@eqn], {τ0,η0,α0,ξ0}][[1]] /. (var_->val_) :> (var -> Simplify[PowerExpand@val])

Solve[Simplify[#[[1,1]]&/@eqn],τ0]

FullSimplify[-(E^(-((α0 (1+ϵ))/(2 ϵ))+(((α1+α2) (1+ϵ))/(2 ϵ)))) (-1+ϵ η2 ξ1)]

sol=Solve[Thread[Flatten/@eqn] /. τ0->(E^(-(((α0-α1-α2) (1+ϵ))/(2 ϵ)))) (1-ϵ η2 ξ1), {η0,α0,ξ0}][[1]] /. (var_->val_) :> (var -> FullSimplify[PowerExpand@val])

FullSimplify@PowerExpand[(E^(-(((α0-α1-α2) (1+ϵ))/(2 ϵ)))) (1-ϵ η2 ξ1) /. α0 -> α1+α2+2 Log[-1+ϵ η2 ξ1]]

Lemma 3 at δ=0. 𝕆((e^(α f + β e))|f e)=𝕆((e^(c h + a e - 2ϵ c l + b f))|e l f), with {a->-(β/(-1+α β ϵ)),b->-(α/(-1+α β ϵ)),c->((Log[1-α β ϵ])/ϵ)}.

Derivation.

ME[α ρf].ME[β ρe] // Simplify // MF

eqn=ME[α ρf].ME[β ρe]==ME[a ρe].ME[c (ρh-2ϵ ρl)].ME[b ρf]

sol=Solve[Thread[Flatten/@eqn], {a,b,c}][[1]]

sol=sol/.C[1]->0