Pensieve header: Calculations following Albert-Harinck-Torossian’s arXiv:0802.2049.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\WKO4"];
<< FreeLie.m;
<< AwCalculus.m;
$SeriesShowDegree=4;

{f=LS[{x,y}, fs], g=f // LieMorphism[LW@x->-LW@y, LW@y->-LW@x]};
msgs=SeriesSolve[{f},
(ℏ^(-1))(LS[LW@x+LW@y]-BCH[LW@y,LW@x]≡f-g-Ad[-LW@x][f]+Ad[LW@y][g])]

Read[msgs]

f@{6}

Read[msgs]

treqn=(
(Subscript[div, x])[f]+(Subscript[div, y])[g]≡(1/2)(Subscript[tr, u])[adSeries[(ad/((E^ad)-1)), LW@x][LW@u]+adSeries[(ad/((E^ad)-1)), LW@y][LW@u]-adSeries[(ad/((E^ad)-1)), BCH[LW@x,LW@y]][LW@u]])

treqn@{6}

{F=LS[{x,y}, Fs], G=F // LieMorphism[LW@x->-LW@y, LW@y->-LW@x]};
SeriesSolve[{F},
(ℏ^(-1))(LS[LW@x+LW@y]-BCH[LW@y,LW@x]≡F-G-Ad[-LW@x][F]+Ad[LW@y][G]) && (Subscript[div, x])[F]+(Subscript[div, y])[G]≡(1/2)(Subscript[tr, u])[adSeries[(ad/((E^ad)-1)), LW@x][LW@u]+adSeries[(ad/((E^ad)-1)), LW@y][LW@u]-adSeries[(ad/((E^ad)-1)), BCH[LW@x,LW@y]][LW@u]]]

F@{6}

2880F@5 // Expand // TopBracketForm

120960F@7 // Expand // TopBracketForm

{A=LS[{x,y}, As], B=LS[{x,y}, Bs]};
SeriesSolve[{A,B},
(ℏ^(-1))(b[LW@x, A]+b[LW@y,B] ≡ LS[0]) && ((Subscript[div, x])[A]+(Subscript[div, y])[B]≡CWS[0])]

A@{10}

A@{16}

{TimeUsed[], MaxMemoryUsed[]}