(* ::Package:: *)

(************************************************************************)\012(* This file was generated automatically by the Mathematica front end.  *)\012(* It contains Initialization cells from a Notebook file, which         *)\012(* typically will have the same name as this file except ending in      *)\012(* ".nb" instead of ".m".                                               *)\012(*                                                                      *)\012(* This file is intended to be loaded into the Mathematica kernel using *)\012(* the package loading commands Get or Needs.  Doing so is equivalent   *)\012(* to using the Evaluate Initialization Cells menu command in the front *)\012(* end.                                                                 *)\012(*                                                                      *)\012(* DO NOT EDIT THIS FILE.  This entire file is regenerated              *)\012(* automatically each time the parent Notebook file is saved in the     *)\012(* Mathematica front end.  Any changes you make to this file will be    *)\012(* overwritten.                                                         *)\012(************************************************************************)\012\012

(* ::Input::Initialization:: *)
Subscript[utm, n_][i_,j_->k_]:=Module[{A,\[Alpha],\[Beta],eqns,vars,sol,\[Zeta]},
\!\(\*SubscriptBox[\(\[DoubleStruckCapitalE]\), \({i, j} -> {k}\)]\)[
A=IdentityMatrix[n];
Do[A=A . MatrixExp[SparseArray[{\[Alpha],\[Beta]}->Subscript[\[Xi], \[Alpha],\[Beta]],{n,n}]],{\[Beta],1,n},{\[Alpha],1,\[Beta]}]; (* This specifies the PBW ordering *)
eqns=And@@Thread[Flatten[CF[(Inverse[A/.Subscript[\[Xi], \[Alpha]\[Beta]__]:>Subscript[\[Zeta], \[Alpha]\[Beta]]]/.Subscript[\[Zeta], \[Alpha]_,\[Alpha]_]:>Subscript[\[Xi], \[Alpha],\[Alpha]][i]+Subscript[\[Xi], \[Alpha],\[Alpha]][j]) . (A/.Subscript[\[Xi], \[Alpha]\[Beta]__]:>Subscript[\[Xi], \[Alpha]\[Beta]][i]) . (A/.Subscript[\[Xi], \[Alpha]\[Beta]__]:>Subscript[\[Xi], \[Alpha]\[Beta]][j])]]==Flatten[IdentityMatrix[n]]];vars=Union@Cases[eqns,Subscript[\[Zeta], __],\[Infinity]];{sol}=Solve[eqns,vars];
CF[(vars/.Subscript[\[Zeta], \[Alpha]\[Beta]__]->Subscript[x, \[Alpha]\[Beta]][k]) . (vars/.sol)+Sum[Subscript[x, \[Alpha],\[Alpha]][k](Subscript[\[Xi], \[Alpha],\[Alpha]][i]+Subscript[\[Xi], \[Alpha],\[Alpha]][j]),{\[Alpha],n}]]
]
];


(* ::Input::Initialization:: *)
Subscript[utm, n_][i_,j_->k_]:=Module[{A,\[Alpha],\[Beta],lhs,rhs,eqns,vars,sol,\[Zeta]},
\!\(\*SubscriptBox[\(\[DoubleStruckCapitalE]\), \({i, j} -> {k}\)]\)[
A=IdentityMatrix[n];
Do[A=A . MatrixExp[SparseArray[{\[Alpha],\[Beta]}->Subscript[\[Xi], \[Alpha],\[Beta]],{n,n}]],{\[Beta],1,n},{\[Alpha],1,\[Beta]}]; (* This specifies the PBW ordering *)
lhs=(A/.Subscript[\[Xi], \[Alpha]\[Beta]__]:>Subscript[\[Xi], \[Alpha]\[Beta]][i]) . (A/.Subscript[\[Xi], \[Alpha]\[Beta]__]:>Subscript[\[Xi], \[Alpha]\[Beta]][j]);
rhs=A/.Subscript[\[Xi], \[Alpha]\[Beta]__]:>Subscript[\[Zeta], \[Alpha]\[Beta]]/.Subscript[\[Zeta], \[Alpha]_,\[Alpha]_]:>Subscript[\[Xi], \[Alpha],\[Alpha]][i]+Subscript[\[Xi], \[Alpha],\[Alpha]][j];
eqns=And@@Thread[Flatten/@CF/@(lhs==rhs)];
vars=Union@Cases[rhs,Subscript[\[Zeta], __],\[Infinity]];{sol}=Solve[eqns,vars];
CF[(vars/.Subscript[\[Zeta], \[Alpha]\[Beta]__]->Subscript[x, \[Alpha]\[Beta]][k]) . (vars/.sol)+Sum[Subscript[x, \[Alpha],\[Alpha]][k](Subscript[\[Xi], \[Alpha],\[Alpha]][i]+Subscript[\[Xi], \[Alpha],\[Alpha]][j]),{\[Alpha],n}]]
]
];


(* ::Input::Initialization:: *)
Subscript[utm, n_][i_,j_->k_]:=Module[{A,\[Alpha],\[Beta],lhs,rhs,eqns,vars,sol,\[Zeta]},
\!\(\*SubscriptBox[\(\[DoubleStruckCapitalE]\), \({i, j} -> {k}\)]\)[
A=IdentityMatrix[n];
Do[A=A . MatrixExp[SparseArray[{\[Alpha],\[Beta]}->Subscript[\[Xi], \[Alpha],\[Beta]],{n,n}]],{\[Beta],1,n},{\[Alpha],1,\[Beta]}]; (* This specifies the PBW ordering *)
lhs=(A/.Subscript[\[Xi], \[Alpha]\[Beta]__]:>Subscript[\[Xi], \[Alpha]\[Beta]][i]) . (A/.Subscript[\[Xi], \[Alpha]\[Beta]__]:>Subscript[\[Xi], \[Alpha]\[Beta]][j]);
rhs=A/.Subscript[\[Xi], \[Alpha]\[Beta]__]:>Subscript[\[Zeta], \[Alpha]\[Beta]]/.Subscript[\[Zeta], \[Alpha]_,\[Alpha]_]:>Subscript[\[Xi], \[Alpha],\[Alpha]][i]+Subscript[\[Xi], \[Alpha],\[Alpha]][j];
sol=TriangularSolve[
Flatten@Table[lhs[[\[Alpha],\[Alpha]+\[Delta]\[Alpha]]]==rhs[[\[Alpha],\[Alpha]+\[Delta]\[Alpha]]],{\[Delta]\[Alpha],1,n-1},{\[Alpha],1,n-\[Delta]\[Alpha]}],
vars=Flatten@Table[Subscript[\[Zeta], \[Alpha],\[Alpha]+\[Delta]\[Alpha]],{\[Delta]\[Alpha],1,n-1},{\[Alpha],1,n-\[Delta]\[Alpha]}]
];
CF[(vars/.Subscript[\[Zeta], \[Alpha]\[Beta]__]->Subscript[x, \[Alpha]\[Beta]][k]) . (vars/.sol)+Sum[Subscript[x, \[Alpha],\[Alpha]][k](Subscript[\[Xi], \[Alpha],\[Alpha]][i]+Subscript[\[Xi], \[Alpha],\[Alpha]][j]),{\[Alpha],n}]]
]
];



