Pensieve header: Verification of the backie commutation relation, $[a_{jk}, a_{kj}]$.

The bracket

On the elements β, a, c, δa, ca, δaa.

Generalities

DQ[is___] := (Sort[{is}]===Union[{is}]);
OQ[is___] := OrderedQ[{is}];

Simp[expr_] := Simplify[expr];
S[β[f_]] := β[Simp[f]];
S[a[i_,j_]] := a[i,j];
S[a[f_, i_,j_]] := a[Simp[f], i,j];
S[c[f_, k_]] := c[Simp[f], k];
S[δa[f_, i_,j_]] := δa[Simp[f], i,j];
S[ca[f_, j_,k_,l_]] := ca[Simp[f], j,k,l];
S[δaa[f_, i_,j_,k_,l_]] := δaa[Simp[f], i,j,k,l];
S[expr_] := expr /. (λ_β|λ_a|λ_δa|λ_c|λ_ca|λ_δaa) :> S[λ];

β[0] :=0;
β /: β[f_] + β[g_] := β[f+g] // S;
β /: g_*β[f_] := β[g f] // S;
a[0,_,_] := 0;
a /: a[f_,j_,k_] + a[g_, j_,k_] := a[f+g, j,k] // S;
a /: g_*a[f_, j_,k_] := a[g f, j,k] // S;
c[0,_] :=0;
c /: c[f_,j_] + c[g_, j_] := c[f+g, j] // S;
c/: g_*c[f_, j_] := c[g f, j] // S;
δa[0,_,_] := 0;
δa /: δa[f_,j_,k_] + δa[g_, j_,k_] := δa[f+g, j,k] // S;
δa /: g_*δa[f_, j_,k_] := δa[g f, j,k] // S;
ca[0,_,_,_] :=0;
ca /: ca[f_,j_,k_,l_] + ca[g_, j_,k_,l_] := ca[f+g, j,k,l] // S;
ca /: g_*ca[f_, j_,k_,l_] := ca[g f, j,k,l] // S;
δaa[0, _,_,_,_] :=0;
δaa /: δaa[f_,i_,j_,k_,l_] + δaa[g_, i_,j_,k_,l_] := δaa[f+g, i,j,k,l] // S;
δaa /: g_*δaa[f_, i_,j_,k_,l_] := δaa[g f, i,j,k,l] // S;

NonCommutativeMultiply

Unprotect[NonCommutativeMultiply];
NonCommutativeMultiply[0,_] =0; NonCommutativeMultiply[_, 0] = 0;
NonCommutativeMultiply[x_,x_] = 0;
NonCommutativeMultiply[x_Plus, y_] := NonCommutativeMultiply[#,y]& /@ x;
NonCommutativeMultiply[x_, y_Plus] := NonCommutativeMultiply[x,#]& /@ y;

β[f_]**a[g_,j_,k_] := a[f g, j,k];
β[f_]**c[g_,j_] := c[f g, j];
c[g_,j_] ** β[f_] := c[f g, j];
β[f_]**δa[g_,j_,k_] := δa[f g, j,k];
β[f_]**ca[g_,i_,j_,k_] := ca[f g, i,j,k];
ca[g_,i_,j_,k_]**β[f_] := ca[f g, i,j,k];
β[f_]**δaa[g_,i_,j_,k_,l_] := δaa[f g, i,j,k,l];
δa[g_,j_,k_] ** β[f_] := δa[f g, j,k];
δ**a[f_,i_,j_] := δa[f,i,j];
c[f_,i_]**a[g_,j_,k_] := ca[f g, i,j,k];
a[f_,i_,j_]**δa[g_,k_,l_] := δaa[f g,i,j,k,l];
δa[f_,i_,j_]**a[g_,k_,l_] := δaa[f g,i,j,k,l];

δ**_c = 0;
δ**_δa = 0;
δ**_ca = 0;
δ**_δaa = 0;
_c**_c=0;
_c**_δa = _δa**_c = 0;
_c**_ca = _ca**_c = 0;
_c**_δaa = _δaa**_c = 0;
_δa**_δa = 0;
_δa**_δaa = _δaa**_δa = 0;
_δa**_ca = _ca**_δa = 0;

NonCommutativeMultiply::ndef = "NonCommuatativeMultiply is not defined on {`1`,`2`}."
NonCommutativeMultiply[x_,y_] := (Message[NonCommutativeMultiply::ndef, x,y]; Undefined);

Bracket Generalities

B[0,_] =0; B[_, 0] = 0;
B[x_,x_] = 0;
B[x_Plus, y_] := B[#,y]& /@ x;
B[x_, y_Plus] := B[x,#]& /@ y;

The γ shortcuts

γ[f_, j_,k_] := δa[f,j,k]-c[(Subscript[b, j])f,k] // S;
γ[f_, j_,k_,l_] /; DQ[j,k,l] := ca[f,l,j,k] - ca[f,k,j,l] // S;
ac[f_, j_,k_,l_] := ca[f, l,j,k] + B[a[1,j,k], c[f,l]];
adown[f_, j_] := a[f, j,j] + β[(Subscript[ϵ, 1])f (Subscript[b, j])]+c[(Subscript[ϵ, 1])f,j];

Fundamental Brackets

a-β, a-c, a-a, AS

B[a[j_,k_],β[g_]]:=γ[(∂_(Subscript[b, j]))g-(∂_(Subscript[b, k]))g,j,k];
B[β[g_], a[j_,k_]] := -B[a[j,k], β[g]];
B[a[j_,k_],a[l_,m_]] /; ({j,k}⋂{l,m}==={}) := 0;B[a[j_,k_],a[j_,l_]]/;DQ[j,k,l]:=γ[1,j,k,l] // S;
B[a[j_,k_],a[i_,k_]]/;DQ[i,j,k]:=a[(Subscript[b, i]),j,k]-a[(Subscript[b, j]),i,k] // S;
B[a[j_,k_],a[k_,l_]]/;DQ[j,k,l]:=a[(Subscript[b, j]),k,l]-a[(Subscript[b, k]),j,l]-γ[1,j,k,l] // S;
B[a[k_,l_], a[j_,k_]]/;DQ[j,k,l]:= -B[a[j,k],a[k,l]];
(* backie *) B[a[j_,k_], a[k_,j_]] /; DQ[j,k] := a[(Subscript[ϵ, 5])(Subscript[b, j]),k,j]-a[(Subscript[ϵ, 5])(Subscript[b, k]),j,k]+a[(Subscript[ϵ, 5])(Subscript[b, j]),k,k]-a[(Subscript[ϵ, 5])(Subscript[b, k]),j,j]+ca[(Subscript[ϵ, 4]),k,k,j]-ca[(Subscript[ϵ, 4]),j,j,k]+ca[(Subscript[ϵ, 4]),k,j,j]-ca[(Subscript[ϵ, 4]),j,k,k]+γ[(Subscript[ϵ, 4]),j,k]-γ[(Subscript[ϵ, 4]),k,j];
(* [tail, selfie] *) B[a[j_,k_], a[j_,j_]] /; DQ[j,k] := γ[-(Subscript[ϵ, 6]),j,k] //S;
B[a[j_,j_], a[j_,k_]] /; DQ[j,k] := -B[a[j,k], a[j,j]];
(* [head, selfie] *) B[a[j_,k_], a[k_,k_]] /; DQ[j,k] := γ[-(Subscript[ϵ, 7]),j,k] //S;
B[a[k_,k_], a[j_,k_]] /; DQ[j,k] := -B[a[j,k], a[k,k]];
B[a[f_,j_,k_], c[g_,j_]] /; DQ[j,k] := γ[-f g,j,k];
B[a[f_,j_,k_], c[g_,k_]] /; DQ[j,k] := γ[f g,j,k];
B[a[f_,j_,k_], c[g_,l_]] /; ({j,k}⋂{l}==={}) := 0;
B[a[f_,j_,j_], c[g_,j_]] = 0;
B[c[g_,l_], a[f_,j_,k_]] := -B[a[f,j,k], c[g,l]];

Vanishing brackets

B[_β, _β|δ|_c|_δa|_ca|_δaa] = 0;
B[_β|δ|_c|_δa|_ca|_δaa, _β] = 0;
B[δ|_c|_δa|_ca|_δaa, δ|_c|_δa|_ca|_δaa] = 0;

Composite Brackets

B[a[f_, j_,k_],β[g_]] :=β[f]**B[a[j,k],β[g]];
B[β[g_], a[f_, j_,k_]] := -B[a[f,j,k], β[g]];
B[a[f_, j_,k_], a[l_,m_]] := B[β[f], a[l,m]]**a[1,j,k] + β[f]**B[a[j,k], a[l,m]];
B[a[f_, j_,k_], a[g_, l_,m_]] := B[a[f,j,k], β[g]]**a[1,l,m] + β[g]**B[a[f,j,k], a[l,m]];
B[a[f_,i_,j_], δa[g_,k_,l_]] := δ**B[a[f,i,j], a[g,k,l]];
B[δa[f_,i_,j_], a[g_,k_,l_]] := δ**B[a[f,i,j], a[g,k,l]];
B[a[f_,i_,j_], ca[g_,k_,l_,m_]] := B[a[f,i,j], c[g,k]]**a[1,l,m]+c[g,k]**B[a[f,i,j],a[l,m]];
B[ca[g_,k_,l_,m_], a[f_,i_,j_]] := -B[a[f,i,j], ca[g,k,l,m]];
B[a[f_,i_,j_], δaa[g_,k_,l_,m_,n_]] := B[a[f,i,j], δa[g,k,l]]**a[1,m,n]+δa[g,k,l]**B[a[f,i,j],a[m,n]];
B[δaa[g_,k_,l_,m_,n_], a[f_,i_,j_]] := -B[a[f,i,j], δaa[g,k,l,m,n]];

B::ndef = "B is not defined on {`1`,`2`}."
B[x_,y_] := (Message[B::ndef, x,y]; Undefined);

Backie Verification

From pensieve://2015-05/Selfie and backie commutation relations:

B[a[j,k],a[k,jj]]

T1=a[(Subscript[b, j]),k,j]+a[-(Subscript[b, k]),j,j]+ac[-1,j,k,j]+ca[1,k,j,j]

B[a[j,kk],a[k,j]]

T2=a[(Subscript[b, j]),k,k]+a[-(Subscript[b, k]),j,k]+ca[-1,j,k,k]+ac[1,k,j,k]

B[a[j,k],a[k,j]]-T1-T2 /. {(Subscript[ϵ, 4|5])->1}